Abstract
In this paper, Adomian Decomposition Method with Adomian polynomials are applied to solve Allen - Cahn equation with the initial condition only, also DuFort-Frankel method is applied with the initial and boundary conditions. The numerical results that are obtained by the Adomian decomposition method have been compared with the exact solution of the equation shown that it is more efficient than the DuFort-Frankel method, that is illustrated through the tables and Figures.
Highlights
Allen-Cahn equation is origin in the gradient theory of phase transitions [8], this equation is a model in which two distinct phases try to coexist in a domain while minimizing their interaction which is proportional to the (N-1)-dimension volume of the inter face
The numerical solution of the generalized Allen-Cahn equation is used by Michal et al in 2004,[23] that proposed an algorithm for image segmentation, this technique is devoted to recovery of pattern boundaries from the original, possibly noisy image or signal
Allen-Cahn equation, is a special type of non-linear partial differential equation, which arises as diffusion – convection equation in computational fluid dynamics or reaction – diffusion problem in material science
Summary
Allen-Cahn equation, is a special type of non-linear partial differential equation, which arises as diffusion – convection equation in computational fluid dynamics or reaction – diffusion problem in material science. These equations are originally used to solve the phase transition problems, transformation of thermodynamic system from one phase to another due to an abrupt change in one or more physical properties. Adomian decomposition method assumes that the unknown function u(x,t) can be expressed by an infinite series of the form: And the non-linear operator F(u) can be decomposed by an infinite series of polynomial given by.
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