Generalized Spline Approximation Method for Solving Ordinary and Partial Differential Equations

The main theme of this paper is to approximate the solution of ordinary differential equations by u sing basis of generalized s pline functions then using tensor product a nalysis to generalize the a pproximation fo r so lving p artial differential equations, (PDE's)


Introduction
Polynomial splines have been extensively used in several areas of applied mathematics such as computer graphics and approximation theory.Therefore, one of the first generalizations in this direction are the so called generalized splines which were introduced in the 50's of the 20 th century by Ahlberg, Nilson and Walsh, [8].In 1964 Schoenberg and Greville suggested that a spline function associated with a general linear differential operator of finite order which is called generalized spline and exhibited smoothness and best approximation results, [1].In 2008, Saleh M. studied the numerical solutions of ordinary fractional differential equations by G-spline functions to solve the linear and nonlinear differential equations, [9].The use of spline approximations for PDE's, and their relationship to finite difference schemes is considered by Hoskins and Sakai in 1970.Hoskins considers the two dimensional Poisson equation and show that a cubic spline approximation leads to a nine-point difference formula.For the heat conduction problem in one space dimension, Sakai studied a cubic spline approximation for the space derivative is combined with a

Definition (2.1), [7]:
Let ∆ : a = x 0 < x 1 < … < x N = b, N∈ be a partition on [a,b].A real valued function S, defined on [a,b] is said to be generalized spline with partition ∆ if the following conditions holds simultaneously:

3-
where 2n [x i-1 ,x i ] class of all functions defined on [x i-1 ,x i ] which possess an absolutely continuous (2n−1) th derivative on [x i-1 ,x i ] whose 2n th derivative is in L 2 (x i-1 ,x i ).

Solution of ODE's Using Generalized Spline
The objective of this section is to search for an approximate solution in the form of generalized spline function of order 2n (since L * LS of order 2n) and class C 2n-2 [a,b] for the first order initial value problems given by: where is the initial condition.If then problem (1) has a unique solution by assuming that f(t,y) is continuous and differentiable in D and Lipschitzian, [4] satisfies where M is a constant called the Lipschitz constant.
The construction of the approximate solution starts by letting: where a j 's are the unknown parameters which must be determined and the u j (t) are the generalized spline basis, and .The first algebraic equation related to the resulting system of S(t) is determined by the initial condition given in eq.( 1), i.e., .Now, S(t) should satisfy eq.( 1) for t = h which gives the equation for the first subinterval [0,h], i.e. : and repeat the same steps for the interval [h,2h], yields to: and continuing in like manner, a spline function S(t) is obtained and satisfying the equation: and then the following linear system of algebraic equations is obtained: Which may be solved uniquely for a j ' s The following example illustrate the above method of solution and the accuracy of the results:

Example (3.1)
Suppose the generalized spline method is be used to approximate the solution of the initial value problem: with h = 0.2 ; t i = 0.2i ; i=1,2,3,4 ; y 0 = 0.5 PDF created with pdfFactory Pro trial version www.pdffactory.comThen Gauss elimination method may be used to solve the above system to find which are found to be and hence: As a comparison, the exact solution of the considered problem is given by, [3]: Figure (1), give the exact solution y(t) and the approximation W(t) for the considered example, in which one can see the accuracy of the obtained results and the applicability of the method.

Solution of PDE ' s Using Generalized Spline and Tensor Product
Generalized spline method will be used here to approximate the solution of PDE's for example of the form: with boundary conditions: and initial condition where g(x) is given function.Let: where are the basis of the generalized spline function for the equation and are the basis of generalized spline function for the equation .From the initial condition given by eq.( 9), one may get: substituting the knot points for the xaxis to get an equation for each knot point, and from the boundary conditions given by eq.( 8), we have: PDF created with pdfFactory Pro trial version www.pdffactory.comand Similarly, substitute the knot points for the t-axis to get an equation for each knot point at x = 0 and at x = 1, and from the eq.( 7), we have: substitute the mesh points , to get an equation for each pair (i,j), then from eqs.( 11)-( 14), and a system with unknown coefficients c ij must be determined to compute eq.(10).
The next example is an illustration to the above method: Example (4.1)Consider the partial differential equation: with boundary and initial conditions: Let be a partition for the x-axis, such that: where h=1/3, then , are the mesh points with respect to the x-axis.Let the differential operator for the generalized spline function be given as Lu=D 2 u+Du-6u, then the fundamental solutions of L * Lu=0 are .Let be a partition for the t-axis such that: where k=0.01, then , will be the mesh points with respect to t-axis.Assume the same differential operator with respect to t, was applied to get the following fundamental solutions Now, from eq.( 11), one may have: = sin((22/7)x) (15) and substituting in eq.( 15) to get the first four algebraic equations.From eq.( 12), we have:

=0
(16) Also substituting: in eq.( 16) to get the second three algebraic equations.From eq.( 13), we have:  17) to get the third three equations.From eq.( 14), we have: Substitute in eq.( 18) to get six equations, then the resulting system of sixteen equations with sixteen algebraic unknown coefficients are obtained.
Gauss elimination method may be used to solve the algebraic equations ( 15)-( 18), to get the approximate solution of the PDE given in the following function: The exact solution is, [3]: The approximate solution z(x,t) and the exact solution u(x,t) are illustrated in table (1) Lw=D 4 w-13D 2 w+36w, with basis functions: which will give a non-polynomial generalized spline function as: Then the algebraic system of equations is: and rewrite this system in the matrix form Ax = b where, │A│≠0: PDF created with pdfFactory Pro trial version www.pdffactory.comEng.& Tech.Journal ,Vol.28,No.