Numerical Solution of Integro-Differential Equations with Local Polynomial Regression

In this paper, we try to find numerical solution of               b  d , . a y x p x y x g x K x t y t t y a a x b a t b              d , . , a y x p x y x g x K x t y t t y a a x b a t b                   d x t y t t y a      a or               x  by using Local polynomial regression (LPR) method. The numerical solution shows that this method is powerful in solving integro-differential equations. The method will be tested on three model problems in order to demonstrate its usefulness and accuracy.

a y x p x y x g x K x t y t t y a a x b a t b , a y x p x y x g x K x t y t t y a a x b a t b

Introduction
In recent years, there has been a growing interest in the Integro-Differential Equations (IDEs) which are a combination of differential and Fredholm-Volterra integral equations.IDEs play an important role in many branches of linear and nonlinear functional analysis and their applications in the theory of engineering, mechanics, physics, chemistry, astronomy, biology, economics, potential theory and electrostatics.The mentioned integro-differential equations are usually difficult to solve analytically, so a numerical method is required.Many different methods are used to obtain the solution of the linear and nonlinear IDEs such as the successive approximations, A domain decomposition, Homotopy perturbation method, Chebyshev and Taylor collocation, Haar Wavelet, Tau and Walsh series methods [1][2][3][4][5][6][7][8].Recently, the authors [9], have used local polynomial regression (LPR) method for the numerical solution of linear and non-linear Fredholm and Volterra integral equations.
In this paper, we consider the linear IDEs, where the upper limit of the integral is constant or vari- are given functions, whereas x 0 needs to be determined.The subject of this paper is to try to find numerical solutions of integro-differential equations by means of local polynomial regression method which is presented firstly by Hikmat Caglar [9].Finally, we show the method to achieve the desired accuracy.Details of the structure of the present method are explained in sections.We apply LPR method for IDEs.In Section 3, it's proved the efficiency of numerical method.Finally, Section 4 contains some conclusions and directions for future expectations and researches.

Numerical Method
In this section, we describe local polynomial regression method to find the approximating solution of Equation (1).The following is the mathematical formulation of the local polynomial regression.

Local Polynomial Regression
First, we introduce the mathematical thoughts of local polynomial regression.This idea was mentioned in [10][11][12][13][14]. Since the form of regression function is not specified, so the data points with long distance from x provide little information to 0   y x .Therefore, we can only use the local data points around 0

 
x .We suppose that y x has p + 1 derivative at 0 x , for point x , located in the neighborhood of this point 0 x , we can use the p-order multivariate polynomials to locally approximate   y x , and the surrounding local point of 0 x , so we model where parameter j  depends on 0 x  so called local pa-rameter.Obviously, the local parameter j fits the local model with local data and it can be minimized, where controls the size of the bandwidth of local area.Using matrix notation to represent the local polynomial regression is more convenient.Below is the design matrix corresponding to (3) with h X and Y : The weighted least squares problem (3) can be written as where,   so the solution vector can be written as Furthermore, we can get the estimation where 1 is a column vector ( the same size of E  ) with the first element equal to 1, and the rest equal to zero, that is, The selection of does not influence the results much.We selected the quadratic kernel as follows:

Illustration of Numerical Method
In this section, the LPR method for solving Equation ( 1) is outlined.Let Equation ( 2) be an approximate solution of IDEs (1): and it is required that the approximate solution (7) satisfies the IDEs at the point x X  .Putting (7) in (1), it follows that This leads to the system (9) Consequently, the matrix form (4) can be written as follows by using expression (9).
Putting expression (10) and expression (11) in Equation (5), then estimated set of coefficients i  are obtained by solving matrix system solution.Therefore, approximate solution (7) can be obtained.

Simulation and Analysis
In this section, we consider some examples of IDEs.To show the efficiency of the present method for our problem in comparison with the exact solution, we report absolute error which is defined by: where LPR is absolute error.Ey LPR is LPR solution.exact is exact solution.Calculations were all performed by using MATLAB 7.0.

y y
Example 1: First we consider the integro-differential equation: For which the exact solution is x y x e  .Some numerical results of these solutions are shown in Table 1.We solve example 1 with n = 20, 30, 50 by choosing p = 3 and various values of parameters h presented in Table 1.
LPR gets up to value 0 which is very accurate at point x = 0 given h = 0.04, n = 50.
For which the exact solution is y x x   .Some numerical results of these solutions are revealed in Table 2.We solve example 2 with n = 20, 30, 50 by choosing p = 3 and various values of parameters h presented in Table 2. LPR achieves value 0 which is so accurate at point x = 0 given h = 0.08, n = 20.x given kinds of parameters h and n.Further, the tabulated results indicate that the absolute errors reduce rapidly when parameter n increases approximately.
Example 3: At last, we consider the FIDE: For which the exact solution is y Table 1.Absolute errors at point x with p = 3, different h, Example 1.
x x   .Some numerical results of these solutions are also shown in Table 3 which is similar to Tables 1 and 2.   In Table 3, some numerical results of these solutions are also opened up.We solve example 2 with n = 20, 30, 50 by choosing p = 3 and different values of parameters h presented in Table 3.  which is very small at point x = 0.0 given h = 0.04, n = 50.It's also showed that small absolute errors at other point x given different parameters h and n.
From Table 3, we can conclude that the absolute errors reduce approximately when parameter n increases.

Conclusion
In this paper, we make use of LPR method to solve the linear integro-differential equations.It's showed that this method is very convergent for solving linear integro-differential equations.Moreover, the numerical results approximate the exact solution very well.The Method can be extended to different parameters p, h and kinds of kernel functions.LPR method can also solve nonlinear or integro-differential equations which can be researched and resolved.
Ey LPR also gets up to value 1.75 × 10 −7 which is very small at point x = 0.0 given h = 0.08, n = 20.Moreover, it's showed small absolute error at other point Ey x given different parameters and .More importantly, the tabulated results indicate that the absolute errors present decreases more rapidly when parameter n increases.

h n Example 2 :
Consider the FIDE: value which is very small at point x = 0.0 given h = 0.04, n = 50.It's also represented small absolute errors at other point