Exact Solutions and Reductions of

Nonlinear Diffusion PDEs of Pantograph Type∗

Andrei D. Polyanin, Vsevolod G. Sorokin

Ishlinsky Institute for Problems in Mechanics RAS,

101 Vernadsky Avenue, bldg 1, 119526 Moscow, Russia

††footnotetext:  ∗ This is a preprint of the article A.D. Polyanin, V.G.
Sorokin, Nonlinear pantograph-type diffusion PDEs: Exact solutions and the
principle of analogy, Mathematics, 2021, 9(5), 511; doi:10.3390/math9050511.

Abstract. We study nonlinear pantograph-type reaction-diffusion PDEs, which,
in addition to the unknown $u=u(x,t)$, also contain the same functions with
dilated or contracted arguments of the form $w=u(px,t)$, $w=u(x,qt)$, and
$w=u(px,qt)$, where $p$ and $q$ are the free scaling parameters (for equations
with proportional delay we have $0<p<1$, $0<q<1$). A brief review of
publications on pantograph-type ODEs and PDEs and their applications is given.
Exact solutions and reductions of various types of such nonlinear partial
functional differential equations are described for the first time. We present
examples of nonlinear pantograph-type PDEs with proportional delay, which
admit traveling-wave and self-similar solutions (note that PDEs with constant
delay do not have self-similar solutions). Additive, multiplicative and
functional separable solutions, as well as some other exact solutions are also
obtained. Special attention is paid to nonlinear pantograph-type PDEs of a
rather general form, which contain one or two arbitrary functions. In total,
more than forty nonlinear pantograph-type reaction-diffusion PDEs with dilated
or contracted arguments, admitting exact solutions, have been considered.
Multi-pantograph nonlinear PDEs are also discussed. The principle of analogy
is formulated, which makes it possible to efficiently construct exact
solutions of nonlinear pantograph-type PDEs. A number of exact solutions of
more complex nonlinear functional differential equations with varying delay,
which arbitrarily depends on time or spatial coordinate, are also described.
The presented equations and their exact solutions can be used to formulate
test problems designed to evaluate the accuracy of numerical and approximate
analytical methods for solving the corresponding nonlinear initial-boundary
value problems for PDEs with varying delay. The principle of analogy allows
finding solutions to other nonlinear pantograph-type PDEs (including nonlinear
wave-type PDEs and higher-order equations).

Keywords: nonlinear reaction-diffusion equations; pantograph-type differential
equations; PDEs with proportional delay; PDEs with varying delay; partial
functional-differential equations; exact solutions; self-similar solutions;
additive and multiplicative separable solutions; functional separable
solutions; generalized separable solutions

## Contents

1\. Introduction

1.1. Differential Equations with Constant Delay

1.2. Pantograph-Type ODEs and PDEs and Their Applications

1.3. Concept of ‘Exact Solution’ for Nonlinear Pantograph-Type PDEs

2\. Solutions of Pantograph-Type PDEs with $u=u(x,t)$ and $w=u(px,qt)$

2.1. Equations Containing Free Parameters

2.2. Equations Linear in Derivatives and Containing One Arbitrary Function

2.3. More Complex Nonlinear Equations Containing One Arbitrary Function

2.4. Nonlinear Equations Containing Two Arbitrary Functions

3\. Solutions of Pantograph-Type PDEs with $u=u(x,t)$ and $w=u(x,qt)$

3.1. Equations Containing Free Parameters

3.2. Equations Linear in Derivatives and Containing One Arbitrary Function

3.3. More Complex Nonlinear Equations Containing One Arbitrary Function

3.4. Nonlinear Equations Containing Two Arbitrary Functions

4\. Solutions of Pantograph-Type PDEs with $u=u(x,t)$ and $w=u(px,t)$

4.1. Equations Linear in Derivatives

4.2. More Complex Nonlinear Equations

5\. Some Generalizations

5.1. Nonlinear Multi-Pantograph Type PDEs

5.2. Nonlinear Pantograph-Type PDEs with Dilation/Contraction of Arguments in
the Derivative

5.3. Principle of Analogy of Solutions

6\. Brief Conclusions

References

##  1 Introduction

###  1.1 Differential Equations with Constant Delay

In natural science, an important role is played by the study of the hereditary
properties of nonlinear systems of different nature, when the rate of change
of the unknown depends not only on the state at a given time, but also on the
previous evolution of the process. In a particular case, the state of the
system is determined not by its entire history, but by the current moment and
some moment in the past. Such systems are called delay systems. In the
simplest case, the differential equations modelling such systems, in addition
to the unknown function $u(t)$, also contain the function $u(t-\tau)$, where
$t$ is the time and $\tau>0$ is the delay [1, 2, 3]. The delay is generally
assumed constant.

When modeling nonlinear systems with constant delay and two independent
variables $x$ and $t$, where $x$ is the spatial coordinate, reaction-diffusion
equations of the following form are most often encountered [4]:

$\displaystyle u_{t}=au_{xx}+F(u,w),\quad\ w=u(x,t-\tau).$ (1)

The presence of delay significantly complicates the analysis of equations of
the form (1). Such equations admit traveling-wave solutions $u=u(z)$, where
$z=x+\lambda t$ (e.g, see [4, 5, 6, 7]), but do not admit self-similar
solutions $u=t^{\beta}\varphi(xt^{\lambda})\,$ (recall that many PDEs without
delay have self-similar solutions).

More complex than traveling-wave, exact solutions of nonlinear delay reaction-
diffusion equations are obtained in [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 23,
24]. Exact solutions of nonlinear delay equations of the Klein—Gordon type and
other nonlinear hyperbolic equations are given in [18, 19, 20, 21, 22, 23,
24]. Some exact solutions of differential-difference equations of a viscous
incompressible fluid (which generalize the Navier—Stokes equations) are
described in [25]. Numerical methods for solving delay PDEs are discussed, for
example, in [26, 27, 28].

It is important to note that delay differential equations have a number of
specific qualitative features [4, 29, 19] that are not inherent in equations
without delay.

###  1.2 Pantograph-Type ODEs and PDEs and Their Applications

In some cases, the delay may be time dependent, i.e. $\tau=\tau(t)$, where
$\tau(t)>0$ is a given function [2].

To illustrate this, consider a first-order linear ODE with varying delay
proportional to the independent variable:

$\displaystyle u^{\prime}_{t}=au+bw,\quad w=u(pt),$ (2)

which for $p>0$ ($p\not=1$) is called the pantograph equation.

For $0<p<1$, the equation describes the dynamics of a current collection
system for an electric locomotive [30] and is a special case of an ODE with
varying delay for $\tau(t)=(1-p)t$. The function $u(pt)$ included in the
pantograph equation (2) differs from the function $u(t)$ by dilation along the
$t$-axis by $1/p$ times.

The pantograph equation and more complex related ODEs and PDEs that contain
the unknown functions with dilation (for $0<p<1$) or contraction (for $p>1$)
of arguments are used in mathematical modeling of various processes in biology
[31, 32, 33, 34, 35], astrophysics [36], electrodynamics [37], population
theory [38], number theory [39], stochastic games [40], graph theory [41],
risk and queue theory [42], and theory of neural networks [43]. Note that in
[31, 36, 34, 35] the equations are derived for the case $p>1$.

Analysis of pantograph-type ODEs and construction of approximate analytical
solutions are carried out, for example, in [44, 45, 46, 47, 30, 48, 49, 50,
51, 52, 53, 54, 55]. Numerical methods for solving such equations are
discussed in [56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71,
72, 73, 74, 75, 76, 81, 77, 78, 79, 80]. Thus, in general, approximate
analytical and numerical methods for solving pantograph-type ODEs can
currently be considered fairly well developed.

Significantly fewer publications are devoted to the analysis and solving of
pantograph-type PDEs. In [34], a linear first-order pantograph-type PDE is
used to model the growth and division of cells structured by size. Its
solution is sought in the form of a series, the terms of which are determined
by solving simpler equations without dilated arguments. In [35], a more
complex linear pantograph-type reaction-diffusion equation is investigated
(obtained by adding a diffusion term to the equation that was analyzed in
[34]). In [82], solutions of linear pantograph-type heat and wave equations
are obtained using the method of separation of variables. In [83, 84, 85], the
problems of unique solvability and smoothness of linear boundary value
problems for elliptic PDEs with dilation or contraction of arguments in higher
derivatives are considered. Analytical methods for solving some linear and
nonlinear PDEs with proportional delays are discussed in [86, 87, 88]. In
[89], a finite-difference scheme for the numerical integration of first-order
PDEs with constant delay in $t$ and proportional delay in $x$ is constructed.
Papers [90, 91, 92] are devoted to numerical methods for solving pantograph-
type PDEs with proportional delay [90, 91] and more complex varying delay
[92].

This article describes different classes of exact solutions to nonlinear
pantograph-type reaction-diffusion equations of the form

$\displaystyle u_{t}=[g(u)u_{x}]_{x}+F(u,w),$ (3)

where $u=u(x,t)$ and $w=u(px,qt)$, $p>0$ and $q>0$ ($p$ and $q$ cannot be
equal to $1$ at the same time).

######  Remark 1.

Exact solutions of ‘ordinary’ nonlinear diffusion type equations (without
dilated or contracted arguments) can be found, for example, in [93, 94, 95,
96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111,
112, 114, 115, 113, 116, 117].

In most cases, exact solutions of pantograph-type PDEs of the form (3) are
obtained as follows: a solution to a pantograph-type equation is sought in the
same form as a solution to a simpler equation without delay, which is obtained
from (3) for $w=u$ (for details, see Section 5.3). More complex exact
solutions are found using various modifications of the functional constraints
method [10, 15]. In addition, to construct some exact solutions of pantograph-
type equations, we use exact solutions of nonlinear PDEs with varying delay of
general form, which were given in [9, 11, 15].

As a result of the analysis, we have managed to find many nonlinear
pantograph-type reaction-diffusion equations, which admit exact traveling-wave
solutions, self-similar solutions, additive, multiplicative, generalized, and
functional separable solutions, as well as some other exact solutions. Special
attention is paid to nonlinear pantograph-type PDEs of a rather general form
that contain arbitrary functions (it is these equations and their solutions
that are of greatest interest for testing numerical and approximate analytical
methods for solving the corresponding nonlinear initial-boundary value
problems for PDEs with varying delay).

###  1.3 Concept of ‘Exact Solution’ for Nonlinear Pantograph-Type PDEs

In this article, the term ‘exact solution’ for nonlinear pantograph-type PDEs
(with dilation or contraction of one or several independent variables) and
equations with varying delay will be used in cases where the solution is
expressed:

(i) in terms of elementary functions, functions included in the equation (this
is necessary when the equation contains arbitrary functions), and indefinite
and/or definite integrals;

(ii) in terms of solutions of ODEs without delay or systems of such equations;

(iii) in terms of solutions of pantograph-type ODEs and ODEs with constant or
varying delay or systems of such equations.

