© AUG 2019 | IRE Journals | Volume 3 Issue 2 | ISSN: 2456-8880
Finite Difference Methods for Two Dimensional Heat Equation KHAING KHAING SOE WAI1, SAN SAN TINT2 1, 2 Department of Engineering Mathematics, Technological University (Myitkyina), Myitkyina, Myanmar Abstract- In this paper, we first consider the initial boundary value problem for the heat equation. And, the finite difference methods for the heat equation in one space dimension, its consistency and stability are studied. Then, the two dimensional heat equation is solved by using finite difference methods.
u (t, x) = 0,
(t, x) (0, T) , T
Where any inhomogeneous boundary conditions have been eliminated. If we look for classical solutions of the IBVP, whose derivatives are continuous on QT , the following compatibility condition must also be satisfied
Indexed Terms- Consistency, stability, difference methods and heat Equation. I.
finite
INTRODUCTION
We consider the heat equation
u t div (a(x) u) 0, t R , x , Where u denotes the temperature and a defines the heat conductivity. The generic domain for the heat equation is the space-time cylinder QT (0,T) . In order to obtain a well-posed problem, we must prescribe spatial boundary condition
u(t, x) g(t, x) x , t (0, T) and an initial condition
u(0, x) u 0 (x), x .
II.
FINITE DIFFERENCE METHODS IN ONE SPACE DIMENSION
First, we consider the initial boundary value problem in one space dimension u t – u xx f , (t, x) (0, T) × (0, 1) u(0, x) u 0 (x) , x (0, 1) (2) u(t, 0) u1 (t) , t (0, T) u(t, 1) u 2 (t), t (0, T). We discretise the equation equidistantly with respect to space and time xi = ih, i = 0, …, N tk = k, k = 0, …, M. k
We use u i to denote the approximation for u(t k , xi)
The equation combined with the initial datum forms an initial boundary value problem. Note that, the operator Lu = – div (a(x) u) is elliptic, and that in general, the equation with an elliptic differential operator L u t Lu 0 Defines a parabolic equation. We consider the initial boundary value problem (IBVP) (1) u t Lu = f (t, x), (t, x)QT u (0, x) = u0(x), x
IRE 1701491
u 0 (x) 0 x .
and write
fik f (t k , xi ).
We require approximations to the first order time derivative and the second order spatial derivative. We use the second difference operator for the spatial variables 1 D D u ik 2 (u ik1 2u ik u ik1 ). h If we choose a 2-point discretisation for the time derivative, we obtain a 6-point scheme, which in general has the form
u ik 1 u ik D D (u ik 1 (1 )u ik ) fik With the additional conditions
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