Numerical Solution Of Freholm-Volterra Integral Equations .
Applied Mathematics, 2013, 4, 204-209http://dx.doi.org/10.4236/am.2013.41A031 Published Online January 2013 (http://www.scirp.org/journal/am)Numerical Solution of Freholm-Volterra IntegralEquations by Using Scaling Function Interpolation MethodYousef Al-Jarrah, En-Bing LinDepartment of Mathematics, Central Michigan University, Mt. Pleasant, USAEmail: enbing.lin@cmich.eduReceived October 15, 2012; revised November 15, 2012; accepted November 23, 2012ABSTRACTWavelet methods are a very useful tool in solving integral equations. Both scaling functions and wavelet functions arethe key elements of wavelet methods. In this article, we use scaling function interpolation method to solve Volterra integral equations of the first kind, and Fredholm-Volterra integral equations. Moreover, we prove convergence theoremfor the numerical solution of Volterra integral equations and Freholm-Volterra integral equations. We also present threeexamples of solving Volterra integral equation and one example of solving Fredholm-Volterra integral equation. Comparisons of the results with other methods are included in the examples.Keywords: Wavelets; Coiflets; Scaling Function Interpolation; Volterra Integral Equation; Fredholm-Volterra IntegralEquation1. IntroductionThe study of finite-dimensional linear systems is welldeveloped. As an infinite-dimensional counter part offinite-dimensional linear systems, one can view integralequations as extensions of linear systems of algebraicequations. An integral equation maybe interpreted as ananalogue of a matrix equation which is easier to solve.There are many different ways to transform integralequations to linear systems. Many different methods havebeen used for solving Volterra integral equations andFreholm-Velterra integral equations numerically.In this paper, we first recall the method of scalingfunction interpolation. Then we solve linear Volterra integral equation of the form:f x k x, t y t dtxa(1)and Fredholm-Volterra integral equations of the form:y x g x a k1 x, t y t dtx a k2 x, t y t dtb(2)where the functions k x, t , k1 x, t and k2 x, t areknown functions and called kernels. The function f x is known, and the function y t is to be determined.One of the motivations in this study arose from equationsin theoretical physics. In fact, there are many applications in several disciplines as well. We will use scalingfunction interpolation method to solve integral equations.As a natural question, one would wonder any possibleCopyright 2013 SciRes.convergence properties and how this method would compare with other methods. We will prove two convergencetheorems and present several examples.2. ApproximationWavelets and scaling functions are a useful tool in approximation methods of solutions of differential and integral equations [1]. We first recall Multiresolution analysis (MRA) [2]. We assume the scaling function andwavelet function , Ψ are sufficiently smooth and satisfyMRA with compact support and Ψ has N vanishing moments (defined below). The scaling function x isdefined as x p 2 j x p p j , p x p(3)pfor some coefficients p , p Z . By using this dilationand translation we defined a nested of sequence spaces V j , j Z which is called MRA of L2 R with thefollowing properties:V j V j 1 , j Z V j 0 (5)is dense in L2 R (6) x V j 2 x V j 1 .(7)V Vjj Z(4)j ZFor the subspace V1 is built by 2x p , p ZAM
Y. Al-JARRAH, E.-B. LINthen V0 x p , p Z and V0 V1 we can write x p 2 x p p 1, p x . In general,pp(9)and the wavelet function can be obtained by x p j , p x . x x dx 0,(11)and in this case the wavelet is said to have a vanishingmoment of order k.Semiorthogonality: i , p x , j , k x i , p x j , k x dx 0;p k ; i , j , p, k Z .(12)The set of scaling functions n, j is orthogonal atthe same level n, which means: n, p x , n, k x n , p x n, k x dx 0;n, p, k Z .M k x x dx 0, k 1, 2, , L 1 .k x x dx 0,k M k k 1, 2, , L 1 .fj1p x, y where the index set is p, q sup p j, p sup p .j, pIn addition, the moment M l satisfiesM l c , l 1, 2, , N 1 .lThen c M 1 , andf fjL2 C f N N 1 . 