2020/01/06 by Andrei D. Polyanin, Polyanin, Andrei D.
Mathematics · Physics and Astronomy · #Analysis of PDEs (math.AP) #Exactly Solvable and Integrable Systems (nlin.SI) #FOS: Mathematics #FOS: Physical sciences #Fractional Differential Equations Solutions #Mathematical Physics (math-ph) #Nonlinear Waves and Solitons #Numerical methods for differential equations
paper · pdf · doi:10.48550/arxiv.2001.01645
openalex publication_date 2020/01/06 · openalex created_date 2022/07/26 · openalex updated_date 2026/07/28
The study gives a brief overview of existing modifications of the method of\nfunctional separation of variables for nonlinear PDEs. It proposes a more\ngeneral approach to the construction of exact solutions to nonlinear equations\nof applied mathematics and mathematical physics, based on a special\ntransformation with an integral term and the generalized splitting principle.\nThe effectiveness of this approach is illustrated by nonlinear diffusion-type\nequations that contain reaction and convective terms with variable\ncoefficients. The focus is on equations of a fairly general form that depend on\none, two or three arbitrary functions (such nonlinear PDEs are most difficult\nto analyze and find exact solutions). A number of new functional separable\nsolutions and generalized traveling wave solutions are described (more than 30\nexact solutions have been presented in total). It is shown that the method of\nfunctional separation of variables can, in certain cases, be more effective\nthan (i) the nonclassical method of symmetry reductions based on an invariant\nsurface condition, and (ii) the method of differential constraints based on a\nsingle differential constraint. The exact solutions obtained can be used to\ntest various numerical and approximate analytical methods of mathematical\nphysics and mechanics.\n