Abstract
Many physical and engineering problems can be modeled using partial differential equations such as heat transfer through conduction process in steady and unsteady state. Perturbation methods are analytical approximation method to understand physical phenomena which depends on perturbation quantity. Homotopy perturbation method (HPM) was proposed by Ji Huan He. HPM is considered as effective method in solving partial differential equations. The solution obtained by HPM converges to exact solution, which are in the form of an infinite function series. Biazar and Eslami proposed new homotopy perturbation method (NHPM) in which construction of an appropriate homotopy equation and selection of appropriate initial approximation guess are two important steps. In present work, heat flow analysis has been done on a rod of length L and diffusivity α using HPM and NHPM. The solution obtained using different perturbation methods are compared with the solution obtained from most common analytical method separation of variables.
Highlights
Partial differential equations play a dominant role in applied mathematics
The analytical approximate solutions of one-dimensional heat conduction equation are obtained by applying new homotopy perturbation method and new homotopy perturbation method
It is found that new homotopy perturbation method (NHPM) converges very rapidly as compared to homotopy perturbation method (HPM) and other traditional methods
Summary
Partial differential equations play a dominant role in applied mathematics. The classical heat conduction equation is second order linear partial differential equation. The two most important steps in application of new homotopy perturbation method to construct a suitable homotopy equation and choose a suitable initial guess, we aim in this work to effectively employ the (NHPM) to establish exact solution for two-dimensional Laplace equation with Dirichlet and Neumann boundary condition, the difference between (NHPM) and standard (HPM) is starts from the form of initial approximation of the solution. The semi analytic solution of one-dimensional heat conduction equation is obtained by means of homotopy perturbation method and new homotopy perturbation method These methods are effectively applied to obtain the exact solution for the problem in hand which reveals the effectiveness and simplicity of the method. The obtained analytic solution for one dimensional heat conduction equation with boundary and initial conditions using NHPM is same as the universally accepted exact solution This tells us about the capability and reliability of this method. The convergence of solution to the exact solution is very rapid
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