2j where C is a constant depending only on N, diameter ofΩ andf N : max x , y , m 0, , N N f x, y . x m y N mFor one-dimensional analogue, we havefj x 12j p f 2 j j , p x , x a, b ,(19)andf fjL a ,b 2 C f N N 1 . 2 (20)wheref is the moment of scaling functions.p c q c , j j, p x j,q y ,j(18)2 2 x, y f 2(14)(15) x f x j , p x dx j , p x .To approximate a given function f, one can use sampling values of f at certain points. It is proved in [3],namely, an interpolation theorem using coiflet, namely, if x and x are sufficiently smooth and satisfy theEquations (10)-(15) and the function f x C k ,where Ω is a bounded open set in R2, k N 2 , j Zthen,(13)Coiflet (of order L) has more symmetries and it is anorthogonal multiresolution wavelet system with,jp(10)for some coefficients p . Some interesting properties ofscaling and wavelet functions make wavelet methodmore efficiently than other methods such as spline approximations in solving an equation. A lot of computational time and storage capacity can be saved since we donot require a huge number of arithmetic operations partlydue to the following properties.Vanishing moments:kf x f(8)p j, p 2 j p ,(17)Hence the Equation (16) becomes as follows:In fact, for each j we define the orthogonal subspaceW j of V j in the subspace V j 1 , the or thogonal basisof W j is denoted bywhere p j , p x , f x f x j , p x dx .p x p 2 j x p p j , p x .205N : max x , y a ,b , m 0, , N N f x . x m3. Scaling Function InterpolationIn MRA, any given function f x L2 R can be interpolated by using the basis functions in the subspaceV j as follows:f x fj x p j , p x (16)4. Solutions of Linear Integral EquationIn this section, Coiflet is used to solve linear integralEquations (1) and (2), where we will explain the methodin terms of matrix notation.pwhere the coefficientsv p are evaluated by using thesemiorthogonality of the scaling functions (12) such thatCopyright 2013 SciRes.4.1. Linear Volterra Integral EquationIn this subsection we will use the interpolation FormulaAM
Y. Al-JARRAH, E.-B. LIN206(19) to solve Volterra integral Equation (1). The unknown function y x in Equation (1) can be expressedin term of scaling functions j , p x in the subspaceV j such thaty j x a p j , p x .(21)one can have the system of linear equations;a B Gwhere a is the vector of unknowns as we introduce inEquation (21),G g x1 , g x2 , , g xn pBy substituting Equation (21) into the Equation (1), wehave the following system, f x k x, t a p j , p t dt0 p and B1 x1 B1 x2 B x B2 x2 B 2 1 Bn x1 Bn x2 x(22)x a p k x , t j , p t dtpwith0B p x j , p x a k1 x, t j , p x dtxTo simplify the system, letxAp x k x, t j , p t dt a k2 x, t j , p x dtb0Then the system (22) becomesf x a p Ap x .(23)pThe coefficients a p , p can be evaluated by substituting the set of real numbers xp, x p 0, X , p , 0 X b a1 A1 x1 a2 A2 x1 an An x1 f x1 a1 A1 x2 a2 A2 x2 an An x2 f x2 a1 A1 xn a2 A2 xn an An xn f xn .If we use the notation a a1 , a2 , , an and An x2 y j x a p j , p x .f f x1 , f x2 , , f xn , then the system (23) isequivalent to the system aA f , and the solution isa fA 1 .(24)This gives raise to coefficients in (23) and we obtaineda numerical solution to Equation (1).4.2. Linear Fredholm-Volterra IntegralEquationTo solve the Fredholm-Volterra integral Equation (2), weuse a similar algorithm as we use in 4.1. The unknownfunction can be approximated by using Equation (1) andCopyright 2013 SciRes.In this section, we provide with the convergence rate ofour method for the numerical solution of solving linearVolterra integral equations and Freholm-Volterra integralequation respectively. We will explain the necessaryconditions for the convergence.Theorem 5.1In Equation (1), suppose that the functionsk x, t C 0, X c, d , 0 x X b ,k x, t m0 0 and the two functions f x , y x are in C 0, X , 0 x X b , for j Z ,A1 xn A2 xn An xn A1 x2 A2 x2 and the set of x1 , x2 , , xn is in the interval a, b which one can be choose equally spaced. In the next section we will discuss the convergence for the method byderiving a convergence theorem of this numerical solution.5. Error Analysisinto the system (23), let n , then the system (23)can be written in the form A1 x1 A x A 2 1 An x1 B1 xn B2 xn Bn xn If an approximate solution of the Equation (1) withcoefficients obtained in (24), and the error at the pointj 1 xi is e xi y j xi y xi . Then e x c , 2 where c is a constant.Proof:We begin with the following equation.x kx,tetdt k x, t p p j , p t y t dt . (25)00 xAt any point xi x j ; j Equation (25) becomes:xi kx,tetdt k x, t p p j , p t y t dt ,00 xiAM
