Using Linear programming to find solution volterra Integral Equation.

Conclusion :This method is based on approximating the solution using a new technique based on the programming methodlinearity by using the Simplex method to find the solution to thisnumerically equations.

The Aim of these studies: The major purpose of this thesis is: 1-To survey as well as to modify L.P. method have been used solve volterra integral equation of convolution type.
2-To write successful program for this method.

Preface
Considers the use of linear method as an approximate method for treating these equations.
For this method a. program is written, example are solved, results are tabulated, and the computer programs is written in turbo Pascal (ver.7)

-Classification of linear integral equations
Before discussing the appearance of the integral equations.
It will be necessary present some definitions and introduce a preliminary classification of linear Integral equation.{2}

Def (1-1)
The general linear integral equations can be represented as:

Methods of linear program. (2-1) Simplex Method
The principle algorithm used in solving linear program is simplex method, this method deals with finding extreme values of linear function when the variable are constrained by inequalities.
Simplex method can be summarized by the following .{2} Step 1: Set up the initial tableau.

 
Step 2 : Finding which element should enter the basis by choosing the most positive value form the objective function (z jc j ) Step 3 : Choose the minimum value of ratio determine which element should leave the basis .

Step 4 :
Repeat the steps (2 and 3 ) until all the values of (z j -c j )  0 to test optimality , if c j α j  0 them c j s satisfy the system of LP simplex method can be explained in the table below :

Basis α 1……………
. α n α n α n+m α n+1 ..  It is also used when new constraints are added to an LP problem for which the optimal solution has already been obtained.{6} The following advantages can be obtained by using dual property 1-If the primal problem continuous are large number of rows (Constraints) and a smaller number of columns (variables), the computational procedure can be considerably reduced be converting into dual and them solving.In order to convert the primal problem to the dual problem, we these steps: Step 1 Convert the problem in to maximize problem it is initially in for.

Step 2
Convert type constraints, if any, in to type by multiplying both sides of such constraints by -1.

Step 3
Convert the inequality constants in to equality by the addition of slack variables and obtaining basic solution.

Step 4
To choose the basic, the most negative value form the objective function in selected.

Step 5
Choos the minimum value of ratio ( ) such that a js > 0 to leave basic .
Step 6 Repeat step (4 and 5) until all the value of objective function be ≥ 0.

Section ( 3)
Technique for solving VIT using L.P Now in this section linear programming method is used to approximate the solution of volterra integral equation.
(3-1) The solution of VIT 2nd using L.P Linear programming method can be used to approximate the solution of equation (1-9) or on the operator notation ,we have Where n  is the residual function so

Minimizing the residual
There fore In a liner programming, code is available in which the available are not required to be nonnegative, then it can be use on problem as formulated in (3-5).
Moreover the scheme (3-5) solves only problems, which have increased or decreased solution on a fixed interval.
For more general case, in which the solution may oscillate often between positive and negative values, the following modified technique can be applied first, introduces The technique for solving VLEs using LP method can be summarized the following steps:

Discussion
In this section the linear programming method for solving Integral equation is discussed.The linear programming method provides a very convenient and useful algorithm and the results obtained using this method conform point have been identified: 1-Linear programming method can be used even if there is on information about the exact solution (form the residual function S n in equation(3-3)) 2-The number of convenient (m) i.e. as m increases the value of the L.S.E approach to zero.

4-
The involved integrals

2 -
Duality in linear programming has certain for reaching consequences of economic nature.3-Calculation of the dual cheeks the accuracy of prime solution.In general if the primal problem.Them the dual problem is given by Using Linear programming ………...………… Ali Hussein shua"a J. OF COL.OF B .ED.NO. ....TO above transform is valid only for symmetric problem.
) =cos x we seek he approximate solution :

Table ( 3
-2) the numerical results with the exact solution as will as least square error and running time