Application of Iterative Method for Solving Higher Order Integro-Differential Equations

The main aim of this paper is to apply a n ew technique suggested by Temimi and Ansari namely (TAM) for solving higher order Integro-Differential Equations. These equations are commonly hard to handle analytically so it is request numerical methods to get an efficient approximate solution . Series solutions of the problem under consideration are presented by means of the Iterative Method (IM). The numerical results show that the method is effective, accurate and easy to implement rapidly convergent series to the exact solution with minimum amount of computation. The MATLAB is used as a software for the calculations.


Introduction
Integro-Differential Equations (IDEs) include in many mathematical formulations of physical phenomena, these problems have a major role of interest and arise in many applications in various fields of science, such as chemical kinetics, fluid dynamics, engineering problems and biological models.In recent years, there exist numerical techniques gained a great interest by many authors such as Lagrange Interpolation Method [1].Wavelet-Galerkin Method (WGM) [2].Adomian's Decomposition Method (ADM) [3][4][5].Modified Adomian Decomposition Method (MADM) [6].Variational Iteration Method (VIM) [7,8].Differential Transform Method [9].Tau Method [10].Generalized Spline Method [11], and Semi Analytical-Numerical Techniques such that Taylor polynomials [12].and Rationalized Haar Functions Method [13].However, none of a fore mentioned methods are successfully solved higher order IDEs.Furthermore, prior studies need more effort to realize the outcomes, they are not precise and commonly they are improved for specific sorts IDEs.Temimi and Ansari (TAM) have been suggested a new iterative method, i.e., Semi Analytic Iterative Method (SAIM) for solving linear and nonlinear functional equations [14].This method has been extensively studied by many researchers recently; it has been successfully applied for solving some linear and nonlinear partial and ordinary differential equations [15][16][17][18].It is worth mentioning, SAIM is not yet used to solve higher order IDEs.This method is accurate and powerful technique, needn't to impose any additional restrictions to get the numerical solution of these problems.It is qualified method for extremely the number of calculations will be reduced while still maintaining the solution is more accurate and efficient.Higher Order Linear Fredholm IDEs was solved by Power Series and Chebyshev Series Approximation Methods in [19].IDEs was solved by Modified Taylor Expansion Method in [20].And Voltera IDEs was solved by using the Laplace Transform Method in [21].In this research, we applied this reliable technique to solve higher order IDEs.Generally, Volteraintegrodifferential equation [21].Given in the form: And Fredholmintegrodifferential equation [19].given in the form: Both of Equations.( 1) & ( 2) with Initial Conditions (ICs).
where y (n) (x) is the n th derivatives, k(x, t)and h(x), are given continuous smooth functions,  is a parameter, (x) unknown function to be determined and , , ω k are constants.Since the results of equation (1) and equation (2) combine the differential and integral operators, then it is necessary to define ICs as in (3).The proposed method was applied to establish series solutions for Equation (1) or Equation (2).We illustrated that this method is effective and perfect in handling to solve higher order IDEs in scientific and engineering problems.Several numerical examples are introduced and comparison with existing methods, the results reveal that the method is accurate and easy to implement.

Fundamental Idea for the Iterative Method
The main steps of Iterative Method (IM).It is rewrite that any differential equation can be written as [14].
With Boundary Conditions B (y, Where x is the independent variable, L is a Linear operator, N is a non-linear operator and the boundary operator is B. The method which proposed as the following way.The initial approximation is the primary step in the IM, by assuming that the initial guess  0 (x) is solution of problem y(x) and solution of equation can be solving: (5) To generate the next iteration of the solution as follows: After several simple iterative steps of the solution, the general form of this Equation which is: Evidently each iteration of the function   (x) represent effectively alone solution for equation (4).
We will implement the steps of method at the equation (1), so Equation (1) can be express as: The differential operator L(y (x)) is the highest order derivative in the equation ( 8), we assume that L is invertible by using the given ICs in equation ( 3) and applying the inverse operator L −1 in both sides of equation ( 8), we get the following Equation: Where the function ψ 0 is arising from integrating the source term, from applying the given ICs in Equation ( 3) which are prescribed.Now, we illustrate the method as the following steps: Step 1: to get  0 () solving L( 0 (x)) -h(x) = 0 (10) with ICs in Equation ( 3) and applying the inverse operator L −1 in both sides of Equation ( 10), we obtain: Step 2: The next iterate is: x 0 d(t) = 0 with ICs in Eq. ( 3) , (11) solving this equation and applying the inverse operator L −1 in both sides of Equation ( 11), leads to get  1 () as: Step (3): After several simple iterative steps of the solution, the general form of this equation given as Solving this equation and applying the inverse operator L −1 in both sides of Equation (12), leads to get  +1 (): evidently each iteration of the function   (x) represents effectively solution for Equation (8).
Similarly, by the same steps we solve Fredholm IDEs.