Combinations of solutions from paragraphs (i)–(iii) are allowed. In case (i),
an exact solution can be represented in explicit, implicit or parametric form.
This definition generalizes the concept of ‘exact solution’ that is used for
nonlinear PDEs with constant delay in [10, 12] and for PDEs without delay in
[93].

In what follows, unless otherwise specified, in nonlinear pantograph-type
PDEs, it is assumed that $p$ and $q$ are the free scaling parameters ($p$ and
$q$ are positive and cannot be equal to $1$ at the same time), $f(z)$ and
$g(z)$ are arbitrary functions, and $a$, $b$, $c$, etc. are arbitrary real
constants. The unknown function is denoted by $u$, and the unknown function
with dilation or contraction of one or more arguments is denoted by $w$.

##  2 Solutions of Pantograph-Type PDEs That Contain Unknown Functions
$u=u(x,t)$ and $w=u(px,qt)$

###  2.1 Equations Containing Free Parameters

Equation 1. The equation with power-law nonlinearity

$\displaystyle u_{t}=au_{xx}+bw^{k},\quad\ w=u(px,qt),$ (4)

for $k\not=1$ admits the self-similar solution

$\displaystyle u(x,t)=t^{\textstyle{\mskip 1.0mu1\mskip 1.0mu\over\mskip
1.0mu1-k\mskip 1.0mu}}U(z),\quad\ z=xt^{-1/2},$ (5)

where the function $U=U(z)$ is described by the nonlinear pantograph-type ODE:

$\displaystyle aU^{\prime\prime}_{zz}+{\textstyle{\mskip 1.0mu1\mskip
1.0mu\over\mskip 1.0mu2\mskip 1.0mu}}zU^{\prime}_{z}-{\mskip 1.0mu1\mskip
1.0mu\over\mskip 1.0mu1-k\mskip 1.0mu}U+bq^{{\mskip 1.0muk\mskip
1.0mu\over\mskip 1.0mu1-k\mskip 1.0mu}}W^{k}=0,\quad\ W=U(sz),\quad\
s=pq^{-1/2}.$ (6)

######  Remark 2.

It is interesting to note that the ODE with proportional delay (4)
for$0<p,q<1$ in the special case $p=q^{1/2}$ has an exact solution expressed
in terms of the solution of the ODE without delay (6) with $s=1$; for
$p<q^{1/2}$, Eq. (4) reduces to the delay ODE with $s<1$; and for $p>q^{1/2}$,
to the ODE with contracted argument for $s>1$. Moreover, a solution of the ODE
(4) for $p,q>1$ for appropriate values of the parameters $p$ and $q$ can also
be expressed in terms of the solution of the ODE with delay ($s<1$), without
delay ($s=1$), and with contracted argument ($s>1$).

The equations and their solutions given below in Section 2 have similar
qualitative features.

Equation 2. The more complex equation with power-law nonlinearity

$\displaystyle u_{t}=au_{xx}+bu^{m}w^{k},\quad\ w=u(px,qt),$

for $k\not=1-m$ admits the self-similar solution

$\displaystyle u(x,t)=t^{\textstyle{\mskip 1.0mu1\mskip 1.0mu\over\mskip
1.0mu1-m-k\mskip 1.0mu}}U(z),\quad\ z=xt^{-1/2},$

where the function $U=U(z)$ is described by the nonlinear pantograph-type ODE:

$\displaystyle aU^{\prime\prime}_{zz}+{\textstyle{\mskip 1.0mu1\mskip
1.0mu\over\mskip 1.0mu2\mskip 1.0mu}}zU^{\prime}_{z}-{\mskip 1.0mu1\mskip
1.0mu\over\mskip 1.0mu1-m-k\mskip 1.0mu}U+bq^{{\mskip 1.0muk\mskip
1.0mu\over\mskip 1.0mu1-m-k\mskip 1.0mu}}U^{m}W^{k}=0,$ $\displaystyle
W=U(sz),\quad\ s=pq^{-1/2}.$

Equation 3. The equation with exponential nonlinearity

$\displaystyle u_{t}=au_{xx}+be^{\lambda w},\quad\ w=u(px,qt),$

admits the exact solution

$\displaystyle u(x,t)=U(z)-{\mskip 1.0mu1\mskip 1.0mu\over\mskip
1.0mu\lambda\mskip 1.0mu}\ln t,\quad\ z=xt^{-1/2},$

where the function $U=U(z)$ is described by the nonlinear pantograph-type ODE:

$\displaystyle aU^{\prime\prime}_{zz}+{\mskip 1.0mu1\mskip 1.0mu\over\mskip
1.0mu2\mskip 1.0mu}zU^{\prime}_{z}+{\mskip 1.0mu1\mskip 1.0mu\over\mskip
1.0mu\lambda\mskip 1.0mu}+{\mskip 1.0mub\mskip 1.0mu\over\mskip 1.0muq\mskip
1.0mu}e^{\lambda W}=0,$ $\displaystyle W=U(sz),\quad\ s=pq^{-1/2}.$

Equation 4. The more complex equation with exponential nonlinearity

$\displaystyle u_{t}=au_{xx}+be^{\mu u+\lambda w},\quad\ w=u(px,qt),$

for $\mu\not=-\lambda$ admits the exact solution

$\displaystyle u(x,t)=U(z)-{\mskip 1.0mu1\mskip 1.0mu\over\mskip
1.0mu\mu+\lambda\mskip 1.0mu}\ln t,\quad\ z=xt^{-1/2},$

where the function $U=U(z)$ is described by the nonlinear pantograph-type ODE:

$\displaystyle aU^{\prime\prime}_{zz}+{\mskip 1.0mu1\mskip 1.0mu\over\mskip
1.0mu2\mskip 1.0mu}zU^{\prime}_{z}+{\mskip 1.0mu1\mskip 1.0mu\over\mskip
1.0mu\mu+\lambda\mskip 1.0mu}+bq^{\textstyle-{\mskip 1.0mu\lambda\mskip
1.0mu\over\mskip 1.0mu\mu+\lambda\mskip 1.0mu}}e^{\mu U+\lambda W}=0,$
$\displaystyle W=U(sz),\quad\ s=pq^{-1/2}.$

Equation 5. The equation with logarithmic nonlinearity

$\displaystyle u_{t}=au_{xx}+u(b\ln u+c\ln w+d),\quad\ w=u(px,qt),$ (7)

admits the multiplicative separable solution

$\displaystyle u(x,t)=\varphi(x)\psi(t),$ (8)

where the functions $\varphi=\varphi(x)$ and $\psi=\psi(t)$ are described by
the nonlinear pantograph-type ODEs:

$\displaystyle
a\varphi^{\prime\prime}_{xx}+\varphi(b\ln\varphi+c\ln\bar{\varphi}+d)=0,\quad\
\bar{\varphi}=\varphi(px);$
$\displaystyle\psi^{\prime}_{t}=\psi(b\ln\psi+c\ln\bar{\psi}),\quad\
\bar{\psi}=\psi(qt).$

######  Remark 3.

Eq. (7) and its solution can be substantially generalized if the pantograph
term $w=u(px,qt)$ is replaced by $w=u(\xi(x),\eta(t))$, where $\xi(x)$ and
$\eta(t)$ are arbitrary functions (the solution is sought in the same form
(8)).

###  2.2 Equations Linear in Derivatives and Containing One Arbitrary Function

Equation 6. The equation containing an arbitrary function

$\displaystyle u_{t}=au_{xx}+u^{n}f(w/u),\quad\ w=u(px,qt),$

for $n\not=1$ admits the self-similar solution

$\displaystyle u(x,t)=t^{\textstyle{\mskip 1.0mu1\mskip 1.0mu\over\mskip
1.0mu1-n\mskip 1.0mu}}U(z),\quad\ z=xt^{-1/2},$

where the function $U=U(z)$ satisfies the second-order pantograph-type ODE:

$\displaystyle{\mskip 1.0mu1\mskip 1.0mu\over\mskip 1.0mu1-n\mskip
1.0mu}U-{\mskip 1.0mu1\mskip 1.0mu\over\mskip 1.0mu2\mskip
1.0mu}zU^{\prime}_{z}=aU^{\prime\prime}_{zz}+U^{n}f(q^{{\mskip 1.0mu1\mskip
1.0mu\over\mskip 1.0mu1-n\mskip 1.0mu}}W/U),$ $\displaystyle W=U(sz),\quad\
s=pq^{-1/2}.$

Equation 7. The equation containing an arbitrary function

$\displaystyle u_{t}=au_{xx}+e^{\lambda u}f(u-w),\quad\ w=u(px,qt),$

admits the exact solution

$\displaystyle u(x,t)=U(z)-{\mskip 1.0mu1\mskip 1.0mu\over\mskip
1.0mu\lambda\mskip 1.0mu}\ln t,\quad\ z=xt^{-1/2},$

where the function $U=U(z)$ is described by the nonlinear pantograph-type ODE:

$\displaystyle aU^{\prime\prime}_{zz}+{\mskip 1.0mu1\mskip 1.0mu\over\mskip
1.0mu2\mskip 1.0mu}zU^{\prime}_{z}+{\mskip 1.0mu1\mskip 1.0mu\over\mskip
1.0mu\lambda\mskip 1.0mu}+e^{\lambda U}f\Bigl{(}U-W+{\mskip 1.0mu1\mskip
1.0mu\over\mskip 1.0mu\lambda\mskip 1.0mu}\ln q\Bigr{)}=0,$ $\displaystyle
W=U(sz),\quad\ s=pq^{-1/2}.$

###  2.3 More Complex Nonlinear Equations Containing One Arbitrary Function

Equation 8. The equation with varying transfer coefficient of power-law type

$\displaystyle u_{t}=a(u^{k}u_{x})_{x}+u^{n}f(w/u),\quad\ w=u(px,qt),$ (9)

for $n\not=1$ admits the self-similar solution

$\displaystyle u(x,t)=t^{\textstyle{\mskip 1.0mu1\mskip 1.0mu\over\mskip
1.0mu1-n\mskip 1.0mu}}U(z),\quad\ z=xt^{\textstyle{\mskip 1.0mun-k-1\mskip
1.0mu\over\mskip 1.0mu2(1-n)\mskip 1.0mu}},$

where the function $U=U(z)$ satisfies the second-order pantograph-type ODE:

$\displaystyle{\mskip 1.0mu1\mskip 1.0mu\over\mskip 1.0mu1-n\mskip
1.0mu}U+{\mskip 1.0mun-k-1\mskip 1.0mu\over\mskip 1.0mu2(1-n)\mskip
1.0mu}zU^{\prime}_{z}=a(U^{k}U^{\prime}_{z})^{\prime}_{z}+U^{n}f(q^{{\mskip
1.0mu1\mskip 1.0mu\over\mskip 1.0mu1-n\mskip 1.0mu}}W/U),$ $\displaystyle
W=U(sz),\quad\ s=pq^{\textstyle{\mskip 1.0mun-k-1\mskip 1.0mu\over\mskip
1.0mu2(1-n)\mskip 1.0mu}}.$

