Analytical Solutions for Nonlinear Convection–Diffusion Equations with Nonlinear Sources

Аннотация

Nonlinear equations of the convection–diffusion type with nonlinear sources are used for the description of many processes and phenomena in physics, mechanics, and biology. In this work, we consider the family of nonlinear differential equations which are the traveling wave reductions of the nonlinear convection–diffusion equation with polynomial sources. The question of the construction of the general analytical solutions to these equations is studied. The steady-state and nonstationary cases with and without account of convection are considered. The approach based on the application of nonlocal transformations generalizing the Sundman transformation is applied for construction of the analytical solutions. It is demonstrated that in the steady-state case without account of convection, the general analytical solution can be found without any constraints on the equation parameters and it is expressed in terms of the Weierstrass elliptic function. In the general case, this solution has a cumbersome form; the constraints on the parameters for which it has a simple form are found, and the corresponding analytical solutions are obtained. It is shown that in the nonstationary case, both with and without account of convection, the general solution to the studied equations can be constructed under certain constraints on the parameters. The integrability criteria for the Liénard-type equations are used for this purpose. The corresponding general analytical solutions to the studied equations are explicitly constructed in terms of exponential or elliptic functions.