Numerical Results
We will be applying the SAIM for solving some examples of the Fredholm IDEs and Voltera IDEs.

Solution
Via implementing same steps as described in the previous section, we first begin by solving the following initial problem to find the initial approximation  0 (x), the SAIM will be applied as So, the primary step is: Then, the general relation as follows: By solving the problem defined in Equation ( 14), we have y 0 = e x − 1934613350591413 6755399441055744 x 3 .The first iteration can be gotten as: Thus, the solution of Equation ( 16) as: y 1 = e x − 644871116863805 54043195528445952 x 3 .
The second iteration is: Then, the solution of Eqaution (17) as y 2 = e x − 26869629869327 54043195528445952 x 3 .Also, by same steps, the other solutions can be generated from calculating these problems via using MATLAB, we obtain y 3 , y 4 , … , we get the solution till y 11 .

Solution
Applying same steps as in the previous example, we first begin by solving the following initial problem in order to find the initial approximation  0 (x), the SAIM will be applied: as L(y 0 ) = Then, the general relation as follows: By solving the problem defined in Equation (19) we have: The first iteration can be gotten as:

Solution
To find y 0 we take dx 4 , N(y) = 0. So, the primary step is: Then, the general relation as: By solving the problem defined in Equation ( 22) we have: 96 − 579905046931621 180143985094819840 x 5   By same way the next step is to find  1 as follows: The first iteration can be gotten as: and has a solution:
y (4) () = sinx + cosx + 2 ∫ sin(x − t) y(t)dt By solving the problem defined in Eq. ( 26) we have y 0 = cosx + sinx + x 2 + x 3   3 as same way, the next step is to find y 1 as follows: The first iteration can be gotten as: y 1 (4) () = sinx + cosx + 2 ∫ sin(x − t) y(t)dt x 0 (28) and has a solution:   The essence of this method, SAIM in comparison with the other analytical methods does not need large computations such as Lagrange multiplier in the VIM or any complex assumptions like nonlinear Adomian's polynomials in the ADM.It also does not need to constrict Homotopy in the HPM.Furthermore, this method proved that it is efficient in overcoming the difficulties in calculating and solving high -order integro-differential equations with easier steps.

Conclusion
Most integro-differential equations are difficult to be solved analytically.For this purpose, the efficient Iterative analytic method (Temimi and Ansari)  mentioned type of these equations.The present iteration method has strength features among other analytic methods.The obtained solutions can be shown as a series form that converges to the exact solution with simple computations.No complicated calculations have been shown with the presented method.Finally, the advantage of this method is to apply some examples to show that the present iterative method is very accurate and suitable for solving the highorder Voltera -Fredholm Integro-Differential Equations.

Figure 2 .
Figure 2. Exact and approximate solution of the illustrative example.

Figure 4 .
Figure 4. Exact and approximate solution of the illustrative example.

Table 1 .
Numerical results of the illustrative example above of Y 11 .
Figure 1.Exact and approximate solution of the illustrative example.

Haitham Jour. for Pure & Appl. Sci. 32 (2) 2019
by same way in the previous example.The next step is to find y 1 as follows:

Table 2 .
Numerical results of the illustrative example above of Y 12 .

Table 3 .
Numerical results of the illustrative example above of Y 7 .Exact and approximate solution of the illustrative example.

Table 4 .
Numerical results of the illustrative example above of Y 2.
is applied to solve such