Equation 9. The equation containing an arbitrary function

$\displaystyle u_{t}=a(e^{\lambda u}u_{x})_{x}+e^{\mu u}f(u-w),\quad\
w=u(px,qt),$ (10)

for $\mu\not=0$ admits the exact solution

$\displaystyle u(x,t)=U(z)-{\mskip 1.0mu1\mskip 1.0mu\over\mskip
1.0mu\mu\mskip 1.0mu}\ln t,\quad\ z=xt^{\textstyle{\mskip
1.0mu\lambda-\mu\mskip 1.0mu\over\mskip 1.0mu2\mu\mskip 1.0mu}},$

where the function $U=U(z)$ is described by the nonlinear pantograph-type ODE:

$\displaystyle{\mskip 1.0mu\lambda-\mu\mskip 1.0mu\over\mskip 1.0mu2\mu\mskip
1.0mu}zU^{\prime}_{z}-{\mskip 1.0mu1\mskip 1.0mu\over\mskip 1.0mu\mu\mskip
1.0mu}=a(e^{\lambda U}U^{\prime}_{z})^{\prime}_{z}+e^{\mu
U}f\Bigl{(}U-W+{\mskip 1.0mu1\mskip 1.0mu\over\mskip 1.0mu\mu\mskip 1.0mu}\ln
q\Bigr{)}=0,$ $\displaystyle W=U(sz),\quad\ s=pq^{\textstyle{\mskip
1.0mu\lambda-\mu\mskip 1.0mu\over\mskip 1.0mu2\mu\mskip 1.0mu}}.$

###  2.4 Nonlinear Equations Containing Two Arbitrary Functions

Equation 10. The equation with varying transfer coefficient of general form

$\displaystyle u_{t}=a[uf^{\prime}_{u}(u)u_{x}]_{x}+{\mskip 1.0mu1\mskip
1.0mu\over\mskip 1.0muf^{\prime}_{u}(u)\mskip 1.0mu}[bf(u)+cf(w)+d],\quad
w=u(px,qt),$ (11)

admits the functional separable solution in implicit form

$\displaystyle f(u)=\varphi(t)x+\psi(t),$ (12)

where the functions $\varphi=\varphi(t)$ and $\psi=\psi(t)$ satisfy the
pantograph-type ODEs:

$\displaystyle\varphi^{\prime}_{t}$
$\displaystyle=b\varphi+cp\bar{\varphi},\quad\ \bar{\varphi}=\varphi(qt),$
$\displaystyle\psi^{\prime}_{t}$
$\displaystyle=b\psi+c\bar{\psi}+d+a\varphi^{2},\quad\bar{\psi}=\psi(qt).$

######  Remark 4.

More general than Eq. (11), the equation in which the pantograph term
$w=u(px,qt)$ is replaced by $w=u(p_{1}x+p_{0},q_{1}t+q_{0})$ also has the
functional separable solution (12). The more complex functional differential
equation in which $w=u(p_{1}x+p_{0},\eta(t))$, where $\eta(t)$ is an arbitrary
function, also has a solution of the same form.

Equation 11. For $q=p$, a nonlinear pantograph-type reaction-diffusion
equation of the general form

$\displaystyle u_{t}=[g(u)u_{x}]_{x}+F(u,w),\quad\ w=u(px,pt),$

admits the traveling-wave solution

$\displaystyle u(x,t)=U(z),\quad\ z=kx-\lambda t,$

where $k$ and $\lambda$ are arbitrary constants and the function $U=U(z)$ is
described by the nonlinear pantograph-type ODE:

$k^{2}[g(U)U^{\prime}_{z}]^{\prime}_{z}+\lambda U^{\prime}_{z}+F(U,W)=0,\quad\
W=U(pz).$

##  3 Solutions of Pantograph-Type PDEs That Contain Unknown Functions
$u=u(x,t)$ and $w=u(x,qt)$

Preliminary remarks. This section does not include equations and their
solutions that can be obtained by setting $p=1$ in the equations and solutions
considered in Section 2.

###  3.1 Equations Containing Free Parameters

Equation 12. The equation with logarithmic nonlinearity

$\displaystyle u_{t}=au_{xx}+u(b\ln u+c\ln w+d),\quad\ w=u(x,qt),$ (13)

admits the functional separable solution

$\displaystyle u(x,t)=\exp[\psi_{2}(t)x^{2}+\psi_{1}(t)x+\psi_{0}(t)],$

where the functions $\psi_{n}=\psi_{n}(t)$ are described by the nonlinear
system of pantograph-type ODEs:

$\displaystyle\psi_{2}^{\prime}=4a\psi_{2}^{2}+b\psi_{2}+c\bar{\psi}_{2},\quad\bar{\psi}_{2}=\psi_{2}(qt),$
$\displaystyle\psi_{1}^{\prime}=4a\psi_{1}\psi_{2}+b\psi_{1}+c\bar{\psi}_{1},\quad\bar{\psi}_{1}=\psi_{1}(qt),$
$\displaystyle\psi_{0}^{\prime}=a[\psi_{1}^{2}+2\psi_{2}]+b\psi_{0}+c\bar{\psi}_{0}+d,\quad\bar{\psi}_{0}=\psi_{0}(qt).$

Equation 13. The equation with logarithmic nonlinearity

$\displaystyle u_{t}=au_{xx}+u(b\ln^{2}u+c\ln u+d\ln w+s),\quad\ w=u(x,qt),$
(14)

depending on the sign of the product $ab$, admits two functional separable
solutions given below.

$1^{\circ}$. The solution for $ab>0$:

$\displaystyle u(x,t)=\exp[\psi_{1}(t)\varphi(x)+\psi_{2}(t)],$
$\displaystyle\varphi(x)=A\cos(\lambda x)+B\sin(\lambda
x),\quad\lambda=\sqrt{b/a},$

where $A$ and $B$ are arbitrary constants and the functions
$\psi_{n}=\psi_{n}(t)$ are described by the nonlinear system of pantograph-
type ODEs:

$\displaystyle\psi_{1}^{\prime}$
$\displaystyle=2b\psi_{1}\psi_{2}+(c-b)\psi_{1}+d\bar{\psi}_{1},\quad\bar{\psi}_{1}=\psi_{1}(qt),$
$\displaystyle\psi_{2}^{\prime}$
$\displaystyle=b(A^{2}+B^{2})\psi_{1}^{2}+b\psi_{2}^{2}+c\psi_{2}+d\bar{\psi}_{2}+s,\quad\bar{\psi}_{2}=\psi_{2}(qt).$

$2^{\circ}$. The solution for $ab<0$:

$\displaystyle u(x,t)=\exp[\psi_{1}(t)\varphi(x)+\psi_{2}(t)],$
$\displaystyle\varphi(x)=A\cosh(\lambda x)+B\sinh(\lambda
x),\quad\lambda=\sqrt{-b/a},$

where $A$ and $B$ are arbitrary constants and the functions
$\psi_{n}=\psi_{n}(t)$ are described by the nonlinear system of pantograph-
type ODEs:

$\displaystyle\psi_{1}^{\prime}$
$\displaystyle=2b\psi_{1}\psi_{2}+(c-b)\psi_{1}+d\bar{\psi}_{1},\quad\bar{\psi}_{1}=\psi_{1}(qt),$
$\displaystyle\psi_{2}^{\prime}$
$\displaystyle=b(A^{2}-B^{2})\psi_{1}^{2}+b\psi_{2}^{2}+c\psi_{2}+d\bar{\psi}_{2}+s,\quad\bar{\psi}_{2}=\psi_{2}(qt).$

For $A=\pm B$, we have $\varphi(x)=Ae^{\pm\lambda x}$. In this case, the
second equation of the system becomes independent and the first one becomes
linear in $\psi_{1}$.

######  Remark 5.

Eqs. (13) and (14) and their solutions can be substantially generalized if the
pantograph term $w=u(x,qt)$ is replaced by $w=u(x,\eta(t))$, where $\eta(t)$
is an arbitrary function (the solutions are sought in the same form, see [9]).

###  3.2 Equations Linear in Derivatives and Containing One Arbitrary Function

Equation 14. The equation containing an arbitrary function

$\displaystyle u_{t}=au_{xx}+f(u-w),\quad\ w=u(x,qt),$

admits the additive separable solution

$\displaystyle u(x,t)=C_{1}x^{2}+C_{2}x+\psi(t),$

where $C_{1}$ and $C_{2}$ are arbitrary real constants and the function
$\psi=\psi(t)$ is described by the nonlinear first-order pantograph-type ODE:

$\displaystyle\psi^{\prime}_{t}=2aC_{1}+f(\psi-\bar{\psi}),\quad\bar{\psi}=\psi(qt).$

Equation 15. The equation containing an arbitrary function

$\displaystyle u_{t}=au_{xx}+bu+f(u-w),\quad\ w=u(x,qt),$ (15)

depending on the sign of the product $ab$, admits two additive separable
solutions given below.

$1^{\circ}$. The solution for $ab<0$:

$\displaystyle u(x,t)=A\cosh(\lambda x)+B\sinh(\lambda
x)+\psi(t),\quad\lambda=\sqrt{-b/a},$

where $A$ and $B$ are arbitrary constants and the function $\psi=\psi(t)$ is
described by the nonlinear first-order pantograph-type ODE:

$\displaystyle\psi^{\prime}_{t}=b\psi+f(\psi-\bar{\psi}),\quad\bar{\psi}=\psi(qt).$
(16)

$2^{\circ}$. The solution for $ab>0$:

$\displaystyle u(x,t)=A\cos(\lambda x)+B\sin(\lambda
x)+\psi(t),\quad\lambda=\sqrt{b/a},$

where $A$ and $B$ are arbitrary constants and the function $\psi=\psi(t)$ is
described by the nonlinear first-order pantograph-type ODE (16).

Note that Eq. (15) and its solutions can be substantially generalized if the
pantograph term $w=u(x,qt)$ is replaced by $w=u(x,\eta(t))$, where $\eta(t)$
is an arbitrary function (the solutions are sought in the same form, see [9,
11]).

Equation 16. The equation containing an arbitrary function

$\displaystyle u_{t}=au_{xx}+uf(w/u),\quad\ w=u(x,qt),$

admits several multiplicative separable solutions given below.

