Exact Solutions of Reaction–Diffusion PDEs with Anisotropic Time Delay
Abstract
:Contents |
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1. Introduction
1.1. Differential Equations with Delay: Brief Overview
- (i)
- PDEs with constant delay (or, usually, PDEs with delay) contain a function w of the form , where is the constant delay time;
- (ii)
- PDEs with proportional delay contain a function w of the form , where q is the scaling parameter, ;
- (iii)
- PDEs with variable delay contain a function w of the form , where is the variable delay.
1.2. PDEs with Spatially Anisotropic Time Delay. Reductions. Exact Solutions
2. Approaches to Constructing Exact Solutions
2.1. Constructing Solutions to Complex Equations Using Solutions of Simpler Equations
2.2. PDEs with Anisotropic Time Delay Allowing Solutions of a Special Form
- (i)
- The original simpler PDEs with constant delay (similar equations are called generating equations later) are reduced to ODEs depending only on the spatial argument x;
- (ii)
- Parameters and functional coefficients of original PDEs with constant delay do not explicitly depend on .
3. Exact Solutions of Reaction–Diffusion Type PDEs
3.1. Reaction–Diffusion Equations with Spatially Anisotropic Time Delay
3.2. Reaction–Diffusion Equations with Proportional Anisotropic Time Delay
4. Exact Solutions of Wave-Type PDEs
4.1. Wave-Type Equations with Spatially Anisotropic Time Delay
4.2. Wave-Type Equations with Proportional Anisotropic Time Delay
5. Table of Generating PDEs with Constant or Proportional Delay
6. Brief Conclusions
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
References
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Type of Equation | Form of Equation | References/Comments |
---|---|---|
First-order ODE with delay | , | [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20] |
Second-order ODE with delay | , | [21] |
First-order ODE with proportional argument | , | [22] [23,24] |
First-order ODE with variable delay | , | [25] |
First-order PDE with delay | , | [26] |
First-order Hopf-type PDE with delay | , | [27] |
First-order PDE with proportional argument | , | : [28] |
First-order PDE with delay and proportional argument | , | : [29] |
First-order PDE with constant time delay and variable space delay | , | [30] |
Reaction–diffusion PDE with delay | , | Exact solutions construction: [31,32,33,34] Traveling waves analysis: [35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53] Other topics: [17,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77] |
Reaction–diffusion PDE with delay and variable transfer coefficient | , | [78,79,80,81] |
Reaction–diffusion PDE with proportional argument | , | [82,83] |
Reaction–diffusion PDE with proportional argument | , | [83] |
Reaction–diffusion PDE with proportional arguments | , | [83,84] |
Reaction–diffusion PDE with variable delay | , | [31,33,84,85,86,87,88,89] |
Fokker–Planck type PDE with proportional argument | , | : [90] |
Wave-type PDE with delay | , | [91,92,93] |
Wave-type PDE with variable delay | , | [94] |
Telegraph type PDE with delay | , | [95] |
Telegraph type PDE with proportional argument | , | : [96] |
Wave-type PDE with proportional argument | , | [97] |
Wave-type PDE with nonlinear speed and proportional arguments | , | [97] |
No. | Form of Equation | References/Comments |
---|---|---|
1 | Equation (5) with | |
2 | [31,33] | |
3 | [31] | |
4 | [31,33] | |
5 | [33] | |
6 | [31] | |
7 | [79] | |
8 | [79] | |
9 | [79] | |
10 | [79,80,81] | |
11 | [79,81] | |
12 | Equation (20) with | |
13 | [83] | |
14 | [83] | |
15 | Equation (25) with | |
16 | [91] | |
17 | Equation (28) with | |
18 | [81] | |
19 | [114] | |
20 | [80,81,114] | |
21 | [81] | |
22 | Equation (32) with | |
23 | [97] | |
24 | [97] |
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Polyanin, A.D.; Sorokin, V.G. Exact Solutions of Reaction–Diffusion PDEs with Anisotropic Time Delay. Mathematics 2023, 11, 3111. https://doi.org/10.3390/math11143111
Polyanin AD, Sorokin VG. Exact Solutions of Reaction–Diffusion PDEs with Anisotropic Time Delay. Mathematics. 2023; 11(14):3111. https://doi.org/10.3390/math11143111
Chicago/Turabian StylePolyanin, Andrei D., and Vsevolod G. Sorokin. 2023. "Exact Solutions of Reaction–Diffusion PDEs with Anisotropic Time Delay" Mathematics 11, no. 14: 3111. https://doi.org/10.3390/math11143111