Numerical Solution of Mixed Volterra – Fredholm Integral Equation Using the Collocation Method

Volterra – Fredholm integral equations (VFIEs) have a massive interest from researchers recently. The current study suggests a collocation method for the mixed Volterra Fredholm integral equations (MVFIEs)."A point interpolation collocation method is considered by combining the radial and polynomial basis functions using collocation points". The main purpose of the radial and polynomial basis functions is to overcome the singularity that could associate with the collocation methods. The obtained interpolation function passes through all Scattered Point in a domain and therefore, the Delta function property is the shape of the functions. The exact solution of selective solutions was compared with the results obtained from the numerical experiments in order to investigate the accuracy and the efficiency of scheme.


Introduction:
Consider the general mixed Volterra - where (s, t) is unknown function should to be found , ( , ) and ( , , , , ( , )) are given analytic functions on = Ω × [ , ] and ( 2 × ) ,respectively and Ω is close subset on R, with norm ||.|| . Equations of this type arise in the main branches of modern mathematics that appear in various applied areas including mechanics, physics and engineering …etc. In recent years different numerical methods have been used to solve (Eq. 1). Hassan (1) investigated a new iterative method to solve the (MVFIEs). In (2) Nili used the Meshless method for solving (MVFIEs) of Urysohn type on nonrectangular regions numerically. Nemati (3) introduced numerical method for the (MVFIEs) using Hybrid Legendre Functions. In (4) Shahooth presented a numerical solution for mixed (MVFIEs) of the second kind using Bernstein polynomial method. Babolian (5) applied block pulse function and the associated operational matrix to solve the (MVFIEs) in 2 -dimensional spaces. In our previous work (6), solution of mixed Volterra -Fredholm integral equation by designing neural network is given . The aim of this paper is to apply collocation method for solving mixed Volterra -Fredholm integral equations (MVFIEs) which have the formula (Eq.1).

Point Interpolation Method (PIM) on (RBF)
Let ( , ) be an approximation function in a domain ( " with an arbitrarily distributed nodes set " ) denoted by ( , ), = 1,2,3, … , . Where is the influence domain nodes number. At the node ( , ) assumed to be the Nodal Function Value.
(4) Where are the coefficient of Ⱥ ( , ) and the coefficient of Ƀ ( , ) (usually, > ) the vectors are defined a The general form of a multiquadrics (MQ) radial basis functions is Where is the distance between the node ( , ) and the interpolating point( , ) in the Euclidean 2 -dimensional space could be described as The terms of polynomial basis function are as following: The coefficients and in Eq. (5) are found by taking the interpolation of the scattered nodal points N in the domain. Ther ef or e," ℎ point, the interpolation is:" The polynomial terms are so important to obtain the optimum approximation. To fulfill that, consider the following conditions: The matrix form is expressed as follows: Where the vector , matrixes Ⱥ 0 and Ƀ 0 ar e defined as: The interpolation finally expressed as (15) The shape functions would depend only on the scattered nodes position when the Radial Basis Functions been calculated, only when the inverse of matrix Ȼ is obtained.

Solution of (MVFIEs) by (RPIM)
When considering the following 2 - 5. Find coefficients of members of numerical solution solving the equations with unknown by inverse matrix method (eq.12) 6. Applied the mean square error (MSE) law to study behavior of the RPIM.

Numerical Examples
Consider

Conclusion:
In this work, we have studied a collocation method to solve linear and non-linear mixed Volterra -Fredholm integral equations (MVFIEs). Our approach is built on combining the radial and polynomial basis functions in a point interpolation collocation method. Zeros of shifts Legendre are used as collocation points. The possible singularity that could associate in collocation methods is overcome by involving the radial basis functions. Many examples are solved by our method for different numbers of term N and M. It can be seen from the results in all the Tables, that it is clear that the approximate solution is in high agreement with the exact solution and the accuracy of RPIM results is in positive relationship with the numbers term (N) of RBF and ( M ) of polynomial ( P ).The efficient of RPIM method is excellent and of high precision . In general the accuracy and efficiency of our method (especially, for linear MVFIEs) are great and this method is easy to compute. Also, it can be easily extended and applied to nonlinear systems.