Abstract
Nonlinear phenomena play a crucial role in applied mathematics and physics. Explicit solutions to the nonlinear equations are of fundamental importance. Various methods for obtaining explicit solution to nonlinear evolution equations have been proposed. In this letter homotopy perturbation method (HPM) is employed for solving one-dimensional non-homogeneous parabolic partial differential equation with a variable coefficient and a system of nonlinear partial differential equations. The final results obtained by means of HPM, were compared with those results obtained from the exact solution and the Adomian Decomposition Method (ADM). The comparison shows a precise agreement between the results, and introduces this new method as an applicable one which it needs less computations and is much easier and more convenient than others, so it can be widely used in engineering too.
Highlights
Nonlinear phenomena play a crucial role in applied mathematics and physics
Explicit solutions to the nonlinear equations are of fundamental importance
In equation (1) the final results obtained from homotopy perturbation method (HPM) were compared with the results of the exact solution, and in equation. (3) they compared with the results of the Adomian Decomposition Method[9] which it is another approximate method for solving nonlinear partial differential equations
Summary
Nonlinear phenomena play a crucial role in applied mathematics and physics. Explicit solutions to the nonlinear equations are of fundamental importance. The nonlinear equations are solved and elegantly with out transforming or linearizing the equation by using the homotopy perturbation method (HPM). It provides an efficient explicit solution with high accuracy, minimal calculations, and avoidance of physically unrealistic assumptions. We implemented the HPM for finding the approximate solutions of one-dimensional nonhomogeneous parabolic partial differential equation with a variable coefficient and a system of nonlinear partial differential equations[8, 9]. These two equations are as fallows: ut = uxx + φ(x,t) = uxx + exp(−x)(cos t − sin t),. Solutions of Eqs. (8) and (9) can be written as a power (4) series in p:
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