Applied Mathematical Sciences, Vol. 7, 2013, no. 85, 4213 - 4221 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ams.2013.33160
The Use of He’s Variational Iteration Method for Solving the One-Dimensional Parabolic Equation with Non-Classical Boundary Conditions Yucheng Liu and Manoj Chand Department of Mechanical Engineering University of Louisiana at Lafayette Lafayette, LA 70503, USA
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[email protected] Copyright © 2013 Yucheng Liu and Manoj Chand . This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract Over the last 15 years, the He’s variational iteration method (HVIM) has been applied to obtain formal solutions to a wide class of differential equations. This method leads to computable, efficient, solutions to linear and nonlinear operator equations. The parabolic partial differential equations with non-classical boundary conditions model various physical problems. The aim of this paper is to investigate the application of HVIM for solving the second-order linear parabolic partial differential equation with non-classical boundary conditions. The HVIM provides a reliable technique that requires less work when compared with the traditional techniques such as the Adomian decomposition method (ADM). The present approach can be used and extended for investigating more scientific applications. Keywords: He’s variational iteration method, one-dimensional parabolic equation, non-classical boundary conditions, numerical algorithm, closed form solutions
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Yucheng Liu and Manoj Chand
1. Introduction One-dimensional parabolic equation is a type of second-order partial differential equation (PDE) which describes a wide family of problems in science and engineering including heat diffusion, ocean acoustic propagation, etc. Such equation with non-classical boundary conditions has many important applications in chemical diffusion, thermo elasticity, heat conduction processes, population dynamics, vibration problems, nuclear reactor dynamics, inverse problems, control theory, medical science, biochemistry, and certain biological processes [2-5, 9, 14, and 15]. In this study, such problem takes the form ∂u ∂ 2 u = , 0 < x < 1, 0 < t ≤ T (1) ∂t ∂x 2 with initial conditions u (x, 0) = f(x), 0 ≤ x ≤ 1 (2) and non-classical boundary conditions t
u (0, t ) = ∫ φ ( x, t ) u ( x, t ) dx,
0
