scholarly journals Comparison of the Effectiveness of Different Methods for Constructing Exact Solutions to Nonlinear PDEs. Generalizations and New Solutions

Mathematics ◽  
2019 ◽  
Vol 7 (5) ◽  
pp. 386 ◽  
Author(s):  
Andrei D. Polyanin
Keyword(s):  
Exact Solutions ◽  
Direct Method ◽  
Separation Of Variables ◽  
Surface Condition ◽  
Reaction Diffusion ◽  
Variable Coefficients ◽  
Nonlinear Pdes ◽  
Boundary Layer Equations ◽  
Compatibility Analysis ◽  
Klein Gordon

The paper shows that, in looking for exact solutions to nonlinear PDEs, the direct method of functional separation of variables can, in certain cases, be more effective than the method of differential constraints based on the compatibility analysis of PDEs with a single constraint (or the nonclassical method of symmetry reductions based on an invariant surface condition). This fact is illustrated by examples of nonlinear reaction-diffusion and convection-diffusion equations with variable coefficients, and nonlinear Klein–Gordon-type equations. Hydrodynamic boundary layer equations, nonlinear Schrödinger type equations, and a few third-order PDEs are also investigated. Several new exact functional separable solutions are given. A possibility of increasing the efficiency of the Clarkson–Kruskal direct method is discussed. A generalization of the direct method of the functional separation of variables is also described. Note that all nonlinear PDEs considered in the paper include one or several arbitrary functions.

scholarly journals Functional Separation of Variables in Nonlinear PDEs: General Approach, New Solutions of Diffusion-Type Equations

Mathematics ◽  
2020 ◽  
Vol 8 (1) ◽  
pp. 90 ◽  
Author(s):  
Andrei D. Polyanin
Keyword(s):  
Exact Solutions ◽  
Applied Mathematics ◽  
Separation Of Variables ◽  
Surface Condition ◽  
Traveling Wave Solutions ◽  
Mathematical Physics ◽  
Variable Coefficients ◽  
Nonlinear Pdes ◽  
Diffusion Type ◽  
Differential Constraint

The study gives a brief overview of existing modifications of the method of functional separation of variables for nonlinear PDEs. It proposes a more general approach to the construction of exact solutions to nonlinear equations of applied mathematics and mathematical physics, based on a special transformation with an integral term and the generalized splitting principle. The effectiveness of this approach is illustrated by nonlinear diffusion-type equations that contain reaction and convective terms with variable coefficients. The focus is on equations of a fairly general form that depend on one, two or three arbitrary functions (such nonlinear PDEs are most difficult to analyze and find exact solutions). A lot of new functional separable solutions and generalized traveling wave solutions are described (more than 30 exact solutions have been presented in total). It is shown that the method of functional separation of variables can, in certain cases, be more effective than (i) the nonclassical method of symmetry reductions based on an invariant surface condition, and (ii) the method of differential constraints based on a single differential constraint. The exact solutions obtained can be used to test various numerical and approximate analytical methods of mathematical physics and mechanics.

scholarly journals Functional separable solutions of two classes of nonlinear mathematical physics equations

2019 ◽  
Vol 486 (3) ◽  
pp. 287-291
Author(s):  
A. D. Polyanin ◽  
A. I. Zhurov
Keyword(s):  
Separation Of Variables ◽  
Functional Differential Equations ◽  
Traveling Wave Solutions ◽  
Reaction Diffusion ◽  
Mathematical Physics ◽  
Variable Coefficients ◽  
Reaction Diffusion Equations ◽  
Equations Of Mathematical Physics ◽  
Klein Gordon ◽  
Functional Differential

The study describes a new modification of the method of functional separation of variables for nonlinear equations of mathematical physics. Solutions are sought in an implicit form that involves several free functions; the specific expressions of these functions are determined in the subsequent analysis of the arising functional differential equations. The effectiveness of the method is illustrated by examples of nonlinear reaction-diffusion equations and Klein-Gordon type equations with variable coefficients that depend on one or more arbitrary functions. A number of new exact functional separable solutions and generalized traveling-wave solutions are obtained.

scholarly journals Lie symmetry and exact solution of (2+1)-dimensional generalized Kadomtsev-Petviashvili equation with variable coefficients

2013 ◽  
Vol 17 (5) ◽  
pp. 1490-1493
Author(s):  
Hong-Cai Ma ◽  
Zhen-Yun Qin ◽  
Ai-Ping Deng
Keyword(s):  
Exact Solution ◽  
Exact Solutions ◽  
Variable Coefficient ◽  
Direct Method ◽  
Symmetry Transformation ◽  
Solution Procedure ◽  
Variable Coefficients ◽  
Lie Symmetry ◽  
Petviashvili Equation ◽  
Non Linear Systems

The simple direct method is adopted to find Non-Auto-Backlund transformation for variable coefficient non-linear systems. The (2+1)-dimensional generalized Kadomtsev-Petviashvili equation with variable coefficients is used as an example to elucidate the solution procedure, and its symmetry transformation and exact solutions are obtained.