$1^{\circ}$. The solution

$\displaystyle u(x,t)=[A\cosh(\lambda x)+B\sinh(\lambda x)]\psi(t),$

where $A$, $B$, and $\lambda$ are arbitrary constants and the function
$\psi=\psi(t)$ is described by the nonlinear first-order pantograph-type ODE:

$\displaystyle\psi^{\prime}_{t}=a\lambda^{2}\psi+\psi
f(\bar{\psi}/\psi),\quad\bar{\psi}=\psi(qt).$

$2^{\circ}$. The solution

$\displaystyle u(x,t)=[A\cos(\lambda x)+B\sin(\lambda x)]\psi(t),$

where $A$, $B$, and $\lambda$ are arbitrary constants and the function
$\psi=\psi(t)$ is described by the nonlinear first-order pantograph-type ODE:

$\displaystyle\psi^{\prime}_{t}=-a\lambda^{2}\psi+\psi
f(\bar{\psi}/\psi),\quad\bar{\psi}=\psi(qt).$

$3^{\circ}$. The degenerate solution

$\displaystyle u(x,t)=(Ax+B)\psi(t),$

where $A$, $B$, and $\lambda$ are arbitrary constants and the function
$\psi=\psi(t)$ is described by the nonlinear first-order pantograph-type ODE:

$\displaystyle\psi^{\prime}_{t}=\psi
f(\bar{\psi}/\psi),\quad\bar{\psi}=\psi(qt).$

Equation 17. The equation containing an arbitrary function

$\displaystyle u_{t}=au_{xx}+bu\ln u+uf(w/u),\quad\ w=u(x,qt),$ (17)

admits the multiplicative separable solution

$\displaystyle u(x,t)=\varphi(x)\psi(t).$

where the functions $\varphi=\varphi(x)$ and $\psi=\psi(t)$ are described by
the nonlinear second-order ODE and the nonlinear first-order pantograph-type
ODE:

$\displaystyle a\varphi^{\prime\prime}_{xx}=C_{1}\varphi-b\varphi\ln\varphi,$
(18) $\displaystyle\psi^{\prime}_{t}=C_{1}\psi+\psi
f(\bar{\psi}/\psi)+b\psi\ln\psi,\quad\bar{\psi}=\psi(qt);$

$C_{1}$ is an arbitrary constant.

The first equation of Eq. (18) is autonomous; its general solution can be
obtained in implicit form. A particular one-parameter solution of this
equation has the form

$\varphi=\exp\biggl{[}-{\mskip 1.0mub\mskip 1.0mu\over\mskip 1.0mu4a\mskip
1.0mu}(x+C_{2})^{2}+{\mskip 1.0muC_{1}\mskip 1.0mu\over\mskip 1.0mub\mskip
1.0mu}+{\mskip 1.0mu1\mskip 1.0mu\over\mskip 1.0mu2\mskip 1.0mu}\biggr{]},$

where $C_{2}$ is an arbitrary constant.

Note that Eq.(17) and its solution can be substantially generalized if the
pantograph term $w=u(x,qt)$ is replaced by $w=u(x,\eta(t))$, where $\eta(t)$
is an arbitrary function (the solution is sought in the same form, see [9,
11]).

###  3.3 More Complex Nonlinear Equations Containing One Arbitrary Function

Equation 18. The equation with varying transfer coefficient of power-law type

$\displaystyle u_{t}=a(u^{k}u_{x})_{x}+uf(w/u),\quad w=u(x,qt)$ (19)

admits the multiplicative separable solution

$u(x,t)=\varphi(x)\psi(t),$

where the functions $\varphi=\varphi(x)$ and $\psi=\psi(t)$ are determined
from the ODE and the pantograph-type ODE:

$\displaystyle a(\varphi^{k}\varphi^{\prime}_{x})^{\prime}_{x}=b\varphi,$
$\displaystyle\psi^{\prime}_{t}=b\psi^{k+1}+\psi
f\bigl{(}\bar{\psi}/\psi\bigr{)},\quad\ \bar{\psi}=\psi(qt);$

$b$ is an arbitrary constant.

Equation 19. The equation with varying transfer coefficient of power-law type

$\displaystyle u_{t}=a(u^{k}u_{x})_{x}+uf(w/u)+bu^{k+1},\quad w=u(x,qt),$ (20)

admits three multiplicative separable solutions given below.

$1^{\circ}$. The solution for $b(k+1)>0$:

$u(x,t)=[C_{1}\cos(\beta x)+C_{2}\sin(\beta x)]^{1/(k+1)}\psi(t),\quad\
\beta=\sqrt{b(k+1)/a},$

where $C_{1}$ and $C_{2}$ are arbitrary constants and the function
$\psi=\psi(t)$ is described by the pantograph-type ODE:

$\psi^{\prime}_{t}=\psi f\bigl{(}\bar{\psi}/\psi\bigr{)},\quad\
\bar{\psi}=\psi(qt).$ (21)

$2^{\circ}$. The solution for $b(k+1)<0$:

$u(x,t)=[C_{1}\exp(-\beta x)+C_{2}\exp(\beta x)]^{1/(k+1)}\psi(t),\quad\
\beta=\sqrt{-b(k+1)/a},$

where $C_{1}$ and $C_{2}$ are arbitrary constants and the function
$\psi=\psi(t)$ is described by the pantograph-type ODE (21).

$3^{\circ}$. The solution for $k=-1$:

$u(x,t)=C_{1}\exp\Bigl{(}-{\mskip 1.0mub\mskip 1.0mu\over\mskip 1.0mu2a\mskip
1.0mu}x^{2}+C_{2}x\Bigr{)}\psi(t),$

where $C_{1}$ and $C_{2}$ are arbitrary constants and the function
$\psi=\psi(t)$ is described by the pantograph-type ODE (21).

######  Remark 6.

Eqs. (19) and (20) and their solutions can be substantially generalized if the
pantograph term $w=u(x,qt)$ is replaced by $w=u(x,\eta(t))$, where $\eta(t)$
is an arbitrary function (the solutions are sought in the same form, see
[15]).

Equation 20. The equation with varying transfer coefficient of power-law type

$\displaystyle u_{t}=a(u^{k}u_{x})_{x}+u^{k+1}f(w/u),\quad\ w=u(x,qt),$

admits the exact solution

$\displaystyle u(x,t)=t^{-1/k}\varphi(z),\quad\ z=x+\lambda\ln t,$

where $\lambda$ is an arbitrary constant and the function $\varphi=\varphi(z)$
satisfies the second-order ODE with constant delay

$\displaystyle
a(\varphi^{k}\varphi^{\prime}_{z})^{\prime}_{z}-\lambda\varphi^{\prime}_{z}+{\mskip
1.0mu1\mskip 1.0mu\over\mskip 1.0muk\mskip
1.0mu}\varphi+\varphi^{k+1}f(q^{-1/k}\bar{\varphi}/\varphi)=0,\quad\
\bar{\varphi}=\varphi(z+\lambda\ln q).$

Equation 21. The equation with varying transfer coefficient of power-law type

$\displaystyle u_{t}=a(u^{k}u_{x})_{x}+b+u^{-k}f(u^{k+1}-w^{k+1}),\quad
w=u(x,qt),$ (22)

admits the functional separable solution

$u(x,t)=\Bigl{[}\psi(t)-{\mskip 1.0mub(k+1)\mskip 1.0mu\over\mskip
1.0mu2a\mskip 1.0mu}x^{2}+C_{1}x+C_{2}\Bigr{]}^{1/(k+1)},$

where $C_{1}$ and $C_{2}$ are arbitrary constants and the function
$\psi=\psi(t)$ is described by the pantograph-type ODE:

$\psi^{\prime}_{t}=(k+1)f(\psi-\bar{\psi}\bigr{)},\quad\ \bar{\psi}=\psi(qt).$

Equation 22. The equation with varying transfer coefficient of exponential
type

$\displaystyle u_{t}=a(e^{\lambda u}u_{x})_{x}+f(u-w),\quad w=u(x,qt),$ (23)

admits the additive separable solution

$u(x,t)={\mskip 1.0mu1\mskip 1.0mu\over\mskip 1.0mu\lambda\mskip
1.0mu}\ln(Ax^{2}+Bx+C)+\psi(t),$

where $A$, $B$, and $C$ are arbitrary constants and the function $\psi(t)$ is
described by the pantograph-type ODE:

$\psi^{\prime}=2a(A/\lambda)e^{\lambda\psi}+f(\psi-\bar{\psi}),\quad\
\bar{\psi}=\psi(qt).$

Equation 23. The equation with varying transfer coefficient of exponential
type

$\displaystyle u_{t}=a(e^{\lambda u}u_{x})_{x}+f(u-w)+be^{\lambda u},\quad
w=u(x,qt),$ (24)

admits two additive separable solutions.

$1^{\circ}$. The solution for $b\lambda>0$:

$u(x,t)={\mskip 1.0mu1\mskip 1.0mu\over\mskip 1.0mu\lambda\mskip
1.0mu}\ln[C_{1}\cos(\beta x)+C_{2}\sin(\beta x)]+\psi(t),\quad\
\beta=\sqrt{b\lambda/a},$

where $C_{1}$ and $C_{2}$ are arbitrary constants and the function
$\psi=\psi(t)$ is described by the pantograph-type ODE:

$\psi^{\prime}_{t}=f\bigl{(}\psi-\bar{\psi}\bigr{)},\quad\
\bar{\psi}=\psi(qt).$ (25)

$2^{\circ}$. The solution for $b\lambda<0$:

$u(x,t)={\mskip 1.0mu1\mskip 1.0mu\over\mskip 1.0mu\lambda\mskip
1.0mu}\ln[C_{1}\exp(-\beta x)+C_{2}\exp(\beta x)]+\psi(t),\quad\
\beta=\sqrt{-b\lambda/a},$

where $C_{1}$ and $C_{2}$ are arbitrary constants and the function
$\psi=\psi(t)$ is described by the pantograph-type ODE (25).

######  Remark 7.

Eqs. (22)–(24) and their solutions can be substantially generalized if the
pantograph term $w=u(x,qt)$ is replaced by $w=u(x,\eta(t))$, where $\eta(t)$
is an arbitrary function (the solutions are sought in the same form, see
[15]).