Y. Al-JARRAH, E.-B. LINthen k x, t e t dt k x, t p p j , p t y t dt . (26)00 xx207And since p p k x, t e t dt0 x m1e t dt m1 e t dt m1 e t 00 Ne t c1 c0 y e t 1m1xi 1m0xixi 0o k x, t p p i , p t y t dto (27) p p i , p t y t dtN 1 1 c2 2 2 jj 1 c . 2 k2 x, t C a, b a, b , and y x , g x are inC a, b , for j Z ,xi y j x a p j , p x o If an approximate solution of the Equation (2) withcoefficients obtained in (24), and the error at the pointxi is e xi y j xi y xi . Thenj 1 e x , where β is a constant 2 Proof:Substitute (21) into Equation (2), we get the followingintegral equation p p i , p t y t dt1 xik x, t dt .m0 0By (19), the unknown function y t can be interpolated by using the coiflet such that:Such that c1 p y j t y j j, p t 2 p(28)If we add and subtract Equation (28) in Equation (27),we get the following inequality:y j x g x k1 x, t y j t dtxae t c1 Theorem 5.2In Equation (2), suppose that the functionsk1 x, t C a, b a, b , a x X b , c1 By using the above results and the orthonomality ofthe scaling functions t , we conclude thatThen, k x , t dtis finite then denote it as p pxi y 2 j p c2 .Forx y 2 j p k 2 x , t y j t dtbxi0 p p j , p t y t y j 2p pa j, p t Subtracts Equation (27) from (1) and substitute x by xito get; p y j j , p t 2 p c1 c1 p y j p 2xi oxi 0 e xi y x y j x i , p t y t dt p y j p 2 i , p t p i , p t dt p p xip j , p t N j , p t y t c0 y Copyright 2013 SciRes. k2 xi , t y t y j t dtba(30)j a k1 xi , t a y t y t dtxi xij a k2 xi , t a y t y t dtbb i ,1 a y t y j t dt i ,2 a y t y j t dt.xiBut by (20), we have that; p y 2j p a y 2 j p 0 k1 xi , t y t y j t dtxi o p y 2 j i , p t y t dtxi(29)N 1 , 2 bAdd and subtract Equation (28) for absolute value inthe previous equation, we get the following equation.AM
Y. Al-JARRAH, E.-B. LIN208e xi i ,1 p p y t y j t y j j , p t y j j , p t dt 2 2 ppxiab p i ,2 y t y j t y ja 2p i ,1 py t y j 2pxia p j , p t dt a p y j 2 p b p i ,1 y t y ja 2pe xi jj 1 1 1 1 2 . 2 2 2 RemarkHere we discuss only the case when the kernel function k x, t is positive. We can generalize our methodfor any given continuous function k x, t in Equation(1);1) If k x, t is positive, we have obtained the convergence theorem.2) If k x, t is negative then let k x, t S x, t ,then the function S x, t is positive and we can applyxour method for the equation f x 0 S x, t y t dtwhich has the same solution as the Equation (1).3) If the function k x, t is neither of the above twocases, the function k x, t can be written as a sum oftwo positive functions wherek x, t k x, t k x, t Then Equation (1) becomesf x 0 k x, t y t dt 0 k x, t y t dtxxAnd hence the result is concluded in a similar fashion.6. Numerical ExamplesIn the following examples, we will solve several linearVolterra integral equations of the first kind and Fredholme-Volterra integral equations using coiflet of order 5and provide the absolute errors. The examples (1-3) arealso shown in [4] and the example 4 is presented in [5].We will compare our results with others and show thatour method has better approximations than other methods.Example 1Consider the integral Equation (1) with;e x e xx t, k x, t e and the exact solution isf x 2y x e x . The numerical results are presented in Table 1.Copyright 2013 SciRes. j , p t dt (31) a j , p t dtxi p j , p t dt a p y j 2 p We use the same idea in the proof of 5.1, and obtainthe following error estimate.j p j, p t y j 2p a j , p t dtbiExample 2Consider the integral Equation (1) with;f x 1 x e x , k x, t 1 x t and b 1, and theexact solution is y x xe x . The numerical results arepresented in Table 2.Example 3Consider the integral Equation (1) with;f x x sin x, k x, t cos x t and the exact solution is y x 2sin x . The numerical result are presented in Table 3.Example 4Consider the integral Equation (2) with;21g x x x 4 , k1 x, t k2 x, t xt ,33and the exact solution is y x x. The numerical results are presented in Table 4.Table 1. The absolute errors