Comments on the paper "Solutions of Multitime Reaction-Diffusion PDE"

TL;DR

This paper critiques a previous work on multitime reaction-diffusion PDEs, arguing that its main results are misleading, overly simplified, and can be derived from earlier solutions using straightforward methods.

Contribution

It clarifies that the solutions presented in the criticized paper are simple generalizations of known solutions and can be obtained more easily through differential constraints.

Findings

Main results are misleading and derivable from earlier work

Exact solutions are simple generalizations of known solutions

Solutions can be obtained via differential constraints

Abstract

The Comments are devoted to the paper ``Solutions of Multitime Reaction-Diffusion PDE'' (Mathematics, vol. 10 (2022), 3623), in which main results are misleading and can be derived in a simple way from those obtained earlier. Moreover, it is shown that the exact solution derived therein are simple generalizations of the known solutions and are easily obtainable by the method of differential constrains. The Comments were submitted to the journal Mathematics.

Equations

$$h^{1}(t)\frac{\partial u}{\partial t^{1}}+\dots+h^{m}(t)\frac{\partial u}{\partial t^{m}}=\mu\frac{\partial^{n}u}{\partial x^{n}}+f\Big{(}u,\dots,\frac{\partial^{n-1}u}{\partial x^{n-1}}\Big{)},$$

$$\frac{\partial u}{\partial\tau^{1}}+\dots+\frac{\partial u}{\partial\tau^{m}}=\mu\frac{\partial^{n}u}{\partial x^{n}}+f\Big{(}u,\dots,\frac{\partial^{n-1}u}{\partial x^{n-1}}\Big{)}.$$

$${\tau }^i=\int \frac{d{t}^i}{h^i(t)},i=1,\dots ,m$$

$$\frac{\partial u}{\partial\tau^{1}}+\dots+\frac{\partial u}{\partial\tau^{m}}=\mu\frac{\partial^{n}u}{\partial x^{n}}-k\frac{\partial u}{\partial x}+f\Big{(}u,\dots,\frac{\partial^{n-1}u}{\partial x^{n-1}}\Big{)},$$

$$\frac{\partial w}{\partial {\tau }^1}+\cdots +\frac{\partial w}{\partial {\tau }^m}=0$$

$$\frac{\partial v}{\partial {\tau }^1}+\cdots +\frac{\partial v}{\partial {\tau }^m}=1,$$

$$u=\phi\Big{(}\tau,\,x+kv(\tau)+P_{1}(\tau)\Big{)}$$

$$\frac{\partial u}{\partial\tau^{1}}+\dots+\frac{\partial u}{\partial\tau^{m}}=\mu\frac{\partial^{n}u}{\partial y^{n}}-k\frac{\partial u}{\partial y}+f\Big{(}u,\dots,\frac{\partial^{n-1}u}{\partial y^{n-1}}\Big{)}$$

$$\frac{\partial u}{\partial {\tau }^1}+\cdots +\frac{\partial u}{\partial {\tau }^m}=0$$

$$u=\psi ({\omega }_1,\dots ,{\omega }_{m-1},y)$$

$$\mu\frac{d^{n}u}{dy^{n}}-k\frac{du}{dy}+f\Big{(}u,\dots,\frac{d^{n-1}u}{dy^{n-1}}\Big{)}=0.$$

$$u=\psi ({\omega }_1,\dots ,{\omega }_{m-1},x+k{\tau }^m)$$

$$u=\phi\Big{(}\tau,\,x+k\tau^{m}+P(\omega_{1},\dots,\omega_{m-1})\Big{)}$$

$$t^1\frac{\partial u}{\partial t^1}+\cdots +t^m\frac{\partial u}{\partial t^m}=\mu \frac{{\partial }^{2}u}{\partial x^2}-a{u}^3+b{u}^2$$

$$\frac{\partial u}{\partial {\tau }^1}+\cdots +\frac{\partial u}{\partial {\tau }^m}=\mu \frac{{\partial }^{2}u}{\partial x^2}-a{u}^3+b{u}^2$$

$${\tau }^i=\ln t^i,i=1,\dots ,m.$$

$$\frac{\partial u_{\ast }}{\partial {\tau }_{\ast }^1}+\cdots +\frac{\partial u_{\ast }}{\partial {\tau }_{\ast }^m}=\frac{{\partial }^2{u}_{\ast }}{\partial x_{\ast }^2}-u_{\ast }^3+u_{\ast }^2$$

