Functional separable solutions of nonlinear reaction-diffusion equations with variable coefficients. (English) Zbl 1428.35171
Summary: The paper presents a number of new functional separable solutions to nonlinear reaction-diffusion equations of the form \[c(x) u_t = [a(x) u_x]_x + b(x) u_x + p(x) f(u),\] where \(f(u)\) is an arbitrary function. It is shown that any three of the four variable coefficients \(a(x), b(x), c(x), p(x)\) of such equations can be chosen arbitrarily, and the remaining coefficient can be expressed through the others. Examples of specific equations and their exact solutions are given. The results obtained are generalized to more complex multidimensional nonlinear reaction-diffusion equations with variable coefficients. Also some functional separable solutions to nonlinear reaction-diffusion equations with delay \[u_t = u_{x x} + a(x) f(u, w),\qquad w = u(x, t - \tau),\] where \(\tau >0\) is the delay time and \(f(u, w)\) is an arbitrary function of two arguments, are obtained. It is important to note that the exact solutions of nonlinear PDEs and delay PDEs that contain arbitrary functions and therefore have sufficient generality are of the greatest practical interest for testing and evaluating the accuracy of various numerical and approximate analytical methods for solving corresponding initial-boundary value problems.
MSC:
| 35K57 | Reaction-diffusion equations |
| 35C05 | Solutions to PDEs in closed form |
| 35A30 | Geometric theory, characteristics, transformations in context of PDEs |
Keywords:
nonlinear reaction-diffusion equations; reaction-diffusion equations with delay; equations with variable coefficients; exact solutions; functional separable solutionsReferences:
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