Reduction Method and New Exact Solutions of the Multidimensional Nonlinear Heat Equation

Kosov, A. A.; Семенов, Е. И.

doi:10.1134/s0012266122020057

Cited by 5 publications

(1 citation statement)

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“…Exact solutions of nonlinear PDEs are most often constructed using the classical method of symmetry reductions [3][4][5][6], the direct method of symmetry reductions [1,[7][8][9], the nonclassical symmetries methods [10][11][12][13], methods of generalized separation of variables [1,9,[14][15][16], methods of functional separation of variables [1,9,17,18], the method of differential constraints [1,9,19,20], and some other exact analytical methods (see, for example, [21][22][23][24][25][26][27]). On methods for constructing exact solutions of nonlinear delay PDEs and functional PDEs, see, for example, [2,[28][29][30][31][32][33][34].…”

Section: Introduction Exact Solutionsmentioning

confidence: 99%

Separation of Variables and Exact Solutions to Nonlinear PDEs

Polyanin¹,

Zhurov²

2021

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Using the principle of structural analogy of solutions, approaches have been developed for constructing exact solutions of complex nonlinear PDEs, including PDEs with delay, based on the use of special solutions to auxiliary simpler related equations. It is shown that to obtain exact solutions of nonlinear nonautonomous PDEs, the coefficients of which depend on time, it is possible to use generalized and functional separable solutions of simpler autonomous PDEs, the coefficients of which do not depend on time. Specific examples of constructing exact solutions to nonlinear PDEs, the coefficients of which depend arbitrarily on time, are considered. It has been discovered that generalized and functional separable solutions of nonlinear PDEs with constant delay can be used to construct exact solutions of more complex nonlinear PDEs with variable delay of general form. A number of nonlinear reaction-diffusion type PDEs with variable delay are described, which allow exact solutions with generalized separation of variables.

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