$$u_{\ast }=\frac{a}{b}u,x_{\ast }=\frac{|b|}{\sqrt{\mu a}}x,{\tau }_{\ast }^i=\frac{b^2}{a}{\tau }^i(i=1,\dots ,m).$$

$$\frac{\partial u}{\partial {\tau }^1}+\cdots +\frac{\partial u}{\partial {\tau }^m}=\frac{{\partial }^{2}u}{\partial x^2}-u^3+u^2$$

$$\frac{\partial u}{\partial \tau }=\frac{{\partial }^{2}u}{\partial x^2}+u(1-u)(u-\delta ).$$

$$u=\frac{\sqrt{2}}{x-\sqrt{2}{\tau }^m+P({\omega }_1,\dots ,{\omega }_{m-1})}$$

$$u=\frac{1}{1+P(\omega_{1},\dots,\omega_{m-1})\exp\Big{(}\frac{x}{\sqrt{2}}-\frac{\tau^{m}}{2}\Big{)}},$$

$$u=\frac{\sqrt{2}}{x-\sqrt{2}\tau +x_0},$$

$$u=\frac{1}{1+C\exp\Big{(}\frac{x}{\sqrt{2}}-\frac{\tau}{2}\Big{)}}.$$

$$u=\frac{1}{2}{1+\tanh\Big{(}\frac{-\sqrt{2}}{4}(x+x_{0})+\frac{1}{4}\tau\Big{)}},$$

$${\tau }^1=H^1(t^1,t^2),{\tau }^2=H^2(t^1,t^2),$$

$$h^{1}(t)\frac{\partial u}{\partial t^{1}}+h^{2}(t)\frac{\partial u}{\partial t^{2}}=\Big{(}h^{1}(t)\frac{\partial H^{1}}{\partial t^{1}}+h^{2}(t)\frac{\partial H^{1}}{\partial t^{2}}\Big{)}\frac{\partial u}{\partial\tau^{1}}+\Big{(}h^{1}(t)\frac{\partial H^{2}}{\partial t^{1}}+h^{2}(t)\frac{\partial H^{2}}{\partial t^{2}}\Big{)}\frac{\partial u}{\partial\tau^{2}}.$$

$$\begin{array}[]{l}\vskip 6.0pt plus 2.0pt minus 2.0pth^{1}(t)\frac{\partial H^{1}}{\partial t^{1}}+h^{2}(t)\frac{\partial H^{1}}{\partial t^{2}}=1,\\ h^{1}(t)\frac{\partial H^{2}}{\partial t^{1}}+h^{2}(t)\frac{\partial H^{2}}{\partial t^{2}}=1.\end{array}$$

$$\begin{array}[]{l}\vskip 6.0pt plus 2.0pt minus 2.0ptH^{1}=\int\frac{dt^{1}}{h^{1}(t)}+W^{1}(I),\\ H^{2}=\int\frac{dt^{1}}{h^{1}(t)}+W^{2}(I),\end{array}$$

$$\frac{d{t}^1}{h^1(t)}=\frac{d{t}^2}{h^2(t)}.$$

Full text

Keywords: exact solution, diffusion-convection-reaction equation, method of differential constrains, scaling transformation.

1 Multitime reaction-diffusion PDE

The recent paper [1] is devoted to search for exact solutions of a special class of PDEs, the so-called multitime reaction-diffusion equations

[TABLE]

where $u(t,x)$ is an unknown smooth function, $f$ and $h^{i}(t)$ are given function from the class $C^{1}$, while $\mu\not=0,\,t=(t^{1},\dots,t^{m})$. In the PDE theory, the class of PDEs (1) with $n=2$ is known as the class of ultra-parabolic equations.

From the very beginning, it should be stressed that the PDE class (1) is not presented in a canonical form. As a result, the paper [1] consists of many cumbersome statements and awkward formulae. Moreover, the technique used for constructing exact solutions is nothing else but a particular realization of the method of differential constrains. This method was suggested by the know Soviet mathematician with the Ukrainian roots Yanenko in 1960s [3]. Although his pioneer work was never translated into English, one is well-known among experts in the field of exact solutions for nonlinear PDEs (see Chapter 5 in [2] and references therein). Notably the method of compatible differential constrains (side conditions) [4] can be considered as a generalization of the method of differential constrains.