Equation 24. The equation with varying transfer coefficient of exponential
type

$\displaystyle u_{t}=a(e^{\lambda u}u_{x})_{x}+b+e^{-\lambda u}f(e^{\lambda
u}-e^{\lambda w}),\quad w=u(x,qt),$ (26)

admits the functional separable solution

$u(x,t)={\mskip 1.0mu1\mskip 1.0mu\over\mskip 1.0mu\lambda\mskip
1.0mu}\ln\Bigl{[}\psi(t)-{\mskip 1.0mub\lambda\mskip 1.0mu\over\mskip
1.0mu2a\mskip 1.0mu}x^{2}+C_{1}x+C_{2}\Bigr{]},$

where $C_{1}$ and $C_{2}$ are arbitrary constants and the function
$\psi=\psi(t)$ is described by the pantograph-type ODE:

$\psi^{\prime}_{t}=\lambda f\bigl{(}\psi-\bar{\psi}\bigr{)},\quad\
\bar{\psi}=\psi(qt).$

Equation 25. The equation with varying transfer coefficient of logarithmic
type

$\displaystyle u_{t}=[(a\ln u+b)u_{x}]_{x}-cu\ln u+uf(w/u),\quad w=u(x,qt),$
(27)

admits two multiplicative separable solutions

$u(x,t)=\exp(\pm\sqrt{c/a}\,x)\psi(t),$

where the function $\psi=\psi(t)$ is described by the pantograph-type ODE:

$\psi^{\prime}_{t}=c(1+b/a)\psi+\psi f\bigl{(}\bar{\psi}/\psi\bigr{)},\quad\
\bar{\psi}=\psi(qt).$

###  3.4 Nonlinear Equations Containing Two Arbitrary Functions

Equation 26. The equation with varying transfer coefficient of general form

$\displaystyle u_{t}=a[g^{\prime}_{u}(u)u_{x}]_{x}+b+{\mskip 1.0mu1\mskip
1.0mu\over\mskip 1.0mug^{\prime}_{u}(u)\mskip
1.0mu}f\bigl{(}g(u)-g(w)\bigr{)},\quad w=u(x,qt),$ (28)

admits the functional separable solution in implicit form

$g(u)=\psi(t)-{\mskip 1.0mub\mskip 1.0mu\over\mskip 1.0mu2a\mskip
1.0mu}x^{2}+C_{1}x+C_{2},$

where the function $\psi=\psi(t)$ is described by the pantograph-type ODE
(25).

Equation 27. The equation with varying transfer coefficient of general form

$\displaystyle u_{t}=a[g^{\prime}_{u}(u)u_{x}]_{x}+bg(u)+{\mskip
1.0mug(u)\mskip 1.0mu\over\mskip 1.0mug^{\prime}_{u}(u)\mskip
1.0mu}f\bigl{(}g(w)/g(u)\bigr{)},\quad w=u(x,qt),$ (29)

admits two functional separable solutions in implicit form.

$1^{\circ}$. The solution for $ab>0$:

$g(u)=\bigl{[}C_{1}\cos(\lambda x)+C_{2}\sin(\lambda x)\bigr{]}\psi(t),\quad\
\lambda=\sqrt{b/a},$

where $C_{1}$ and $C_{2}$ are arbitrary constants and the function
$\psi=\psi(t)$ is described by the pantograph-type ODE (21).

$2^{\circ}$. The solution for $ab<0$:

$g(u)=\bigl{[}C_{1}\exp(-\lambda x)+C_{2}\exp(\lambda x)\bigr{]}\psi(t),\quad\
\lambda=\sqrt{-b/a},$

where $C_{1}$ and $C_{2}$ are arbitrary constants and the function
$\psi=\psi(t)$ is described by the pantograph-type ODE (21).

######  Remark 8.

Eqs. (26)–(29) and their solutions can be substantially generalized if the
pantograph term $w=u(x,qt)$ is replaced by $w=u(x,\eta(t))$, where $\eta(t)$
is an arbitrary function (the solutions are sought in the same form, see
[15]).

##  4 Solutions of Pantograph-Type PDEs That Contain Unknown Functions
$u=u(x,t)$ and $w=u(px,t)$

Preliminary remarks. This section does not include equations and their
solutions that can be obtained by setting $q=1$ in the equations and solutions
considered in Section 2.

###  4.1 Equations Linear in Derivatives

Equation 28. The equation containing an arbitrary function

$\displaystyle u_{t}=au_{xx}+f(u-w),\quad\ w=u(px,t),$ (30)

admits the additive separable solution

$\displaystyle u(x,t)=Ct+\varphi(x),$

where $C$ is an arbitrary constant and the function $\varphi=\varphi(x)$ is
described by the nonlinear second-order pantograph-type ODE:

$\displaystyle
a\varphi^{\prime\prime}_{xx}-C+f(\varphi-\bar{\varphi})=0,\quad\bar{\varphi}=\varphi(px).$

Equation 29. The equation containing an arbitrary function

$\displaystyle u_{t}=au_{xx}+bu+f(u-w),\quad\ w=u(px,t),$ (31)

admits the additive separable solution

$\displaystyle u(x,t)=Ce^{bt}+\varphi(x),$

where $C$ is an arbitrary constant and the function $\varphi=\varphi(x)$ is
described by the nonlinear second-order pantograph-type ODE:

$\displaystyle
a\varphi^{\prime\prime}_{xx}+b\varphi+f(\varphi-\bar{\varphi})=0,\quad\
\bar{\varphi}=\varphi(px).$

Equation 30. The equation containing an arbitrary function

$\displaystyle u_{t}=au_{xx}+uf(w/u),\quad\ w=u(px,t),$ (32)

admits the multiplicative separable solution

$\displaystyle u(x,t)=e^{\lambda t}\varphi(x),$

where $\lambda$ is an arbitrary constant and the function $\varphi=\varphi(x)$
is described by the nonlinear second-order pantograph-type ODE:

$\displaystyle
a\varphi^{\prime\prime}_{xx}+\varphi[f(\bar{\varphi}/\varphi)-\lambda]=0,\quad\
\bar{\varphi}=\varphi(px).$

Equation 31. The equation containing an arbitrary function

$\displaystyle u_{t}=au_{xx}+bu\ln u+uf(w/u),\quad\ w=u(px,t),$ (33)

admits the multiplicative separable solution

$\displaystyle u(x,t)=\exp(Ce^{bt})\varphi(x),$

where $C$ is an arbitrary constant and the function $\varphi=\varphi(x)$ is
described by the nonlinear pantograph-type ODE:

$\displaystyle a\varphi^{\prime\prime}_{xx}+b\varphi\ln\varphi+\varphi
f(\bar{\varphi}/\varphi)=0,\quad\ \bar{\varphi}=\varphi(px).$

######  Remark 9.

Eqs. (30)–(33) and their solutions can be substantially generalized if the
pantograph term $w=u(px,t)$ is replaced by $w=u(\xi(x),t)$, where $\xi(x)$ is
an arbitrary function (the solutions are sought in the same form).

###  4.2 More Complex Nonlinear Equations

Equation 32. The equation with varying transfer coefficient of power-law type

$\displaystyle u_{t}=a(u^{k}u_{x})_{x}+uf(w/u),\quad w=u(px,t)$ (34)

admits the exact solution

$u(x,t)=e^{2\lambda t}U(z),\quad\ z=e^{-k\lambda t}x,$

where $\lambda$ is an arbitrary constant and the function $U=U(z)$ is
determined from the pantograph-type ODE:

$\displaystyle 2\lambda U-k\lambda
zU^{\prime}_{z}=a(U^{k}U^{\prime}_{z})^{\prime}_{z}+Uf(W/U),\quad\ W=U(pz).$

Equation 33. The equation with varying transfer coefficient of power-law type

$\displaystyle u_{t}=a(u^{k}u_{x})_{x}+u^{k+1}f(w/u),\quad\ w=u(px,t).$ (35)

admits the multiplicative separable solution

$u(x,t)=t^{-1/k}\varphi(x),$

where the function $\varphi=\varphi(x)$ is described by the pantograph-type
ODE:

$a(\varphi^{k}\varphi^{\prime}_{x})^{\prime}_{x}+{\mskip 1.0mu1\mskip
1.0mu\over\mskip 1.0muk\mskip
1.0mu}\varphi+\varphi^{k+1}f(\bar{\varphi}/\varphi)=0,\quad\
\bar{\varphi}=\varphi(px).$

Equation 34. The equation with varying transfer coefficient of power-law type

$\displaystyle u_{t}=a(u^{k}u_{x})_{x}+bu^{-k}+f(u^{k+1}-w^{k+1}),\quad\
w=u(px,t),$ (36)

admits the functional separable solution

$u(x,t)=\bigl{[}b(k+1)t+\varphi(x)\bigr{]}^{\textstyle{\mskip 1.0mu1\mskip
1.0mu\over\mskip 1.0muk+1\mskip 1.0mu}},$

where the function $\varphi=\varphi(x)$ is described by the pantograph-type
ODE:

$a\varphi^{\prime\prime}_{xx}+(k+1)f(\varphi-\bar{\varphi})=0,\quad\
\bar{\varphi}=\varphi(px).$

Equation 35. The equation with varying transfer coefficient of exponential
type

$\displaystyle u_{t}=a(e^{\lambda u}u_{x})_{x}+e^{\lambda u}f(u-w),\quad\
w=u(px,t),$ (37)

admits the additive separable solution

$u(x,t)=-{\mskip 1.0mu1\mskip 1.0mu\over\mskip 1.0mu\lambda\mskip 1.0mu}\ln
t+\varphi(x),$

where the function $\varphi=\varphi(x)$ is described by the pantograph-type
ODE:

$a(e^{\lambda\varphi}\varphi^{\prime}_{x})^{\prime}_{x}+{\mskip 1.0mu1\mskip
1.0mu\over\mskip 1.0mu\lambda\mskip
1.0mu}+e^{\lambda\varphi}f(\varphi-\bar{\varphi})=0,\quad\
\bar{\varphi}=\varphi(px).$

Equation 36. The equation with varying transfer coefficient of general form

$\displaystyle u_{t}=[f^{\prime}_{u}(u)u_{x}]_{x}+{\mskip 1.0mua\mskip
1.0mu\over\mskip 1.0muf^{\prime}_{u}(u)\mskip 1.0mu}+g(f(u)-f(w)),\quad\
w=u(px,t),$ (38)

admits the generalized traveling wave solution in implicit form

$f(u)=at+\varphi(x),$

where the function $\varphi=\varphi(x)$ is described by the pantograph-type
ODE:

$\varphi^{\prime\prime}_{xx}+g(\varphi-\bar{\varphi})=0,\quad\
\bar{\varphi}=\varphi(px).$

######  Remark 10.

Eqs. (35)–(38) and their solutions can be substantially generalized if the
pantograph term $w=u(px,t)$ is replaced by $w=u(\xi(x),t)$, where $\xi(x)$ is
an arbitrary function (the solutions are sought in the same form).

##  5 Some Generalizations

###  5.1 Nonlinear Multi-Pantograph Type PDEs

The equations and their solutions considered above can be generalized to the
case of nonlinear multi-pantograph type PDEs of the form

$\displaystyle u_{t}=[g(u)u_{x}]_{x}+F(u,w_{1},\dots,w_{m}),\quad\
w_{j}=u(p_{j}x,q_{j}t),\ \ j=1,\,\dots,\,m.$

Two such equations with exact solutions are given below.