New exact solutions of the reaction–diffusion equation with variable coefficients via the mathematical computation

2018 ◽  
Vol 11 (04) ◽  
pp. 1850051 ◽  
Author(s):  
Jin Hyuk Choi ◽  
Hyunsoo Kim
Keyword(s):  
Diffusion Equation ◽  
Exact Solutions ◽  
Reaction Diffusion ◽  
Variable Coefficients ◽  
Time Dependent ◽  
Reaction Diffusion Equation ◽  
Painlevé Test ◽  
New Exact Solutions ◽  
Mathematical Computation

In this paper, we construct new exact solutions of the reaction–diffusion equation with time dependent variable coefficients by employing the mathematical computation via the Painlevé test. We describe the behaviors and their interactions of the obtained solutions under certain constraints and various variable coefficients.

A scaling transformation method and exact solutions of nonlinear reaction–diffusion model

2020 ◽  
Vol 34 (31) ◽  
pp. 2050356
Author(s):  
Xin Wang ◽  
Yang Liu
Keyword(s):  
Exact Solutions ◽  
Reaction Diffusion ◽  
Integral Form ◽  
Transformation Method ◽  
Variable Coefficients ◽  
Reaction Diffusion Equation ◽  
Polynomial Method ◽  
Nonlinear Odes ◽  
Reaction Diffusion Model ◽  
Scaling Transformation

We find a very simple scaling transformation method by which a kind of rank non-homogenous second order nonlinear ODEs is reduced to an integral form. Comparing with the canonical-like transformation method and the first integral method, our method provides a simple mechanism of reduction of such equations. As an application, a reaction–diffusion equation with variable coefficients is integrated, and its exact cnoidal wave solution is obtained. Furthermore, by using the complete discrimination system for polynomial method, more exact solutions are obtained.

scholarly journals Conditional Lie-Bäcklund Symmetry Reductions and Exact Solutions of a Class of Reaction-Diffusion Equations

2018 ◽  
Vol 2018 ◽  
pp. 1-6
Author(s):  
Xinyang Wang ◽  
Junquan Song
Keyword(s):  
Dynamical Systems ◽  
Exact Solutions ◽  
Separation Of Variables ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Financial Mathematics ◽  
Symmetry Reductions ◽  
Finite Dimensional ◽  
Wide Range

The method of conditional Lie-Bäcklund symmetry is applied to solve a class of reaction-diffusion equations ut+uxx+Qxux2+Pxu+Rx=0, which have wide range of applications in physics, engineering, chemistry, biology, and financial mathematics theory. The resulting equations are either solved exactly or reduced to some finite-dimensional dynamical systems. The exact solutions obtained in concrete examples possess the extended forms of the separation of variables.

scholarly journals Exact solutions of the 2D Dunkl–Klein–Gordon equation: The Coulomb potential and the Klein–Gordon oscillator

2021 ◽  
pp. 2150171
Author(s):  
R. D. Mota ◽  
D. Ojeda-Guillén ◽  
M. Salazar-Ramírez ◽  
V. D. Granados
Keyword(s):  
Angular Momentum ◽  
Energy Spectrum ◽  
Exact Solutions ◽  
Coulomb Potential ◽  
Gordon Equation ◽  
Separation Of Variables ◽  
Point Of View ◽  
Algebraic Point ◽  
Klein Gordon ◽  
Klein Gordon Equation

We introduce the Dunkl–Klein–Gordon (DKG) equation in 2D by changing the standard partial derivatives by the Dunkl derivatives in the standard Klein–Gordon (KG) equation. We show that the generalization with Dunkl derivative of the z-component of the angular momentum is what allows the separation of variables of the DKG equation. Then, we compute the energy spectrum and eigenfunctions of the DKG equations for the 2D Coulomb potential and the Klein–Gordon oscillator analytically and from an su(1, 1) algebraic point of view. Finally, we show that if the parameters of the Dunkl derivative vanish, the obtained results suitably reduce to those reported in the literature for these 2D problems.

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