Each PDE belonging to class (1) is reducible to its canonical form

[TABLE]

The corresponding transformations is very simple and reads as

[TABLE]

provided each function $h^{i}(t)$ depends only on the variable $t^{i}$ (equation (12)[1] contains exactly such functions) and $h^{i}(t)\not=0$ (obviously this inequality is true at least in some sub-domain where the functions $h^{i}(t)$ are continuously differentiable). If some functions $h^{i}(t)$ depend on two or more variables then an appropriate transformation still exists and it is shown in Appendix how that can be constructed.

Transformation (3) (or its analogue, see Appendix) simultaneously reduce equations (2), (3) and (4) from [1] to much simpler forms

[TABLE]

[TABLE]

and

[TABLE]

respectively. According to the method of differential constrains, the linear differential equations (5) and (6) are additional constrains used for constructing exact solutions of the nonlinear PDE (1). Notably, the idea of application of linear differential equations for solving nonlinear PDEs was used earlier in many papers. For example, the method of additional generating conditions [5, 6] is based on additional constrains in the form of linear ordinary differential equations.

Thus, one should study the class of PDEs (2) with the additional constrains (5) and (6) instead of PDEs used in [1] (see formulae (1), (3) and (4) therein).

Now it can be easily demonstrated that Propositions 1–4, which form the theoretical core of the technique developed in [1], are rather trivial statements. Consider, for example, Proposition 1, which simply says that the function

[TABLE]

is an exact solution of Eq.(2) provided the functions $\phi(\tau,x)$, $P_{1}(\tau)$ and $v(\tau)$ are the exact solutions of Eqs.(4), (5) and (6), respectively. Of course, sufficient smoothness of the above functions in the relevant domains is assumed as well. On the other hand, if one substitutes formula (7) into Eq.(2) and calculates derivatives then immediately Eq.(4) for the function $\phi$ is obtained. But the latter is a solution of Eq.(4) by the above assumption. As a result, the authors propose to find exact solutions of Eq.(2) using the given solution of Eqs.(4), which is more complicated. It is a useless technique.

However, using the method of differential constrains, the more constructive result is obtainable in a straightforward way. In fact, Eq.(2) can be rewritten in the form

[TABLE]

by introducing the variable $y=x+k\tau^{m}$. Now we consider the linear PDE

[TABLE]

as the differential constrain for Eq.(4). The general solution of Eq.(9) is readily constructed:

[TABLE]

where $\psi$ is an arbitrary smooth function, $\omega_{j}=\tau^{m}-\tau^{j},\,(j=1,\dots,m-1)$.

Thus, formula (10) is an exact solution of Eq.(4) provided the function $\phi(\tau,y)$ (with the variables $\tau=(\tau^{1},\dots,\tau^{m})$ to-be-determined as parameters) is a solution of the ordinary differential equation (ODE)

[TABLE]

Although (11) is still the nonlinear ODE, many such equations with correctly-specified function $f$ are integrable and the corresponding exact solutions can be found either in the well-known handbooks like [11], or by using computer algebra solvers from Maple, Mathematica etc. Finally, each exact solution (10) of Eq.(4) is automatically the solution

[TABLE]

of the nonlinear PDE (2). In the case $m=1$, one obtains the plane wave solutions, which are common in real-world applications provided they are bounded and non-negative, i.e., they are traveling fronts [7, 2].

Proposition 2 in [1] is a straightforward generalization of Proposition 1. On the other hand, there is no need to introduce $r+1$ particular solutions of Eg.(5) in order to generalize Proposition 1 because Eg.(5) is integrable and its general solution can be simply used. Moreover, a $r$-parameter family of exact solutions of Eg.(4) is needed in order to obtain the relevant family of solutions of Eg.(2). Obviously, this statement cannot be used for real applications because the authors again try to solve the given PDE using solutions of the more complicated PDE. On the other hand, using the general solutions of Egs.(5) and (6), formula (7) can be rewritten in the form

[TABLE]

where $P(\omega_{1},\dots,\omega_{m-1})$ is an arbitrary smooth function and $\phi(\tau,\,x)$ is a solution of Eg.(4). In particular, one obtains the plane wave solutions in the case $m=1$ provided $\phi(\tau,\,x)$ does not depend on $\tau=\tau^{1}$.

2 Exact solutions of a specific multitime reaction-diffusion equation

Exact solutions in explicit forms are presented in Section 4 [1] only for the nonlinear PDE

[TABLE]

with the constants $\mu>0,\,a>0$ and $b\not=0$. All the exact solutions of PDE (14) constructed in [1] are nothing else but a direct generalizations of the known solutions of this equation in the case $m=1$. Moreover, the solutions in the case $m>1$ are obtainable from those with $m=1$ in a straightforward way using the method of differential constrains.

