Abstract
An Alternating Directions Implicit method is presented to solve the homogeneous heat diffusion equation when the governing equation is a bi-harmonic equation (X) based on Alternative Direction Implicit (ADI). Numerical results are compared with other results obtained by other numerical (explicit and implicit) methods. We apply these methods it two examples (X): the first one, we apply explicit when the temperature .
Highlights
A Bi –Harmonic equation is a kind of partial differential equations, the general form of the bi- harmonic equation is:
If ∇ u x, y 0, it is called homogenous bi- harmonic function. This equation appears in many boundary value problems:, when the governing equation for the boundary value problems is a bi harmonic equation
The Bi-harmonic equation is the governing for many (X) problems, for example: it is the governing for the heat equation
Summary
A Bi –Harmonic equation is a kind of partial differential equations, the general form of the bi- harmonic equation is:∇ u x, y F x, y (1)Where ∇ is a bi –harmonic operator in the form: ∇ (2)If ∇ u x, y 0 , it is called homogenous bi- harmonic function.This equation appears in many boundary value problems: (fluid mechanics, elasticity problems, heat diffusion, etc...), when the governing equation for the boundary value problems is a bi harmonic equation. There are many (X) numerical methods to solve this problem. One of these methods is Alternating Direct Implicit Method (ADI). This method is to minimize the two dimensions for a partial differential equation to a linear equation with one dimension. (X) presented the (a) comparative study of (a) 2D asymmetric diffusion problem With convection on the wall using the theta method. Utilized ADI to solve the 2D time dependent heat equations depending on a constant coefficient. (X) used ADI methods for solving elliptic problems. Hyperbolic diffusion equation with convection used an alternating direction implicit method for a second-order. The main purpose of this work is to study the analytic and numerical solution for solving homogeneous heat diffusion equation using ADI method
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