Abstract
This paper describes a number of simple but quite effective methods for constructing exact solutions of nonlinear partial differential equations that involve a relatively small amount of intermediate calculations. The methods employ two main ideas: (i) simple exact solutions can serve to construct more complex solutions of the equations under consideration and (ii) exact solutions of some equations can serve to construct solutions of other, more complex equations. In particular, we propose a method for constructing complex solutions from simple solutions using translation and scaling. We show that in some cases, rather complex solutions can be obtained by adding one or more terms to simpler solutions. There are situations where nonlinear superposition allows us to construct a complex composite solution using similar simple solutions. We also propose a few methods for constructing complex exact solutions to linear and nonlinear PDEs by introducing complex-valued parameters into simpler solutions. The effectiveness of the methods is illustrated by a large number of specific examples (over 30 in total). These include nonlinear heat equations, reaction–diffusion equations, wave type equations, Klein–Gordon type equations, equations of motion through porous media, hydrodynamic boundary layer equations, equations of motion of a liquid film, equations of gas dynamics, Navier–Stokes equations, and some other PDEs. Apart from exact solutions to ‘ordinary’ partial differential equations, we also describe some exact solutions to more complex nonlinear delay PDEs. Along with the unknown function at the current time, u=u(x,t), these equations contain the same function at a past time, w=u(x,t−τ), where τ>0 is the delay time. Furthermore, we look at nonlinear partial functional-differential equations of the pantograph type, which, in addition to the unknown u=u(x,t), also contain the same functions with dilated or contracted arguments, w=u(px,qt), where p and q are scaling parameters. We propose an efficient approach to construct exact solutions to such functional-differential equations. Some new exact solutions of nonlinear pantograph-type PDEs are presented. The methods and examples in this paper are presented according to the principle “from simple to complex”.
Highlights
Exact solutions of nonlinear partial differential equations and methods for their construction are necessary for the development, analysis, and verification of various mathematical models used in natural and engineering sciences, as well as for testing approximate analytical and numerical methods
This paper describes a number of simple, but quite effective, methods for constructing exact solutions of nonlinear partial differential equations, which do not require much special training and lead to a relatively small amount of intermediate calculations
The term ‘exact solution’ for nonlinear partial differential equations will be used in cases where the solution is expressed: (i) in terms of elementary functions, functions included in the equation, and indefinite or/and definite integrals; through solutions of ordinary differential equations or systems of such equations
Summary
Exact solutions of nonlinear partial differential equations and methods for their construction are necessary for the development, analysis, and verification of various mathematical models used in natural and engineering sciences, as well as for testing approximate analytical and numerical methods. Clarkson–Kruskal method [11,12,13,14], methods for generalized separation of variables [13,14,15], methods for functional separation of variables [14,16,17,18], the method of differential constraints [13,14,19], the method of truncated Painlevé expansions [13,20,21], and use of conservation laws to obtain exact solutions [22,23,24] The application of these methods requires considerable special training and, as a rule, is accompanied by time-consuming analysis and a large volume of analytical transformations and intermediate calculations. This paper describes a number of simple, but quite effective, methods for constructing exact solutions of nonlinear partial differential equations, which do not require much special training and lead to a relatively small amount of intermediate calculations. For the convenience of a wide audience with different mathematical backgrounds, the authors tried to do their best, wherever possible, to avoid special terminology
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