for example 1.xiExact 0.4065710.367837j 28.384E 71.911E 64.223E 61.897E 55.633E 71.522E 64.381E 63.106E 61.021E 54.466E 7Absolute errorsj 15.59E 71.638E 61.503E 81.562E 61.272E 62.271E 72.291E 61.647E 72.139E 67.265E 7j 06.788E 64.726E 61.799E 64.971E 72.261E 63.781E 62.572E 62.186E 61.111E 64.201E 5Table 2. The absolute errors for example 2.xiExact 90.3659150.367799j 21.381E 58.994E 61.052E 55.413E 54.37E 61.639E 51.066E 55.727E 63.425E 55.7E 6Absolute errorsj 11.28E 61.488E 61.815E 62.242E 61.689E 61.42E 74.466E 71.692E 68.073E 71.094E 6j 09.954E 67.303E 63.265E 61.059E 63.129E 64.349E 63.997E 64.136E 62.444E 68.08E 5AM
Y. Al-JARRAH, E.-B. LINTable 3. The absolute errors for example 3.xiExact 8850.7049940.841471j 28.384E 71.911E 64.223E 61.897E 55.633E 71.522E 64.381E 63.106E 61.021E 54.662E 7Absolute errorsj 12.019E 66.551E 61.739E 72.082E 62.535E 68.047E 75.533E 64.201E 73.581E 62.985E 6j 04.092E 54.573E 51.081E 62.818E 51.028E 62.235E 61.412E 59.272E 62.303E 53.251E 6holm-Volterra integral equations. We also prove convergence theorem for the numerical solution of Volterraintegral equations and Freholm-Volterra integral equations respectively. It would be interesting to extend theresults to two-dimensional case for the above mentionedequations and apply to some imaging problems.REFERENCES[1]E. B. Lin and N. Liu, “Legendre Wavelet Method forNumerical Solutions of Partial Differential Equations,”Numerical Methods of Partial Differential Equation, Vol.26, No. 1, 2010, pp. 81-94. doi:10.1002/num.20417[2]C. K. Chui, “In Introduction to Wavelets,” AcademicPress, Boston, 1992.[3]E. B. Lin and X. Zhou, “Coiflet Interpolation and Approximate Solutions of Elliptic Partial Differential Equations,” Methods for Partial Differential Equations, Vol.13, No. 4, 1997, pp. 302-320.[4]M. T. Rashad, “Numerical Solution of the Integral Equations of the First Kind,” Applied Mathematics and Computation, Vol. 145, No. 2-3, 2003, pp. 413-420.doi:10.1016/S0096-3003(02)00497-6[5]A. S. Shamloo, S. Shaker and A. Madadi, “NumericalSolution of Fredholm-Volterra Integral Equation by theSunc Function,” American Journal Computation Mathematics, Vol. 2, No. 2, 2012, pp. 136-142.doi:10.4236/ajcm.2012.22019Table 4. The absolute errors for example 4.xiExact 50.60.70.80.91j 23.348E 71.263E 71.905E 72.564E 81.316E 81.876E 76.735E 72.064E 72.589E 75.887E 7Absolute errorsj 11.032E 75.75E 83.789E 81.758E 78.553E 85.004E 73.977E 74.912E 74.063E 72.745E 7j 02.817E 72.971E 74.913E 84.506E 81.323E 71.243E 75.035E 84.879E 82.472E 77.36E 8209In the above tables, we use the notation E n whichde- notes 10 n and j denotes the level of MRA.7. Concluding RemarkIn this paper we have shown a better method in solvingVolterra integral equations of the first kind, and Fred-Copyright 2013 SciRes.AM
equations. An integral equation maybe interpreted as an analogue of a matrix equation which is easier to solve. There are many different ways to transform integral equations to linear systems. Many different methods have been used for solving Volterra integral equations and Freholm-
methods are available for approximating the Volterra integral equation. In particular, Huang[3] used the Taylor expansion of unknown function and obtained an approximate solution. Yang[4] proposed a method for the solution of integral equation using the Chebyshev polynomials, while Yousefi[5] presented a nu
2 Volterra Equations We need to develop solution methods for Volterra integral equations — we begin with an example. Example 2.1 (Inventory control) Suppose a shopkeeper knows that if goods are purchased at any given time then a fraction k(t) of the goods will remain tdays later. At wh
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of Volterra integral equations, called systems of Abel integral equations are studied. Historically, Abel is the first person who had studied integral equations, during the 1820 decade (Jerri, 1999; Linz, 1985). He obtained the following equation, when he was g
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Predator -Prey Model (Lotka -Volterra) Rabbits (in thousands) Foxes (in thousands) Lotka-Volterra (1 st order nonl
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