Numerical Solutions of Volterra Integral Equations Using Galerkin method with Hermite Polynomials

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1 Proceedings of the 3 International Conference on Applied Mathematics and Computational Methods in Engineering Numerical of Volterra Integral Equations Using Galerkin method with Hermite Polynomials M. M. Rahman Department of Mathematics, Islamic University, Kushita-73, Bangladesh mizan_iu@yahoo.com Abstract-In the present paper, we solve numerically Volterra integral equations of second kind, by the well known Galerkin method. For this, we derive a simple and efficient matri formulation using Hermite polynomials as trial functions. Numerical eamples are considered to verify the effectiveness of the proposed derivations and numerical solutions are compared with the eisting methods available in the literature. Keywords- Volterra integral equations, Galerkin method, Hermite polynomials.. INTRODUCTION Many problems of mathematical physics can be started in the form of integral equations. These equations also occur as reformulations of other mathematical problems such as partial differential equations and ordinary differential equations. Numerical simulation in engineering science and in applied mathematics has become a powerful tool to model the physical phenomena, particularly when analytical solutions are not available then very difficult to obtain. Therefore, the study of integral equations and methods for solving them are very useful in application. In recent years, there has been a growing interest in the Volterra integral equations arising in various fields of physics and engineering [], e.g., potential theory and Dirichlet problems, electrostatics, the particle transport problems of astrophysics and reactor theory, contact problems, diffusion problems, and heat transfer problems. Some valid numerical methods, for solving Volterra equations using various polynomials [], have been developed by many researchers. Very recently, Maleknejad et al [3] and Tari and Shahmorad [4] used computational method for solving two-dimensional linear Fredholm integral equations of the second kind, Shahsavaran [5] solved by Block Pulse functions and Taylor Epansion method. Taylor polynomials were also used by Bellour and Rawashdeh [6] and Wang [7] with computer algebra. Bernstein polynomials were used for the solution of second order linear and first order non-linear differential equations by Bhatti and Bracken [8]. These polynomials have been also used for solving Fredholm integral equations of second kind by Shirin and Islam [9]. Amaratunga [] describe an augmented Galerkin technique for the numerical solution for one-dimension partial differential equation. Collocation and Fredholm integral equations are studied by Hanna and Kucera []. In [] a solution of Integral Equation via Laguerre Polynomials is proposed. Rashed [3] studied a numerical solution of the integral equations of the first kind. However, in this paper a very simple and efficient Galerkin numerical method is proposed with Hermite polynomials as trial functions. The formulation is derived to solve the linear Volterra integral equations of second kind having regular as well as weakly singular kernels, in details, in Section 3. In Section, we give a short introduction of Hermite polynomials. Finally, four eamples of different kinds of Volterra integral equations are given to verify the proposed formulation. The results of each numerical eample indicate the convergence and error analysis are discussed illustrate the efficiency of the method.. Hermite Polynomials The general form of the Hermite polynomials of nth degree is defined by Using MATLAB code, the first few Hermite polynomials are given below:,, Now the first si Hermite polynomials over the interval [, ] are shown in Fig.., 76

2 Proceedings of the 3 International Conference on Applied Mathematics and Computational Methods in Engineering (4) Then the Galerkin equations are obtained by multiplying both sides of by and then integrating with respect to from we have Fig.. Graph of first 6 Hermite polynomials over the interval [-, ] 3. Formulation of Integral Equation in Matri Form Let us consider the Volterra integral equation (VIE) of the first kind Since in each equation, there are two integrals. The inner integrand of the left side is a function of, and and is integrated with respect to from As a result the outer integrand becomes a function of only and integration with respect to from yields a constant. Thus for each we have a linear equation with unknowns. Finally represents the system of linear equations in unknowns, are given by () is the unknown function, to be determined, the kernel, is a continuous or discontinuous and square integrable function being the known function satisfying Now we use the technique of Galerkin method, to find an approimate solution of. For this, we assume that Now the unknown parameters are determined by solving the system of equations and substituting these values of parameters in we get the approimate solution of the integral equation (). Now, we consider the Volterra integral equation (VIE) of the second kind (3) are Hermite polynomials of degree is the unknown function to be determined, the kernel, is a continuous or discontinuous and square integrable function and being the known function and is the constant. Then applying the same procedure as described above, we obtain defined in equation and are unknown parameters, to be determined. Substituting into, we get Now the unknown parameters solving the system of equations are determined by and substituting 77

3 Proceedings of the 3 International Conference on Applied Mathematics and Computational Methods in Engineering these values of parameters in we get the approimate solution of the integral equation (7). The maimum absolute error for this formulation is defined by Maimum absolute error 4. Numerical Eamples In this paper, we illustrate the Galerkin method with the help of four numerical eamples, which include second kind with regular kernels and weakly singular kernels, available in the eisting literature [- 4]. The computations, associated with the eamples, are performed by MATLAB. The convergence of each linear Volterra integral equations is calculated by E = ~ ϕ ~ n+ ( ) ϕ ( ) δ n ~ ϕ n ( ) denotes the approimate solution by the proposed method using nth degree polynomial 6 approimation and δ varies from for n. Eample : Consider the Volterra integral equations of second kind ) = e + t) dt (9) The eact solution is ϕ ( ) = e ( + ) been shown in Table for n = 4. Results have. Also Fig. shows the eact and approimate solution for n =, 4 and 5. The maimum absolute errors obtain in the order of 4 n = are shown in table. for 4 TABLE. Computed of eamples Eample for n=4 Eact Appro E E E E E E E E E E-3 Eample : Consider the Volterra integral equations of second kind ϕ ( ) + 3 t ) =, t dt 3 () ϕ ( ) = 3 ( e The eact solution is ). Using the formula derived in the previous section and solving the system for, we get the approimate solution is ~ ϕ ( ) = 3 ( e ), which is similar with eact solution. On the eperimental, we found that the accuracy is for n = 4. Fig.. Eact solution and Numerical solution of eample for n =, 4, 5 78

