 Open Journal of Statistics, 2012, 2, 352-355 http://dx.doi.org/10.4236/ojs.2012.23043 Published Online July 2012 (http://www.SciRP.org/journal/ojs) Numerical Solution of Integro-Differential Equations with Local Polynomial Regression Liyun Su1*, Tianshun Yan1, Yanyong Zhao1, Fenglan Li2, Ruihua Liu1 1School of Mathematics and Statistics, Chongqing University of Technology, Chongqing, China 2Library, Chongqing University of Technology, Chongqing, China Email: *cloudhopping@163.com Received April 18, 2012; revised May 20, 2012; accepted June 3, 2012 ABSTRACT In this paper, we try to find numerical solution of  bd, .ayxpxyxgxKxtyttyaax ba tb d,. ,ayxpxyxgxKxtyttyaax ba tb  dxtyttya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. Keywords: Integro-Differential Equations; Local Polynomial Regression; Kernel Functions 1. Introduction In recent years, there has been a growing interest in the Integro-Differential Equations (IDEs) which are a com- bination of differential and Fredholm-Volterra integral equations. IDEs play an important role in many branches of linear and nonlinear functional analysis and their ap- plications in the theory of engineering, mechanics, phys- ics, chemistry, astronomy, biology, economics, potential theory and electrostatics. The mentioned integro-differ- ential equations are usually difficult to solve analytically, so a numerical method is required. Many different meth- ods are used to obtain the solution of the linear and non- linear IDEs such as the successive approximations, A domain decomposition, Homotopy perturbation method, Chebyshev and Taylor collocation, Haar Wavelet, Tau and Walsh series methods [1-8]. Recently, the authors , have used local polynomial regression (LPR) method for the numerical solution of linear and non-linear Fred- holm and Volterra integral equations. In this paper, we consider the linear IDEs,  xayxpxyx gxK (1) where the upper limit of the integral is constant or vari- able,   are constants, gxpx and the kernel Kxty are given functions, whereas x0 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 . Finally, we show the method to achieve the desired accuracy. Details of the structure of the present method are explained in sec- tions. We apply LPR method for IDEs. In Section 3, it’s proved the efficiency of numerical method. Finally, Sec- tion 4 contains some conclusions and directions for fu- ture expectations and researches. 2. 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. 2.1. Local Polynomial Regression First, we introduce the mathematical thoughts of local polynomial regression. This idea was mentioned in [10- 14]. Since the form of regression function is not specified, so the data points with long distance from x provide *Corresponding author. Copyright © 2012 SciRes. OJS L. Y. SU ET AL. 353little information to 0yx. Therefore, we can only use the local data points around 0x. We suppose that yx has p + 1 derivative at 0x, for point x, located in the neighborhood of this point 0x, we can use the p-order multivariate polynomials to locally approximate yx, and the surrounding local point of 0x, so we model yx as: .00pjjjyxxx (2) where parameter j depends on 0x so called local pa-rameter. Obviously, the local parameter j fits the local model with local data and it can be minimized, 0jyx!j20jij10pijniiXxhYKXx (3) 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 ma- trix corresponding to (3) with hX and Y: 1000ppxXY,X12nYYYWY10nnXx XXx Xx11. (4) minThe weighted least squares problem (3) can be written as TYX0,nx1T where, 10K xhhKXWdiag X so the solution vector can be written as .TXWY0ˆXWX (5) Furthermore, we can get the estimation yx1,, yx01EˆTTXWXX WY 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, 111p10 0E. The selection of does not influence the results much. We selected the quadratic kernel as follows: K23if40otherwiseu11uKu00.p (6) 2.2. 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): jjjyxxx (7) where, 12 nXaXX b iand it is required that the approximate solution (7) satisfies the IDEs at the point xX. Putting (7) in (1), it follows that 1001000dppjjjjjjpxjajjpxxxxxKxtt gxtx 10110001,,0,,,2, ,,1,,,,dijjijij ijijiiXjij iaiiiaj pXxyyainaj jpXxbpXXxcKXtttxygX    10 1112020 2021 21 2122230 303031 3131333000 111pppppppnnnn n nnpnpnpXaa aabc abcabcabc abcabcabcabc abc   (8) This leads to the system (9) Consequently, the matrix form (4) can be written as follows by using expression (9).    121nnyyYyy (10)  (11) Putting expression (10) and expression (11) in Equa- tion (5), then estimated set of coefficients i are ob- tained by solving matrix system solution. Therefore, ap- proximate solution (7) can be obtained. 3. Simulation and Analysis In this section, we consider some examples of IDEs. To show the efficiency of the present method for our prob- lem in comparison with the exact solution, we report Copyright © 2012 SciRes. OJS L. Y. SU ET AL. 354 absolute error which is defined by: LPREy exact LPRy y, where LPR is absolute error. EyLPR is LPR solution. exact is exact solution. Calculations were all performed by using MATLAB 7.0. yyExample 1: First we consider the integro-differential equation: 13301321301,xyx eexy 3dxtyt t3 For which the exact solution is xyx 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 pre- sented in Table 1. LPR gets up to value 0 which is very accurate at point x = 0 given h = 0.04, n = 50. EyLPR 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 Eyx given different parameters and . More importantly, the tabulated results indicate that the absolute errors present decreases more rapidly when parameter n increases. hnExample 2: Consider the FIDE:  ln 1d1201121121ln00,yxyx xxtyxxyt t ln 1 For which the exact solution is yxx. 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 pre- sented in Table 2. LPR achieves value 0 which is so accurate at point x = 0 given h = 0.08, n = 20. EyLPR gets up to 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 pointEy12287 10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:   10cos 2π2πsin 2π1sin 4πsin 4π2201,yx yxxπdxxxtytt ln 1y  For which the exact solution is yTable 1. Absolute errors at point x with p = 3, different h, Example 1. xx. Some numerical results of these solutions are also shown in Table 3 which is similar to Tables 1 and 2. x008, 20LPREyhn 0 055,30LPREyhn 004, 50LPREyhn  0.0 7175 10 10501 10 0 0.2 5391 10 652810 10511 104812 10 0.4 6118 10 10887 105211 10 0.6 6622 10 11521 105732 10 0.8 535810 9364 10 4659 10 1.0 5883 10 9639 10  Table 2. Absolute errors at point x with p = 3, different h, Example 2. x008, 20LPREyhn 0 055,30LPREyhn 004, 50LPREyhn 0.00 9349 10 12287 10 0.2 4512 10 5487 10 9934 10 5832 10 0.4 5172 10 8611 10 5674 10 0.6 6596 10 10827 105512 10 0.8 6412 10 10951 10335510 1.0 588510 9306 10  Table 3. Absolute errors at point x with p = 3, different h, Example 3. x008, 20LPREyhn 0 055,30LPREyhn 004, 50LPREyhn 0.00 0 13175 10 0.2 552310 6719 10 10287 105187 10 0.4 5871 10 10876 104562 10 0.6 6732 10 1193310 5911 10 0.8 6158 10 931810 5496 10 1.0 5219 10 10682 10 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. LPR achieves value 0 at point x = 0 given h = 0.08, n = 20. The equivalent result applys to h = 0.0055, n = 30. EyLPR gets up to value Ey13175 10 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. Copyright © 2012 SciRes. OJS L. Y. SU ET AL. Copyright © 2012 SciRes. 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