GREEN’S FUNCTION OF A CENTERED PARTIAL DIFFERENCE EQUATION

Applying a variation of Jacobi iteration we obtain the Green’s function for the centered partial difference equation �wwu(xw−1, yz) + �zzu(xw, yz−1) + f(u(xw, yz)) = 0, which is the result of applying the finite difference method to an associated nonlinear partial differential equation of the form uxx + uyy + h(u) = 0. We show that approximations of the partial differential equation can be found by applying fixed point theory instead of the standard techniques associated with solving a system of nonlinear equations.


Introduction
The finite difference method is a classical technique used to approximate the solution of a partial differential equation.An application of the technique yields a nonlinear system of equations which can be considered as a nonlinear partial difference equation.Instead of attempting to solve or approximate the solution of the nonlinear system, we will convert the partial difference equation to a fixed point problem whose inversion involves the Green's function of the partial difference equation.The Green's function can be interpreted combinatorially through an adjacency matrix.Finding the Green's function for a partial difference equation has been done; for example, see [1,2,3,4,5,6,7,8,12,13], however none of the techniques are simple nor are the formulas easy to work with when finding solutions which correspond to approximations to partial differential equations.Also, it is difficult to find bounds on the Green's function to establish the existence of solutions for the classical partial difference equation when these techniques are used.In this paper we apply an elementary iterative technique based on Jacobi iteration to obtain the Green's functions.A technique that is easy to computationally apply to approximate the solution of a partial differential equation using partial difference equations techniques.Since the Green's function involves the sum of nonnegative terms (with a combinatorial interpretation) it is easy to establish elementary bounds which can be used in existence of solutions arguments.A basic understanding of difference equations, Green's functions, and discretization of partial differential equations is assumed; see Kelley and Peterson [10] for background results.
The partial difference equation , with boundary conditions u(x, y) = 0 for all x = x 0 , x = x N , y = y 0 , or y = y N , is the result of applying the finite element method to the nonlinear partial differential equation with boundary conditions u(0, y) = 0, 0 ≤ y ≤ 1, and u(x, 0) = 0, 0 ≤ x ≤ 1, where Thus, this is a problem that is of interest to a large audience extending beyond mathematicians.Note, the arguments throughout can be modified to suit any bounded domain.The Green's function is the function whose g(w, z, r, s)f (u(x r , y s )).
3 Finding the Green's Function , which we can write as Thus, for each fixed (r, s) ∈ [1, N − 1] 2 , the Green's function is a solution of a linear system of equations which we can write as .
The column matrix G r,s consists of the Green's function terms, that is, for each (r, s) ∈ [1, N −1] 2 , the element in the ith row of G r,s is g(w, z, r, s), where i = (w−1)(N −1)+z.
Similarly, for each (r, then the (i, j)th entry of A is 1, if (w i , z i ) is adjacent to (w j , z j ), and 0 otherwise.The vertices (w i , z i ) and (w j , z j ) are adjacent if Since A is an adjacency matrix, for all natural numbers k, the (i, j)th entry of A k is the number of distinct paths of length k from node i to node j that never leave the bounded lattice [1, N − 1] 2 (see [11] for details), where node i corresponds to the point (w i , z i ) in the lattice, and the node j corresponds to the point (w j , z j ) in the lattice, again where i = (w i − 1)(N − 1) + z i and j = (w j − 1)(N − 1) + z j .
EJQTDE Spec.Ed.I, 2009 No. 4 For each natural number t and (i, i,j represent the i, jth entry of A t .Then the maximum row sum matrix norm is defined by In the theorem below we verify that I − A 4 is invertible.
Theorem 3.1 If A is the adjacency matrix corresponding to the centered Laplacian with zero boundary conditions, then I − A 4 is invertible.Moreover, ] and (w i , z i ) be the lattice point corresponding to the i th node.For any natural number t, the number of walks in an infinite lattice of length t from node i is and if t ≥ ⌈ N 2 ⌉, then at least one of these walks has left the bounded lattice [1, N − 1] 2 , since a walk in the direction (all the steps either left, right, up or down) of the closest boundary will leave the bounded lattice [1, that is, the number of walks from node i to any other node in the bounded lattice in t steps that does not leave the bounded lattice is at most 4 t − 1.Therefore, which implies that 2 ⌉ guarantees the invertibility of I − A 4 as well as the invertibility of Now that we have that I − A 4 is invertible and that Note, that given that I − A 4 is invertible, then the Green's function could be found by Jacobi Iteration.That is, if we let X 0 = Dr,s 4 , and for n ≥ 1, let then by iteration, we have and hence G r,s = lim n→∞ X n .

Application
In this section we will state the definitions that are used in the remainder of the paper.
Definition 5.1 Let E be a real Banach space.A nonempty closed convex set P ⊂ E is called a cone if it satisfies the following two conditions: Every cone P ⊂ E induces an ordering, ≤, in E given by x ≤ y if and only if y − x ∈ P.

Definition 5.2 An operator is called completely continuous if it is continuous and maps bounded sets into pre-compact sets.
Definition 5.3 Let P be a cone in a real Banach space E and D ⊆ E. Then the operator A : D → E is said to be increasing on Definition 5.4 A cone P of a real Banach space E is said to be normal if there exists a positive constant δ such that x + y ≥ δ for all x, y ∈ P with x = y = 1.
EJQTDE Spec.Ed.I, 2009 No. 4 The following theorem is an elementary fact about normal cones.A proof can be found in [9].Theorem 5.1 Let P be a cone in a real Banach space E. The cone P is normal if and only if the norm of the Banach space E is semi-monotone; that is, there exists a constant N > 0 such that 0 ≤ x ≤ y implies that x ≤ N y .
The next two theorems concerning the convergence of Picard iterates are restatements of theorems that can be found in [9].Theorem 5.2 Let P be a normal cone in a real Banach space E and A : P → E be a completely continuous operator.If u ∈ P with Au ≤ u and there exists a v ∈ P such that v ≤ A n u for all n ∈ N and A is increasing on [v, u], then {A n u} ∞ n=1 is a decreasing sequence bounded below by v ∈ P , and there exists a fixed point u * ∈ P of A such that We now present our solutions result as a fixed point application.

and ∆ ww g(w − 1 ,
z, r, s) + ∆ zz g(w, z − 1, r, s) = −δ w,r δ z,s for all w ∈ [1, N − 1] and z ∈ [1, n − 1].Applying classical techniques it can be shown that any solution of the partial difference equation is a fixed point of the operator T EJQTDE Spec.Ed.I, 2009 No. 4 defined by T u(x w , y z ) =