Equation 37. Consider the nonlinear multi-pantograph type PDE:

$\displaystyle u_{t}=a(u^{k}u_{x})_{x}+u^{n}f(w_{1}/u,\dots,w_{m}/u),\quad\
w_{j}=u(p_{j}x,q_{j}t),$ (39)

which generalizes Eq.(9). For $n\not=1$, Eq. (39) admits a self-similar
solution of the same form as Eq. (9):

$\displaystyle u(x,t)=t^{\textstyle{\mskip 1.0mu1\mskip 1.0mu\over\mskip
1.0mu1-n\mskip 1.0mu}}U(z),\quad\ z=xt^{\textstyle{\mskip 1.0mun-k-1\mskip
1.0mu\over\mskip 1.0mu2(1-n)\mskip 1.0mu}},$

where the function $U=U(z)$ satisfies the second-order multi-pantograph type
ODE:

$\displaystyle{\mskip 1.0mu1\mskip 1.0mu\over\mskip 1.0mu1-n\mskip
1.0mu}U+{\mskip 1.0mun-k-1\mskip 1.0mu\over\mskip 1.0mu2(1-n)\mskip
1.0mu}zU^{\prime}_{z}=a(U^{k}U^{\prime}_{z})^{\prime}_{z}+U^{n}f\bigl{(}q_{1}^{\textstyle{\mskip
1.0mu1\mskip 1.0mu\over\mskip 1.0mu1-n\mskip
1.0mu}}W_{1}/U,\dots,q_{m}^{\textstyle{\mskip 1.0mu1\mskip 1.0mu\over\mskip
1.0mu1-n\mskip 1.0mu}}W_{m}/U\bigr{)},$ $\displaystyle W_{j}=U(s_{j}z),\quad\
s_{j}=p_{j}q_{j}^{\textstyle{\mskip 1.0mun-k-1\mskip 1.0mu\over\mskip
1.0mu2(1-n)\mskip 1.0mu}},\ \ j=1,\,\dots,\,m.$

Equation 38. The nonlinear multi-pantograph type PDE, more general than Eq.
(34),

$\displaystyle u_{t}=a(u^{k}u_{x})_{x}+uf(w_{1}/u,\dots,w_{m}/u),\quad
w_{j}=u(p_{j}x,t),\quad\ w_{j}=u(p_{j}x,t),$

admits a solution of the same form as Eq. (34):

$u(x,t)=e^{2\lambda t}U(z),\quad\ z=e^{-k\lambda t}x,$

where $\lambda$ is an arbitrary constant and the function $U=U(z)$ is
determined from the multi-pantograph type ODE:

$\displaystyle 2\lambda U-k\lambda
zU^{\prime}_{z}=a(U^{k}U^{\prime}_{z})^{\prime}_{z}+Uf(W_{1}/U,\dots,W_{m}/U),\quad\
W_{j}=U(p_{j}z).$

###  5.2 Nonlinear Pantograph-Type PDEs Containing Unknown Functions with
Dilation or Contraction of Arguments in the Derivative

One can also consider more general than Eq. (3) nonlinear pantograph-type
reaction-diffusion equations of the form

$\displaystyle u_{t}=[G(u,w)u_{x}]_{x}+F(u,w).$

Below are three examples of exact solutions to such equations.

Equation 39. Consider the pantograph-type diffusion equation with varying
transfer coefficient

$\displaystyle u_{t}=[G(u,w)u_{x}]_{x},\quad\ w=u(px,qt).$ (40)

$1^{\circ}$. In the general case, this equation admits the self-similar
solution

$\displaystyle u(x,t)=U(z),\quad\ z=xt^{-1/2},$

where the function $U=U(z)$ is described by the nonlinear pantograph-type ODE:

$\displaystyle[G(U,W)U^{\prime}_{z}]^{\prime}_{z}+{\textstyle{\mskip
1.0mu1\mskip 1.0mu\over\mskip 1.0mu2\mskip 1.0mu}}zU^{\prime}_{z}=0,\quad\
W=U(sz),\quad s=pq^{-1/2}.$

$2^{\circ}$. For $q=p$, Eq. (40) admits the traveling-wave solution

$u(x,y)=U(z),\quad\ z=kx-\lambda t,$

where $k$ and $\lambda$ are arbitrary constants and the function $U=U(z)$ is
described by the first-order pantograph-type ODE:

$k^{2}G(U,W)U^{\prime}_{z}+\lambda U=C,\quad\ W=U(pz);$

$C$ is an arbitrary constant.

Equation 40. Consider now the equation

$\displaystyle u_{t}=a(e^{\lambda w}u_{x})_{x}+e^{\mu u}f(u-w),\quad\
w=u(px,qt),$ (41)

which is obtained from Eq. (10) by replacing the function $e^{\lambda u}$ with
$e^{\lambda w}$. For $\mu\not=0$, this equation, like Eq. (10), admits an
exact solution of the form

$\displaystyle u(x,t)=U(z)-{\mskip 1.0mu1\mskip 1.0mu\over\mskip
1.0mu\mu\mskip 1.0mu}\ln t,\quad\ z=xt^{\textstyle{\mskip
1.0mu\lambda-\mu\mskip 1.0mu\over\mskip 1.0mu2\mu\mskip 1.0mu}},$ (42)

where the function $U=U(z)$ is described by the nonlinear pantograph-type ODE:

$\displaystyle{\mskip 1.0mu\lambda-\mu\mskip 1.0mu\over\mskip 1.0mu2\mu\mskip
1.0mu}zU^{\prime}_{z}-{\mskip 1.0mu1\mskip 1.0mu\over\mskip 1.0mu\mu\mskip
1.0mu}=aq^{-\lambda/\mu}(e^{\lambda W}U^{\prime}_{z})^{\prime}_{z}+e^{\mu
U}f\Bigl{(}U-W+{\mskip 1.0mu1\mskip 1.0mu\over\mskip 1.0mu\mu\mskip 1.0mu}\ln
q\Bigr{)},$ $\displaystyle W=U(sz),\quad\ s=pq^{\textstyle{\mskip
1.0mu\lambda-\mu\mskip 1.0mu\over\mskip 1.0mu2\mu\mskip 1.0mu}}.$

Equation 41. The equation, more general than Eq. (41),

$u_{t}=[e^{\lambda u}g(u-w)u_{x}]_{x}+e^{\mu u}f(u-w),\quad\ w=u(px,qt),$

depending on two arbitrary functions $f(y)$ and $g(y)$ also has a solution of
the form (42).

###  5.3 Principle of Analogy of Solutions

A fairly general method for constructing exact solutions of partial functional
differential pantograph-type equations is based on the use of the following
principle.

The principle of analogy of solutions. The structure of exact solutions of
functional differential equations of the form

$\displaystyle
F(x,t,u,u_{x},u_{t},u_{xx},u_{xt},u_{tt},\dots,w,w_{x},w_{t},w_{xx},w_{xt},w_{tt},\dots)=0,$
(43) $\displaystyle w=u(px,qt),$

is often (but not always) determined by the structure of solutions of simpler
PDEs:

$\displaystyle
F(x,t,u,u_{x},u_{t},u_{xx},u_{xt},u_{tt},\dots,u,u_{x},u_{t},u_{xx},u_{xt},u_{tt},\dots)=0.$
(44)

Equation (44) does not contain the unknown function with dilated or contracted
arguments; it is obtained from (43) by formally replacing $w$ with $u$.

Most of the solutions to nonlinear pantograph-type reaction-diffusion
equations described in this article have been obtained using the principle of
analogy. Let us illustrate what has been said with specific examples.

Example 1. Consider the pantograph-type PDE with power-law nonlinearities

$\displaystyle u_{t}=au_{xx}+bu^{m}w^{k},\quad\ w=u(px,qt).$ (45)

Following the principle of analogy of solutions, we set $w=u$ in Equation
(45). As a result, we arrive at the simpler PDE

$\displaystyle u_{t}=au_{xx}+bu^{m+k}.$ (46)

This equation admits a self-similar solution of the form [94]:

$\displaystyle u(x,t)=t^{\textstyle{\mskip 1.0mu1\mskip 1.0mu\over\mskip
1.0mu1-m-k\mskip 1.0mu}}U(z),\quad\ z=xt^{-1/2},\quad\ k\not=1-m.$

Using the principle of analogy, we seek the solution of the pantograph-type
PDE (45) also in the form (46). As a result, for the function $U=U(z)$ we get
the nonlinear pantograph-type ODE:

$\displaystyle aU^{\prime\prime}_{zz}+{\mskip 1.0mu1\mskip 1.0mu\over\mskip
1.0mu2\mskip 1.0mu}zU^{\prime}_{z}-{\mskip 1.0mu1\mskip 1.0mu\over\mskip
1.0mu1-m-k\mskip 1.0mu}U+bq^{{\mskip 1.0muk\mskip 1.0mu\over\mskip
1.0mu1-m-k\mskip 1.0mu}}U^{m}W^{k}=0,$ $\displaystyle W=U(sz),\quad\
s=pq^{-1/2}.$

Example 2. Let us now consider the pantograph-type PDE equation with
exponential nonlinearities

$\displaystyle u_{t}=au_{xx}+be^{\mu u+\lambda w},\quad\ w=u(px,qt).$ (47)

Following the principle of analogy of solutions, we set $w=u$ in Equation
(47). As a result, we arrive at the simpler PDE

$\displaystyle u_{t}=au_{xx}+be^{(\mu+\lambda)u}.$ (48)

This equation admits an invariant solution of the form [94]:

$\displaystyle u(x,t)=U(z)-{\mskip 1.0mu1\mskip 1.0mu\over\mskip
1.0mu\mu+\lambda\mskip 1.0mu}\ln t,\quad\ z=xt^{-1/2},\quad\
\mu\not=-\lambda.$

Using the principle of analogy, we seek the solution of the pantograph-type
PDE (47) in the form (48). As a result, for the function $U=U(z)$ we obtain
the nonlinear pantograph-type ODE:

$\displaystyle aU^{\prime\prime}_{zz}+{\mskip 1.0mu1\mskip 1.0mu\over\mskip
1.0mu2\mskip 1.0mu}zU^{\prime}_{z}+{\mskip 1.0mu1\mskip 1.0mu\over\mskip
1.0mu\mu+\lambda\mskip 1.0mu}+bq^{\textstyle-{\mskip 1.0mu\lambda\mskip
1.0mu\over\mskip 1.0mu\mu+\lambda\mskip 1.0mu}}e^{\mu U+\lambda W}=0,$
$\displaystyle W=U(sz),\quad\ s=pq^{-1/2}.$

Note that the principle of analogy of solutions can also be successfully
applied to construct exact solutions of functional differential wave-type PDEs
[118] and PDEs of higher orders that contain unknown functions with dilated or
contracted arguments.