First of all, PDE (14) is reducible to its canonical form

[TABLE]

by the transformation

[TABLE]

Further, it can be noted that PDE (15) can be simplified to the form

[TABLE]

using the scaling transformations

[TABLE]

So, one can examine the nonlinear PDE (the symbol star is omitted in what follows)

[TABLE]

instead of (14) without losing a generality.

Now one realizes that PDE (19) with $m=1$ is the known Huxley equation and its exact solutions were constructed, for example, in [8, 9] using non-classical symmetries. This equation can be also thought as a particular case of the famous Fitzhugh-Nagumo equation

[TABLE]

Exact solutions of the latter were constructed in many studies [10, 8, 9]. In the book [7], traveling waves of Eqs.(20) are presented. It can be noted that all the exact solutions constructed in Section 4 [1] is nothing else but direct generalizations of those of the Huxley equation.

Let us show this. First of all, it should be noted that an arbitrary solution $u(\tau,x)$ of PDE (19) can be multiplied to the form $u(\tau+\tau_{0},\pm x+x_{0})$ with an arbitrary constant vector $\tau_{0}$ and a constant $x_{0}$. It is a trivial consequence of invariance of PDE (19) w.r.t. the time and space translations and the discrete transformation $x\to-x$. In particular it means that exact solutions $u_{1},\,u_{2}$ and $u_{3}$ are equivalent to those $u_{2},\,u_{4}$ and $u_{6}$, respectively (see pages 9–10 in [1]). Notably the authors should present also the solution $u_{8}$ replacing $x$ by $-x$ in $u_{7}$.

The most interesting from the applicability point of view are the solutions $u_{1}$ and $u_{3}$ [1], which are reducible to much simpler form (see the above transformations):

[TABLE]

and

[TABLE]

respectively. Here $P(\omega_{1},\dots,\omega_{m-1})$ is an arbitrary smooth function of the variables $\omega_{j}=\tau^{m}-\tau^{j},\,(j=1,\dots,m-1)$.

In the case $m=1$, solution (21) takes the form

[TABLE]

which is nothing else but a very particular case of the exact solution (5.25)[8]. Similarly, solution (22) reduces to

[TABLE]

Assuming that the constant $C=e^{x_{0}}>0$, one arrives at the traveling front

[TABLE]

which was identified in [10]. In the case $C<0$, the exact solution (25) with the function $\coth$ is obtained.

Finally, it should be stressed that the method of differential constrains with using the linear PDE (9) as the given constrain allow us easily to identify a set of exact solutions of PDE (19), which are reducible to those obtained in [1]. Actually, the authors do not apply Propositions 1–4 for finding exact solutions but ODE (15)–(16) [1] are used, therefore they implicitly use the method of differential constrains for finding exact solutions. By the way, ODE (16) [1] is fully integrable in terms of elliptic functions, hence its general solution involving $y_{5},\,y_{6}$ and $y_{7}$ [1] as particular cases could be presented therein.

In conclusion, I would like to stress that nowadays there are many papers devoted to search for exact solutions of nonlinear PDEs, in which new methods are suggested without knowing the state-of-art. As a result, the newly suggested methods very often are not new and the solutions obtained are straightforward generalizations of those derived earlier. Study [1] is a typical example.

3 Appendix

If PDE (1) involves the smooth functions $h^{i}(t),\ i=1,\dots,m$ of the more general form than it was assumed above then a transformation reducing PDE (1) to PDE (2) still exists. Let us construct the transformation in the case $m=2$ in order to avoid cumbersome formulae (the case $m>2$ can be examined in the same way). Assuming that a non-degenerate transformation in question possesses the form

[TABLE]

one easily calculates

[TABLE]

So, the local transformation (26) reduces PDE (1) to PDE (2) with $m=2$ provided the functions $H^{1}$ and $H^{2}$ form an arbitrary solution of the linear decoupled system of the first-order PDE

[TABLE]

According to the theory of the linear first-order PDEs the solutions of (28) possess the form

[TABLE]

where $W^{1}$ and $W^{2}$ are functions of the first integral $I(t^{1},t^{2})$ of the ODE

[TABLE]

In the case of real-world applications, the functions $h^{1}(t)$ and $h^{2}(t)$ are such that the first integrals of ODE (29) can be written down in an explicit form, otherwise those can be presented in an implicit form. Obviously, the functions $W^{1}$ and $W^{2}$ should be linearly independent, otherwise a degenerate transformation is obtained.