Solving volterra integral equation via nonlinear programming
 Abstract
 Keywords
 References

Abstract
In this paper we propose an approach to find approximate solution to the nonlinear Volterra integral equation of the second type through a nonlinear programming technique by firstly converting the integral equation into a least square cost function as an objective function for an unconstrained nonlinear programming problem which solved by a nonlinear programming technique (The preconditioned limited memory quasiNewton conjugates, gradient method) and as far as we read this is a new approach in the ways of solving the nonlinear Volterra integral equation. We use Maple 11 software as a tool for performing the suggested steps in solving the examples.

Keywords
Volterra integral equations; Optimization ; Non linear programming

References
[1] A.M. Wazwaz, Linear and Nonlinear Integral Equations Methods and Applications, springer, 2011.
[2] B. B. E. Hashemizadeha, K. Maleknejada, Hybrid functions approach for the nonlinear volterrafredholm integral equations, Procedia Computer Science 3 (2011) 1189–1194.
[3] R. E. K. Maleknejad, E. Hashemizadeh, A new approach to the numerical solution of volterra integral equations by using bernstein’s approximation, Communications in Nonlinear Science and Numerical Simulation 16 (2011) 647–655.
[4] B. N. M. Subhra Bhattacharya, Use of bernstein polynomials in numerical solutions of volterra integral equations, Applied Mathematical Sciences 2 (2008) 1773 – 1787.
[5] A. Shirin, M. S. Islam, Numerical solutions of fredholm integral equations using bernstein.
[6] A. Shahsavaran, Computational method to solve nonlinear integral equations using block pulse functions by collocation method, Applied Mathematical Sciences 5 (2011) 3211 – 3220.
[7] E. A. R. Azzedine Bellour, Numerical solution of first kind integral equations by using taylor polynomials.
[8] W. Wang, A mechanical algorithm for solving the volterra integral equation, Applied Mathematics and Computation 172 (2006) 1323–1341.
[9] J. A. Othman, R. S. Kareem, Solving nonlinear fredholm integral equation of the second type via nonlinear programming techniques, International Journal of Applied Mathematics 29 (2012) 1285–1262.
[10] A. T. Lonseth, Approximate solution of fredholmtype integral equations, Bulletin of the American Mathematical Society 60 (1954) 415– 430.
[11] A. J. Jerri, Introduction to Integral Equations with Applications, John Wiley Sons Inc, 1999.
[12] M. K. H. M. K. A. L. N. A. M. M. Rahman, M. A. Hakim, Numerical solutions of volterra integral equations of second kind with the help of chebyshev polynomials, Annals of Pure and Applied Mathematics 1 (2012) 158–167.
[13] M. T. K. M. Ghasemi, E. Babolian, Numerical solutions of the nonlinear volterrafredholm integral equations by using homotopy perturbation method, Appl. Math. Comput. 188 (2007) 446–449.
[14] C. Minggen, D. Hong, Representation of exact solution for the nonlinear volterrafredholm integral equations, Appl. Math. Comput. 182 (2006) 1795– 1802.
[15] M.Razzaghi, Y.ordokhani, Solution of nonlinear volterrohammerstien integral equation via rationalized haar function, Mathematical Problem in Engineering 7 (2001) 205–219.
[16] S. Yalcinbas, Taylor polynomial solutions of nonlinear volterrafredholm integral equations, Appl. Math. Comput. 127 (2002) 195– 206.
[17] M. Zarebnia, A numerical solution of nonlinear volterrafredholm integral equation, Journal of Applied Analysis and Computation 3 (2013) 95–104.
[18] M. R. Sepehrian, Singleterm walsh series method for the volterra integrodifferential equations, Engi. Anal. Boun. Elem. 28 (2004) 1315–1319.
[19] M. Rashed, Numerical solution of functional differential, integral and integrodifferential equations, Appl. Numer. Math. 156 (2004) 485–492.
[20] R. M. H. Brunner, A. Makroglou, Mixed interpolation collocation methods for first and second volterra integrodifferential equations with periodic solution, Appl. Numer. Math. 23 (1997) 381–402.
[21] S. X. E. Deeba, S.A. Khuri, An algorithm for solving a nonlinear integrodifferential equation, Appl. Numer. Math. 115 (2000) 123–131.
[22] W. P. Gerald. C. F., Applied Numerical Analysis, Addison Wesley, 1984.
[23] J. N. . S. J. Wright, Numerical Optimization, SpringerVerlag, 1999.
[24] W. W.Hager, H. Zhang, The limited memory conjugate gradient method, SIAM J. OPTIM. 23 (2013) 2150 – 2168.
[25] A. S. Igor Griva, Stephen G. Nash, Linear and Nonlinear Optimization, 2nd Edition, SIAM BOOK, 2008.

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