##  6 Brief Conclusions

For the first time, we have described exact solutions of various classes of
nonlinear pantograph-type reaction-diffusion PDEs that, in addition to the
unknown function $u=u(x,t)$, also contain functions with dilated or contracted
arguments of the form $u(px,t)$, $u(x,qt)$, and $u(px,qt)$, where $p$ and $q$
are the free scaling parameters. We have presented examples of nonlinear
pantograph-type PDEs that admit traveling-wave solutions, self-similar
solutions, additive and multiplicative separable solutions, functional
separable solutions, and some other exact solutions. Special attention is paid
to nonlinear pantograph-type PDEs of a rather general form, which contain one
or two arbitrary functions. A number of exact solutions of more complex
nonlinear functional differential equations with varying delay, which
arbitrarily depends on time or spatial coordinate, are described. We have
formulated the principle of analogy of solutions that allows effectively
constructing exact solutions of nonlinear pantograph-type PDEs. The presented
exact solutions can be used as test problems to evaluate the accuracy of
numerical and approximate analytical methods for solving the corresponding
nonlinear initial-boundary value problems for PDEs with varying delay of
pantograph type.

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# Finite Representation Property for Relation Algebra Reducts

Jaš Šemrl

(November 2021)

###### Abstract

The decision problem of membership in the Representation Class of Relation
Algebras (RRA) for finite structures is undecidable. However, this does not
hold for many Relation Algebra reduct languages. Two well known properties
that are sufficient for decidability are the Finite Axiomatisability (FA) of
the representation class and the Finite Representation Property (FRP).
Furthermore, neither of the properties is stronger that the other, and thus,
neither is also a necessary condition. Although many results are known in the
area of FA, the FRP remains unknown for the majority of the reduct languages.
Here we conjecture that the FRP fails for a Relation Algebra reduct if and
only if it contains both composition and negation, or both composition and
meet. We then show the right-to-left implication of the conjecture holds and
present preliminary results that suggest the left-to-right implication.

##  1 Introduction

Relation Algebra and its relational semantics are a neat algebraic tool for
reasoning about various concepts, including program behaviour and correctness
[DJS09] as well as temporal and spatial reasoning [Dün05].

This is why the tractability of decision problems in the area is vital for the
feasibility of these approaches. Two properties that guarantee the
decidability of membership in the representation class are its finite
axiomatisability and the finite representation property.

Neither of these properties hold for the full signature of Relation Algebras,
in fact, the decision problem itself is undecidable [HH02]. This is why we
drop some operations in the signature, to obtain a reduct language, and look
for better behaviour there.

A number of results regarding finite axiomatisability of Relation Algebra
reduct languages exist, though, less is known about the finite representation
property. It is known, however, that neither property is stronger than the
other. For example, the meet-lattice semigroups $\\{\cdot,;\\}$ are a known
example of a finitely axiomatisable representation class with no finite
representation property [BS78, Neu16], whereas the demonically refined
semigroups have been recently shown to be the first example with composition
that has the finite representation property, but a non-finitely axiomatisable
representation class [HŠ21].

Here we examine the finite representation property in detail. We first
conjecture that a Relation Algebra reduct language fails to have the finite
representation property if and only if it includes both composition and
negation, or both composition and meet.

Then we show that if negation and composition are in the signature, the finite
representation property fails. This result, together with [Neu16], shows the
left-to-right implication of the conjecture. We continue by showing that a
finite structure in any other reduct signature is finitely representable if
and only if it embeds into a finite Relation Algebra. Although this does not
prove the right-to-left implication of the conjecture outright, it does
suggest that well known counterexamples to the finite representation property
have a finite representation in these signatures.

##  2 Preliminaries

We now define the concepts discussed in the introduction formally. We do,
however, assume that the reader is familiar with first order logic and Boolean
Algebra.

We begin by defining relational composition (;), converse ($\smile$), and
identity ($1^{\prime}$). Let $R,S\subseteq X\times X$ be a pair of binary
relations over the base $X$. We say

$\displaystyle R;S$ $\displaystyle=\\{(x,z)\mid\exists y:(x,y)\in R,(y,z)\in
S\\}$ $\displaystyle\breve{R}$ $\displaystyle=\\{(y,x)\mid(x,y)\in R\\}$
$\displaystyle 1^{\prime}$ $\displaystyle=\\{(x,x)\mid x\in X\\}$

Together with the Boolean Operations, these define the class of proper
relation algebras as follows

######  Definition 1 (Proper Relation Algebra (PRA)).

The class of Proper Relation Algebras is the class of all
$\\{0,1,-,+,\cdot,1^{\prime},\smile,;\\}$-structures $\mathcal{S}$ that have
some base set $X$, such that if $a\in\mathcal{S}$ then $a\subseteq X\times X$,
$\\{0,1,-,+,\cdot\\}$ are interpreted as proper Boolean operations, and
$\\{1^{\prime},\smile,;\\}$ are interpreted as proper relational identity,
converse, and composition.

This gives rise to the class of Representable Relation Algebras (RRA), defined
as

######  Definition 2 (Representable Relation Algebra (RRA)).

The class of Representable Relation Algebras is the class of Proper Relation
Algebras, closed under isomorphic copies.

######  Remark 3.

Much like with the language of Relation Algebras, we may consider looking at
relation algebra reduct languages
$\tau\subseteq\\{0,1,-,+,\cdot,1^{\prime},\smile,;\\}$, the class of all
proper $\tau$-structures $P(\tau)$, and the class of all representable
$\tau$-structures $R(\tau)$.

######  Definition 4 ((Finite) Representation).

A representation is an isomorphism that maps a structure in the representation
class to a proper structure. It is finite if the elements of the image are
relations over a finite base $X$.

######  Definition 5 (Finite Representation Property (FRP)).

A representation class is said to have the finite representation property if
all its finite members have a finite representation.

######  Definition 6 (Finite Axiomatisability (FA)).

The representation class is finitely axiomatisable if membership can be
axiomatised by finitely many first order formulas.

It is known that the language of Relation Algebras has neither FA, nor the
FRP. However, it is the case that the equational theory of Relation Algebras
is finitely axiomatisable. This gives rise to the class of Relation Algebras
(RA), which are all the $\\{0,1,-,+,\cdot,1^{\prime},\smile,;\\}$-structures
that model the equational theory of representable relation algebras.
Alternatively, we can axiomatise the class as follows

  1. 1.

$\\{0,1,-,+,\cdot\\}$ is a Boolean Algebra

  2. 2.

$;,\smile$ are additive over $+$ (and thus monotone over $\leq$)

  3. 3.

$-(\breve{a})=(-a)^{\smile}$ and $a;1=1\vee(-a);1=1$

  4. 4.

$(a;b)^{\smile}=\breve{b};\breve{a}$, $\breve{\breve{a}}=a$ , and
$\breve{1^{\prime}}=1^{\prime}$

  5. 5.

$a\cdot(b;c)=0\Longleftrightarrow b\cdot(a;\breve{c})=0$

It is easy to see that $PRA\subseteq RRA\subseteq RA$. Furthermore, it is
known that all of the inclusions are proper and thus $PRA\subset RRA\subset
RA$.

Finally, we define the concept of an atom. As Relation Algebras are extensions
of Boolean Algebras, we know that they will inevitably have atoms, which are
all elements $a\neq 0$ such that for all $b$ if $b\leq a$ it is true that
$b=0\vee b=a$. Thus every element of a relation algebra can be expressed as a
unique join of atoms.

##  3 The Conjecture and Related Work

In this section we conjecture what reduct signatures of the language of
Relation Algebras have the finite representation property. Then we survey the
known results in the area and put the findings presented in the later sections
into context. We begin by stating our conjecture regarding the FRP for
Relation Algebra reduct languages.

######  Conjecture 7.

Let $\tau$ be a Relation Algebra reduct signature. $\tau$ has the finite
representation property if and only if

$\\{-,;\\}\not\subseteq\tau\not\supseteq\\{\cdot,;\\}$

Let us start by examining the known results in the area of the finite
representation property. We summarise the results in the table below.

No FRP | FRP
---|---
$\\{\cdot,;\\}\subseteq\tau$ [Neu16] | $\\{;\\},\\{1^{\prime},;\\}$
$\\{-,;\\}\subseteq\tau$ [Section 4] |  $\\{\leq,;\\}$ [Zar59]
|  $\\{D,\smile,;\\}\subseteq\tau\not\supseteq\\{-,+,\cdot\\}$ [HE13, Šem21]
|  $\\{\leq,\setminus,/,\smile,;\\}$ [Rog20]
|  $\\{\sqsubseteq,;\\}$ [HŠ21]

We see that the left-to-right implication of Conjecture 7 is proven, when we
combine our new result from Section 4 with the result from [Neu16]. The
results on the right-to-left implication are, however, more sparse.

Representable $\\{;\\}$\- and $\\{1^{\prime},;\\}$-structures are known to be
finitely representable using the Cayley representation for groups. This result
is extended by [Zar59], to allow for inclusion of partial ordering (provided
there is no relational identity in the signature). In [HE13], an explicit
representation of representable
$\\{0,1,\leq,D,R,1^{\prime},\smile,;\\}$-structures is defined, and in [Šem21]
we show that this construction works for a wider similarity class of
signatures. [Rog20] defines finite representations for ordered residuated
semigroups by embedding them into relational quantales. We also mention our
FRP result for demonically refined semigroups where we define an explicit
construction of a finite representation in [HŠ21]. Although strictly not a
Relation Algebra reduct language, it was the first signature containing
composition where the FA fails, but the FRP holds.

Similarly, there exist finitely signatures with both FA and FRP, neither FA
nor FRP, and FA but no FRP. Examples are summarised in the table below.

| FRP | No FRP
---|---|---
FA |  $\\{\leq,;\\}$[Zar59] |  $\\{\cdot,;\\}$ [BS78, Neu16]
No FA |  $\\{\sqsubseteq,;\\}$ [HŠ21] |  $\\{+,\cdot,;\\}$ [HJ12]

##  4 Failure of FRP for Negation and Composition

In this section we prove that if a RA reduct language contains composition and
negation, the finite representation property will fail. We do that by showing
that the Point Algebra does not have a mapping to a proper relational
structure over a finite base that correctly preserves negation and
composition. This result, together with that in [Neu16] shows the FRP fails in
all cases where Conjecture 7 suggests it will fail, showing its right-to-left
implication.

We begin by looking at the Point Algebra and define it as follows

######  Definition 8 (Point Algebra).

Point Algebra $\mathcal{P}$ is a Proper Relation Algebra over the base of
$\mathbb{Q}$ with eight elements

$\\{0,1,=,\neq,<,\leq,>,\geq\\}$

where $0$ is the empty relation and $1=\mathbb{Q}\times\mathbb{Q}$ and the
rest of the elements are the arithmetic binary predicates defined for
$\mathbb{Q}$. Observe that The Point Algebra is closed under all the
operations in the language of RA.

######  Lemma 9.

For any mapping $\theta:\mathcal{P}\rightarrow\wp(X\times X)$ such that
$(-a)^{\theta}=-(a^{\theta})$ and $(a;b)^{\theta}=a^{\theta};b^{\theta}$,
there will exist some $x,y\in X$ such that $(x,y)\in 1^{\theta}$.

