Numerical Solution of Linear Volterra Integral Equations by using Hermite Numerical Solution of Linear Volterra Integral Equations by using Hermite Polynomials

: In this research, Hermite series method is used to find the convergent numerical solutions, this method is tested on linear Volterra integral equation of the second kind, the result was highly accurate. The algorithm and example is given to explain the solution procedures. The comparison of numerical solutions was compatible with exact solutions.


Introduction:
The Volterra integral equations are a special type of integral equations, and they are divided into the first kind and second kind Alinear Volterra equation of the first kind has the form: a (1) A linear Volterra equation of the second kind has the form of (2) Where f and M are known continuous functions, is a constant parameter and u is unknown function. This kind has been described analytically and numerically [1]and [6].

Msc.Wafaa Abd Ibrahem
The central objective of this survey is to secure and survey the approximate solution for (2) in the space of continuous maps on the interval [a,b] by means of using function series and we assume that the considered equations have at least one solution.
In the study of solution of Volterra integral equations, different methods of solving Volterra integral equations has been developed in recent years, like Bernestein approximation ,which has been pressented to solve Volterra integral equation [ This paper is organized as follows.In section two the definition of Hermite polynomials is given In section three a matrix impersonation for Hermite series is stated.
In section four, solution of linear Volterra integral equation with Hermite polynomials. In section five algorithm. Applications is given in part six. lastly, Abrief of the conclusion and discussion is stated in the final section.

2.Hermite Polynomials
Definition: The Hermite polynomials of nth degree over the interval [a,b] is a set of orthogonal polynomials and it is defined by the recurrence relation[5]: (3)

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Numerical Solution of Linear Volterra Integral Equations by using Hermite Polynomials

A matrix Impersonation For Hermite Series
On much enforcement a matrix shape to the Hermite series is helpful. This is straight forward to develop if only looking at a linear combination in expression of dot products.Given a series written as a linear combination of the Hermite standard function: It is easy to write this as a dot product of two vectors: This can transform to the next form: where are the coefficients of the force standard that are applied to limit the particular Hermite series.
We note that the matrix in this case is lower triangular.

Msc.Wafaa Abd Ibrahem
In this section, Hermite series are applied to find the numerical solution for linear Volterra integral equation, as follows.: Rendering linear Volterra integral equation of the second kind.
(7) Let = By using equation (3) and applying the Hermite polynomials method for equation (5), we obtain the next form.
By using equation (4),which can be transform to the next formula :

Numerical Solution of Linear Volterra Integral Equations by using Hermite Polynomials
Msc.Wafaa Abd Ibrahem (9) Now, to find all integration in equation (7).
Thereafter in order to locate we want n equations. Now, select in the interval [a,b], whose give (n) equations.
Solve the (n) equations by Gauss elimination to find the values and substitute in numerical method we get the numerical solution.
The following algorithm summarises the proceedings for finding the numerical solution for linear Volterra integral equation of the second kind.

Algorithm
Step1: select n the degree of Hermite polynomials Step2: Put the Hermite polynomials in linear Volterra integral equation of the second kind.
Step3: Calculate the integration Numerical Solution of Linear Volterra Integral Equations by using Hermite Polynomials

Msc.Wafaa Abd Ibrahem
Step4: select Step5: Solve the linear system by Gauss elimination to find the unknown coefficients ( ) .
Step6 : Substitute in we get the numerical solution u(t).

Solve three equations by Gauss elimination to find the values as follows
Then the solution is:

7.Conclusion and Discussion
The effect acquired from the application, has shown that Hermite series is a powerful and active technicality in finding approximate solution of linear Volterra integral equation of the second kind .  Sciences,Vol.8,no.11,pp.525-530,2014.