4 Proceedings of the 3 International Conference on Applied Mathematics and Computational Methods in Engineering < Abs-diff >.5.5 Eact Approimation for n= Approimation for n=4 Eample 3: Consider the weakly singular Volterra integral equations of second kind ) t) dt = 7 ( ( t) () The eact solution is. Using the above system (8) for n 8, we get the approimate solution ) < > Fig. 3. Eact solution and Numerical solution of eample for n =, 4 is, which is the eact solution of the integral equation. From table 3, the accuracy is 3 found nearly the order of for n = 6.. TABLE. Computed of eamples Eample for n=4 Eact Appro E E- < Abs-diff > Eact Approimation for n=4 Approimation for n= E E E E E E E E < > Fig. 4. Eact solution and Numerical solution of eample 3 for n = 4, 6 TABLE3. Computed of eamples 3 Eample 3, for n=6 Eact Appro E E E+ 79

5 Proceedings of the 3 International Conference on Applied Mathematics and Computational Methods in Engineering E- Eact Appro E E E E E E E E E E E E-5 Eample 4: Consider the weakly singular Volterra integral equations of second kind 6 ϕ ( ) + t) dt = + 5 ( t) () The eact solution is. We have been shown that the result in Table 4 for n =. Also Fig. 5 shows the eact and approimate solution for n =, 4 and 6. The 5 maimum absolute errors obtain in the order of for n = are found in Table 4. < Abs-diff > Eact Approimation for n= Approimation for n=4 Approimation for n= < > Fig. 5. Eact solution and Numerical solution of eample 4 for n =, 4 and 6 TABLE4. Computed of eamples 4 Eample 4 for n= E E E E-4 5. Conclusions In this paper, a very simple and efficient Galerkin method based on the Hermite polynomial basis tool has been developed to solve second kind Volterra integral equations. The numerical results obtained by the proposed method are in good agreement with the eact solutions. We have shown that the numerical solutions coincide with the eact solutions even a few numbers of polynomials are used to find the approimation. We also notice that the accuracy increase with increase the number of polynomials in the approimations, which is shown in Table, Table, Table 3 and Table 4. We may realize that this method may be applied to solve integral to find the desired accuracy. REFERENCES [] E. Babolian and L. M. Delves, An augmented Galerkin method for first kind Fredholm equations, Journal of the Institute of Mathematics and Its Applications, Vol. 4, No., pp , 979. [] N. Saran, S. D. Sharma and T. N. Trivedi, Special Functions, Seventh edition, Pragati Prakashan, ". [3] K. Maleknejad, E. Hashemizadeh and R. Ezzati, A new approach to the numerical solution of Volterra integral equations by using Bernstein s approimation, Commun. Nonlinear Sci. Numer. Simulat, Vol. 6, pp ,. [4] A Tari and S. Shahmorad, A computational method for solving two-dimensional linear Fredholm integral equations of the second kind, ANZIAM J, Vol. 49, No. 4, pp , 8. [5] A. Shahsavaran, Numerical approach to solve second kind Volterra integral equations of Abel type using Block-Pulse functions and Taylor epansion by collocation method, Appl. Mathe. Sci., Vol. 5, pp ,. [6] Azzeddine Bellour and E. A. Rawashdeh, Numerical solution of first kind integral equations by using Taylor polynomials, J. Inequal. Speci. Func, Vol., pp. 3-9,. 8

6 Proceedings of the 3 International Conference on Applied Mathematics and Computational Methods in Engineering [7] Weiming Wang, A mechanical algorithm for solving the Volterra integral equation, Appl. Mathe. Comput., Vol. 7, pp , 6. [8] M. Idrees Bhatti and P. Bracken, of differential equations in a Bernstein polynomial basis, J. Comput. Appl. Mathe., Vol. 5, pp. 7 8, 7. [9] A. Shirin and M. S. Islam, Numerical of Fredholm integral equations using Bernstein polynomials, J. Sci. Res., Vol., No., pp. 64 7,. [] K. Amaratunga, Wevlet-Galerkin solution for onedimension partial differential equation, Int. j.numer.methods eng., Vol. 37, pp , 994. [] G. Hanna, J. Roumeliotis, and A. Kucera, Collocation and Fredholm integral equations of the first kind, Journal of Inequalities in Pure and Applied Mathematics, Vol. 6, No. 5, pp. 8, 5. [] C. Hwang, Y. P. Shih, Solution of Integral Equation via Laguerre Polynomials, J. Comput. Elect. Engin, Vol. 9, pp. 3-9, 98. [3] M. T. Rashed, Numerical solution of the integral equations of the first kind, Applied Mathematics and Computation, Vol. 45, No., pp. 43-4, 3. 8