Expansion Method For Solving Linear Delay Integro-Differential Equation Using B-Spline Functions

The main goal of this paper lies briefly in submitting and modifying some methods for solving linear delay integro-differential equations (L-DIDEs) containing three types (retarded, neutral and mixed) numerically by employing expansion method (collocation and partition) with the aid of B-spline polynomials as basis functions to compute the numerical solutions of (L-DIDEs). Three numerical examples are given for determining the results of this method.


Delay
integro-differential equations (DIDEs) are equations having delay argument.They arise in many realistic models of problems in science, engineering and medicine [6].Only in the last few years has much effort in behavior of solution of delay differential and delay integro-differential equations, i.e. equations in which the highest order derivative of unknown function appear with delay which is called neutral delay integro-differential equation as well as retarded differential equation.Many researchers used linear delay integro-differential equation in some subjects such as Hopkins, T [1] found the numerical solution of stochastic DIDEs in population dynamics as well as Xuyang lou [14] studied delay integro differential equation, modeling neural field.

B-Spline Polynomials and
Properties:-B-splines are the standard representation of smooth non linear geometry in numerical calculation.Schoendery [12] first introduced the B-spline in 1949.He defines the basis functions using integral convolution.B-spline means spline basis and letter B in B-spline stands for basis.

Definition (1)
The p i , i=0, 1, …, m+1 are called control points or anchor points or de Boor point.Apolygon can be constructed by connecting the de Boor points with lines starting with p 0 and finishing with p n this polygon is called the de Boor polygon.
The m-n basis B-spline of degree n can be defined using the cox-de Boor recursion formula.
when the knots are equidistant we say the B-spline is uniform otherwise we call it non-uniform.
The B-spline can be defined in another way like:- PDF created with pdfFactory Pro trial version www.pdffactory.comwhere There are n+1 n th degree B-spline polynomials for mathematical convenience, we usually set B k, 0 (t) =0 if k<0 or k>n.Some important types of B-spline polynomials which are used in this work are introduced as follows:-

Constant B-spline B k, 0 (t)
The constant B-spline is the simplest spline.It is defined on only one knot span and is not even continues on the knots.It is a just indicator function for the different knot spans.

Linear B-spline B k, 1 (t)
The linear B-spline is defined on two consecutive knot spans and is continues on the knots, but not differentiable.

Quadratic
B-spline with uniform knot-vector is a commonly used form of B-spline.The blending function can easily recalculate, and is equal to each segment in this case:-

Cubic B-spline B k,3 (t)
Cubic B-spline with uniform knot-vector is the most commonly used form of B-spline.The blending function can easily be recalculated and is equal to each segment in this case:- Some important properties of Bspline polynomials are given as follows:-

Property (1):-
Derivatives of the n th degree Bspline polynomial are polynomials of degree n-1.Using the definition of the B-spline polynomial we can show that this derivative can be written as a linear combination of Bspline polynomials [12].
In particular That derivative of a B-spline polynomial can be expressed as the degree of the polynomial, multiplied PDF created with pdfFactory Pro trial version www.pdffactory.comby the difference of two B-spline polynomial of degree n-1.

Property (2):-
The B-spline polynomials of order n form a basis for the space of polynomials of degree less than or equal to n because they span the space of polynomials-any polynomial of degree less than or equal to n can be written as a linear combination of the B-spline polynomials and they are linearly independent that is if there exist constants c 0 , c 1 , …, c n that the identity holds for all t, then all the c i , s must be zero.

Expansion Method:-
This method illustrate important approaches coming frome the field of approximation theory, in which the unknown solution f(t) is expanded in terms of a set of known functions (B-spline polynomials) as follows The unknown then being the expansion coefficients c i .An algorithm based on the above approximation is an expansion method in this work collocation and partition methods are considered as expansion method.

Solution of L-DIDEs Using
Collocation Method with the Aid of B-spline:-First consider the linear retarded integro-differential equation of the form By approximating the unknown function f(t) using B-spline polynomial as a basis function we have Substituting eq.( 8) into eq.(7) for f(t) and with t=t j , and by using property (1) for Second consider the linear neutral integro-differential equation of the form Substituting eq.( 8) into eq.(10) we get the equation then PDF created with pdfFactory Pro trial version www.pdffactory.com Third consider the L-MIDE's of the form By Substituting eq. ( 8) into eq.(12) and follows the previous steps we have the following equation as a results:- for j=0,1,...,N These equations involve n+1 unknown coefficients [c i ], then we may select n-1 points {t 1 ,t 2 ,...,t N-1 }in the range of integration and require f N (t) to satisfy the delay integrodifferential equation at just these n-1 points.This method requires us to solve just the system of n+1 linear equations for each type of delay integro differential equations (retarded, neutral and mixed).thesesystem of n+1 Linear equations for the coefficients c i' s can be written in matrix form as follows:- k=1,2 the existence of (τ k )depends on the type of the equation (retarded, neutral and mixed) and By solving this system by Gauss elimination procedure to find c i 's for i=0,1,...,N.

Solution of L-DIDEs Using Partition Method with the Aid of B-spline:-
In this method we divide the domain R into P non over lapping sub domains R j , j=1,2,...,p, if the weighting functions are chosen as follows then the delay integro -differential equation is satisfied on the average in each of the P sub domains R j , the required equation for partition method become where E(t)is called residue equation and hopefully it approaches zero on R j .
or which we have the residue equation for (retarded, neutral and mixed) integro-differential equation respectively as follows:for retarded equation we have PDF created with pdfFactory Pro trial version www.pdffactory.com for neutral equation we get for mixed equation we get Substituting these equations into eq.(15) we get: These equations give system of (N+1) Linear equations in N+1 unknown coefficients c i , i=0,1,...,N.Rewriting the above equations in a matrix form as follows:- using Gauss elimination procedure we find the values of c i 's.

Example (1):
Consider the following retarded Volterra integro-differential equations: Assume the approximate solution is: This problem is solved using collection method and partition method with the aid of B-spline functions as basis functions, the solution of f(t) is obtained as shown in table (1),with N= 10, h=0.1 and t i =ih, i=0,1,...,N.

Example (2):
Consider the following neutral integro-differential equations: 2) lists the results obtained by achieving collection method and partition method with the aid of Bspline function.

Example (3):
PDF created with pdfFactory Pro trial version www.pdffactory.comConsider the following mixed Fredholem integro-differential equations: Assume the approximate solution is: The results of this example list in table (3) which obtained by using collection and partition method with the aid of B-spline.

Conclusions
1.The expansion method including (collocation and partition) methods with the aid of B-spline polynomials as a basis function which are used in this paper have proved their effectiveness in solving (L-DIDEs) numerically and finding accurate results.
2. B-spline function depends on N as N increased, the error term is decreased.
3. The results show a marked improvement in the least square errors from which we conclude that.[14] Xuyang lou (1) ; Baotong cui (1) .Boundedness and stability for integrodifferential equations modeling neural field with time delay.

com Eng &Tech. Journal,Vol.27, No.10, 2009 Expansion Method For Solving Linear Delay Integro-Differential Equation Using B-Spline Functions
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Table (
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Eng &Tech. Journal,Vol.27, No.10, 2009 Expansion Method For Solving Linear Delay Integro-Differential Equation Using B-Spline Functions 1974
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