eu SOLUTION OF THE TWO-DIMENSIONAL SECOND-ORDER DIFFUSION EQUATION WITH NONLOCAL BOUNDARY CONDITION

Abstract: In the mathematical modeling of many physical phenomena, the diffusion equations with nonlocal boundary condition can be appeared. In this paper, we focus on the two-dimensional inhomogeneous diffusion equations subject to a nonlocal boundary condition. We transform the model of partial differential equation (PDE) into a system of first order, linear, ordinary differential equations (ODEs).


Introduction
The partial differential equations with supplementary conditions are one of the Received: August 29, 2013 c 2014 Academic Publications, Ltd. url: www.acadpubl.eu§ Correspondence author most important branches of the applied sciences and many authors paid much attention to solving this case of equations.
In the last decades, the development of numerical methods for the solution of nonlocal boundary value problems has been an important research area in many branches of science such as chemical diffusion, thermoelasticity, heat conduction processes, population dynamics, inverse problems, control theory and certain biological processes.For example, certain chemicals absorb light at various frequencies can be formulated at the special Heat equation with a nonlocal boundary condition.In [28] an example from meteorology for the evolution of the temperature distribution of air near the ground during calm clear nights has given.The other special partial differential equations with nonlocal boundary condition describe the quasi-static flexure of a thermoelastic rod of unit length [29] and the entropy per unit volume [30].
In 1963, Cannon [8] and Batten [9], discussed on the nonlocal boundary equation independently.thereafter, Kamynin [2] and Ionkin [3] investigated parabolic initial-boundary problems with integral conditions for parabolic equations.Beilin [6] investigated the non-local analogue to classical mixed problems, which involve initial, boundary integral conditions.Gordeziani and Avalishvili [7] discussed hyperbolic equations with non-local boundary conditions.A new matrix formulation technique with arbitrary polynomial bases has been proposed for the numerical/analytical solution of the Heat [1] and Telegraph with nonlocal boundary condition and Two matrix formulation techniques based on the shifted standard and shifted Chebyshev bases are proposed for the numerical solution of the wave equation with the non-local boundary condition [10].
There are some papers that discussed the two-dimensional parabolic partial differential equations with nonlocal boundary conditions and Dirichlet boundary conditions [32,33].In this paper we focus on the following diffusion equation in two space variables while the initial condition is and the dirichlet time-dependent boundary conditions are assumed to be of the form u(0, y, t) = q(y, t), and with the nonlocal boundary condition The functions f (x, t), p(x, y), q(x, t), h(x, t), r(y, t), g(x, t) and m(t) are known functions and the constants α, β, a and b are known constants.M. Siddique presented Pade schemes [31] and a third order L 0 −stable numerical scheme [27] for the numerical solution of problem ( 1)- (7).Authors of [24,25,26], proposed some numerical solution to the (1)-( 7).The method is based on finding a solution in the form of a polynomial in three There are some similar methods, such as differential transformation method [4,5,11,12,22,23,13,20,21].Other similar schemes can be seen in [14,15,16,17,18,19].

Three-Dimensional Differential Transform
Consider a function of two variables w(x, y, t), and suppose that it can be represented as a product of two single-variable functions, i.e., w(x, y, t) = ϕ(x)φ(y)ψ(t).Then the function w(x, y, t) can be represented as where W (i, j, k) is called the spectrum of w(x, y, t).Now we introduce the basic definitions and operations of three-dimensional DT as follows [22].
Definition 2.1.Given a w function which has three components such as x, y, t.Three-dimensional differential transform of w(x, y, t) is defined where the spectrum function W (i, j, k) is the transformed function, which is also called the T-function.let w(x, y, t) as the original function while the uppercase W (i, j, k) stands for the transformed function.Now we define The differential inverse transform of W (i, j, k) as follows: Using Eq. ( 9) in ( 10), we have Now we give the fundamental theorem for the three-dimensional case of DTM by using the following theorem Theorem 2.1.Assume that W (i, j, k), U (i, j, k) and V (i, j, k) are the differential transforms of the functions w(x, y, t), u(x, y, t) and v(x, y, t), respectively; then: 1− If w(x, y, t) = u(x, y, t)±v(x, y, t), then W (i, j, k) = U (i, j, k)±V (i, j, k), 2− If w(x, y, t) = cu(x, y, t), where c ∈ R, then W (i, j, k) = cU (i, j, k) Proof.See [22].

Reformulation of the Problem
In this section, we convert the problem (1)-(7) into a system of first order, linear, ordinary differential equation.
In Eqs. ( 1)-( 7), the functions f (x, y, t), p(x, y), g(x, y), h(x, t), r(y, t) and m(t) generally are not polynomials.We assume that these functions are polynomial or they can be approximated by polynomials to any degree of accuracy.
Then if we suppose that Therefore we consider approximate solution of the form The U n (x, y, t) is the approximation of u(x, y, t).It means that lim n→∞ U n (x, y, t) = u(x, y, t) If we find the values of U (i, j, k), for i, j, k = 0, 1, 2, ..., n, then U n (x, y, t) can be found by using Eq. ( 13).To find these unknowns, we proceed as follows.Firstly, by utilizing Theorem 2.1 and Eqs. ( 12) ,(13) into we get consider the initial condition u(x, y, 0) = p(x, y), By using Eq. ( 12)-( 13) to the above equation we have P (i, j)x i y j , i, j = 0, 1, ..., n.

Conclusions
In this article, the solution of the second order two space dimensional diffusion equation has been discussed by using differential transformation method.Converting the model of partial differential equation to a system of linear equations, is the main part of this paper.The computational difficulties of the other methods can be reduced by applying this process.