A variational-perturbation method for solving the time-dependent singularly perturbed reaction-diffusion problems

Abstract

In this paper, we combine the variational iteration method and perturbation theory to solve a time-dependent singularly perturbed reaction-diffusion problem. The problem is considered in the boundary layers and outer region. In the boundary layers, the problem is transformed by the variable substitution, and then the variational iteration method is employed to solve the transformed equation. In the outer region, we use the perturbation theory to obtain the approximation equation and the approximation solution. The final numerical experiments show that this method is accurate.

A new modification of the homotopy perturbation method

“…46 Homotopy perturbation method combines least square technique 46 and variational theory. 47 Laplace transform has also adopted the homotopy perturbation method. 48,49 These coupled methods are simple in theory and application.…”

Homotopy perturbation method with an auxiliary parameter for nonlinear oscillators

“…The main development of the homotopy perturbation method was to adopt the parameter-expansion technology [29][30][31][32][33] to effectively deal with the zero-order approximate and He's polynomials to deal with the nonlinear terms. [34][35][36][37][38][39][40] Now there is a trend to combination of the homotopy perturbation method with other methods, 18 for examples combination of the method with variational theory 41 and least squares technology. 42 The prevailing trend is to adopt the Laplace transform in the homotopy perturbation method.…”

“…Now there is a trend to combination of the homotopy perturbation method with other methods, 18 for examples combination of the method with variational theory 41 and least squares technology. 42 The prevailing trend is to adopt the Laplace transform in the homotopy perturbation method.…”