Abstract
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.
Highlights
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 methods of generalized and functional separation of variables are based on setting a priori a structural form of u that depends on several free functions
To construct exact solutions of nonlinear partial differential equations, this paper proposes to use a direct method based on a special transformation with an integral term as well as the generalized splitting principle
Summary
The studies [33,34,35] described a new direct method for constructing exact solutions with functional separation of variables It is based on an implicit integral representation of solutions in the form ζ(u) du = ξ1(x)η(t) + ξ2(x),. To construct exact solutions of nonlinear partial differential equations, this paper proposes to use a direct method based on a special transformation with an integral term as well as the generalized splitting principle. This approach is technically simpler and more convenient than finding a solution in the form (5); it generalizes the dependence (4) and allows one to find various solutions in a uniform manner without specifying their structure a priori
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