###### Proof.

As $-$ is represented correctly and $0=-1$, it must hold that $1^{\theta}$ is
not an empty relation (i.e. there exists some $(x,y)\in 1^{\theta}$) or that
$0^{\theta}$ is not an empty relation (i.e. there must exist some $(x,z)\in
0^{\theta}$). In the latter case observe that since $\theta$ represents
composition correctly and $0=1;0$, there must exist some $y$ such that
$(x,y)\in 1^{\theta}$. ∎

######  Lemma 10.

For any mapping $\theta:\mathcal{P}\rightarrow\wp(X\times X)$ such that
$|X|<\omega$, $(-a)^{\theta}=-(a^{\theta})$, and
$(a;b)^{\theta}=a^{\theta};b^{\theta}$, there will exist some $x\in X$ such
that $(x,x)\in 1^{\theta}$.

###### Proof.

By Lemma 9, there must exist some $(y,z)\in 1^{\theta}$. Observe that
$1=1;1=...=1^{|X|+2}=...$, so, in order for $\theta$ to represent ; correctly,
there must exist $x_{1},x_{2},...,x_{|X|+1}\in X$ such that
$\\{(y,x_{1}),(x_{1},x_{2}),(x_{2},x_{3}),...,(x_{|X|},x_{|X|+1}),$
$(x_{|X|+1},z)\\}\subseteq 1^{\theta}$.

As $|X|<\omega$, we know that there exits some $i<j<|X|+2$ such that
$x_{i}=x_{j}=x$. Observe that since
$\\{(x_{i},x_{i+1}),(x_{i+1},x_{i+2}),...,(x_{j-1},x_{j})\\}\subseteq
1^{\theta}$, and $1=1^{j-i}$, it must also hold that $(x,x)\in 1^{\theta}$, as
$\theta$ represents ; correctly. ∎

######  Lemma 11.

For any mapping $\theta:\mathcal{P}\rightarrow\wp(X\times X)$ such that there
exist some $x\in X$ such that $(x_{0},x_{0})\in 1^{\theta}$,
$(-a)^{\theta}=-(a^{\theta})$, and $(a;b)^{\theta}=a^{\theta};b^{\theta}$,
there must exist, for any $n<\omega$, $x_{1},x_{2},...,x_{n}\in X$ such that
$x_{i}\neq x_{j},(x_{i},x_{j})\in\leq^{\theta}$, and
$(x_{j},x_{i})\in>^{\theta}$, for all $0\leq i<j\leq n$.

###### Proof.

We show this by induction. In the base case $n=1$. Since $(x_{0},x_{0})\in
1^{\theta}$, $1=\leq;>$, and $\theta$ represents ; correctly, we know that
there must exist $x_{1}$ such that
$(x_{0},x_{1})\in\leq^{\theta},(x_{1},x_{0})\in>^{\theta}$. If it were true
that $x_{1}=x_{0}$, then both $(x_{0},x_{0})\in\leq^{\theta}$ and
$(x_{0},x_{0})\in>^{\theta}$ and since $>=-\leq$, $\theta$ would not represent
$-$ correctly.

Now assume we have $x_{0},x_{1},...,x_{n}$ for some $0<n<\omega$ points for
which the induction hypothesis holds. Observe that since
$(x_{n},x_{0})\in>^{\theta},(x_{0},x_{n})\in\leq^{\theta}$, and $\theta$
represents composition, it must hold that
$(x_{n},x_{n})\in(>;\leq)^{\theta}=1^{\theta}$. Thus, again to represent ;,
there must exist $x_{n+1}$ such that
$(x_{n},x_{n+1})\in\leq^{\theta},(x_{n+1},x_{n})\in>^{\theta}$, and as $>;>=>$
and $\leq;\leq=\leq$ it also holds that for all
$i<n:(x_{i},x_{n+1})\in\leq^{\theta},(x_{n+1},x_{i})\in>^{\theta}$. Observe
that if $x_{n+1}$ was the same as $x_{i}$ for some $i\leq n$, it would hold
that $(x_{i},x_{n})=(x_{n+1},x_{n})\in\leq^{\theta}$ as well as in
$>^{\theta}$. Thus the induction hypothesis is preserved. ∎

######  Theorem 12.

Any RA-reduct signature containing $\\{-,;\\}$ fails to have the FRP.

###### Proof.

Suppose a signature $\\{-,;\\}\subseteq\tau$ had the FRP. Then the
$\tau$-reduct of the Point Algebra would have a representation $\theta$ over a
finite base $X$. By Lemma 10, there must exist some $x\in X$ such that
$(x,x)\in 1^{\theta}$. This, together with Lemma 11 is sufficient for $X$ to
have $|X|+1$ distinct points and we have reached a contradiction. ∎

##  5 Embedding into RA and FRP

In this section we examine why the Point Algebra, a counterexample to the FRP
for signatures above $\\{\cdot,;\\}$ and $\\{-,;\\}$, has a finite
representation in all other Relation Algebra reduct signatures. In fact, any
structure in a signature not above $\\{\cdot,;\\}$ nor $\\{-,;\\}$ that embeds
into a finite relation algebra (not necessarily representable) will have a
finite representation.

Take a relation algebra $\mathcal{S}$ with the set of atoms $A$ and define a
mapping

$\theta:\mathcal{S}\rightarrow\wp(A\times A)$

where

$(a,b)\in\mathcal{S}^{\theta}\Longleftrightarrow a;s\geq b$

and observe the following

######  Lemma 13.

For any $s\not\leq t\in\mathcal{S}$, there exists a pair $(a,b)\in
s^{\theta}-t^{\theta}$.

###### Proof.

Observe that since $s\not\leq t$, there must exist an atom $a\leq s-t$. If $a$
is an atom, so is $1^{\prime}\cdot(a;\breve{a})=D(a)$. Thus $D(a);s\geq a$, by
monotonicity, but not $D(a);t\geq a$ and thus $(D(a),a)\in
s^{\theta}-t^{\theta}$. ∎

######  Lemma 14.

$\theta$ represents $;,1^{\prime},\smile$ correctly.

###### Proof.

Observe how by associativity and monotonicity if $a;s\geq b$ and $b;t\geq c$,
we have $a;s;t\geq c$, so $s^{\theta};t^{\theta}\leq(s;t)^{\theta}$. For
$s^{\theta};t^{\theta}\geq(s;t)^{\theta}$, observe that if $a;(s;t)\geq b$,
then by associativity $(a;s);t\geq b$. By additivity and distributivity, there
must thus exist an atom $c\leq a;s$ such that $c;t\geq b$.

Observe that for all $a$ it is true by the identity law and monotonicity that
$a;1^{\prime}\geq a$. Furthermore, if $a,b$ are atoms such that
$a;1^{\prime}=a\geq b$ then it must be the case that $a=b$.

Finally, if $a;s\geq b$, then $a;s\cdot b\neq 0$, and, by Percian triangle
law, $b;\breve{s}\cdot a\neq 0$. Since $a$ is an atom, this implies $a\leq
b;\breve{c}$ and thus it holds that if $(a,b)\in s^{\theta}$ then
$(b,a)\in(\breve{s})^{\theta}$. ∎

######  Lemma 15.

$0,1,+$ are correctly represented by $\theta$.

###### Proof.

As $a;0=0$, there is no pair of atoms $a,b$ such that $a;0\geq b$ and thus $0$
is represented correctly. Furthermore, if there exists some $s$ such that
$a;s\geq b$, then $a;1\geq b$, by monotonicity, so $1$ is represented
correctly as well.

Let us show that $s^{\theta}+t^{\theta}\leq(s+t)^{\theta}$. Without loss,
suppose $a;s\geq b$. Then $a;(s+t)\geq b$ as well, by monotonicity. Finally,
if $a;(s+t)\geq b$, then by distribuitivity of the lattice, it must hold that
$a;s\geq b$ or $a;t\geq b$, so $s^{\theta}+t^{\theta}\geq(s+t)^{\theta}$. ∎

Thus we can conclude the following

######  Proposition 16.

If $\tau$ is above neither $\\{-,;\\}$ nor $\\{\cdot,;\\}$ and a finite
$\tau$-structure embeds into a finite relation algebra, then it is finitely
representable.

###### Proof.

By Lemmas 13-15, $\theta$ is a finite representation for a $\tau$-reduct of a
Relation Algebra and all its substructures. ∎

Although this result does not prove the right-to-left implication of
Conjecture 7 outright, it does suggest that finding a counterexample to the
FRP for any of the signatures, conjectured to have the FRP, is going to be a
difficult undertaking. Point Algebra, or the Anti-Monk Algebra, two well known
counterexamples to the FRP in certain signatures are both Relation Algebras
and, as such, have a finite representation in signatures that do not contain
both negation and composition, nor both meet and composition.

However, there do exist signatures where it is known that not all finite
representable structures embed into a finite Relation Algebra. It is important
to add, though, that these proofs heavily rely on the signature in question to
be above $\\{+,\cdot,;\\}$ [HJ12] or $\\{-,;\\}$ [Neu16].

##  6 Problems

Finally, we look at some open problems in the area of the Finite
Representation Property. We have shown that the left-to-right implication of
Conjecture 7 holds and provided a proposition that suggests the right-to-left
implication may hold as well. However, it does remain an open question whether
or not these signatures have the FRP.

######  Problem 17.

Do signatures $\tau$ that are above neither $\\{\cdot,;\\}$ nor $\\{-,;\\}$
have the Finite Representation Property?

Answering this question, as a result of Proposition 16, is equivalent to the
following

######  Problem 18.

For signatures $\tau$ above neither $\\{\cdot,;\\}$ nor $\\{-,;\\}$, does
every finite representable structure embed into a $\tau$-reduct of a finite
Relation Algebra?

However, defining such an embedding is not a trivial undertaking, as closing
the structure of meet-negation completions under an associative composition
operation gives rise to a number of challenges.

As discussed in Section 1, FA and FRP are both sufficient conditions for the
decidability of the membership in the representation class for finite
structures. However, the following is not known.

######  Problem 19.

Does there exist a Relation Algebra reduct language with decidable membership
in representation class for finite structures decision problem, but neither FA
nor FRP?

A related decision problem remains open for the Language of Relation algebras
and a number of its reducts.

######  Problem 20 ([Mad94]).

Is determining whether a finite Relation Algebra has a finite representation
decidable?

We have touched on the Finite Representation Property of demonically refined
semigroups in Section 3. The demonic refinement predicate is part of the
Demonic Relational Calculus, defined to model the behaviour of
nondeterministic programs when the Demon is in control of the nondeterministic
choice. This gives rise to the following problems.

######  Problem 21.

Does every operation in the language of Relation Algebra have a demonic
coutnerpart? Does this allow us to define the Demonic Relation Algebra?

######  Problem 22.

What Demonic Relation Algebra Reduct Signatures have FA representation class?
Which have the FRP? What about mixed signatures?

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