Document Type : Original Article

Author

Department of Applied Mathematics and Computer Science, Faculty of Mathematical Sciences, University of Guilan, Rasht, Iran

Abstract

In this paper an approximate analytical method for solving a class of two-point boundary value problems for fourth order integro-differential equations is presented. The method is based upon the Laplace transform, perturbation technique and polynomial series. Theoretical considerations are discussed. Illustrative examples have been discussed to demonstrate the validity and applicability of the technique. The results show that the method is of high accuracy and efficient for solving integro-differential equations.

Keywords

[1] Kythe, P.K., Puri, P. (1992). Computational methods for linear integral equations, University of New Orleans, New Orleans.
[2] Wazwaz, A.M. (2006). A First Course in Integral Equations, World Scientific, River Edge, NJ, 1997.
[3] M. Dehghan, Solution of a partial integro-differential equation arising from viscoelasticity, Int. J. Comput. Math. 83, 123-129.
[4] Forbes, L.K., Crozier, S. Doddrell, D.M. (1997). Calculating current densities and fields produced by shielded magnetic resonance imaging probes, SIAM J. Appl. Math. 57, 401-425.
[5] Saadatmandi, A. Dehghan, M. (2010).  Numerical solution of the higher-order linear Fredholm integro-differential-difference equation with variable coefficients, Computers and Mathematics with Applications, 59, 2996-3004.
[6] Biazar, J., Aminikhah, H., Eslami, M. (2009). He’s homotopy perturbation method for systems of integro-differential equations, Chaos, Solitons and Fractals, 39,1253-1258.
[7] Yildirim, A. (2008) . Solution of BVPs for fourth-order integro-differential equations by using homotopy perturbation method, Computers and Mathematics with Applications, 56, 3175-3180.
[8] He, J.H. (1999). Homotopy perturbation technique, Computational Methods in Applied Mechanics and Engineering, 178, 257-262.
[9] Maleknejad, K., Mirzaee, F., Abbasbandy, S. (2005). Solving linear integro-differential equations system by using rationalized Haar function method, Applied Mathematics and Computation, 155, 317-328.
[10] Avudainayagam, A., Vani, C. (2000). Wavelet-Galerkin method for integro-differential equations, Applied Numerical Mathematics, 32, 247-254.
[11] Arikoglu, A., Ozkol, I. (2005). Solution of boundary value problems for integro-differential equations by using differential transform method, Applied Mathematics and Computation, 168, 1145-1158.
[12] Sweilam, N.H. (2007). Fourth order integro-differential equations using variational iteration method, Computers and Mathematics with Applications, 54, 1086-1091.
[13] He, J.H. (2007). Variational iteration method: New development and applications, Computers and Mathematics with Applications, 54, 881-894.
[14] Rashed, M. T. (2004). Lagrange interpolation to compute the numerical solutions of differential, integral and integro-differential equations, Applied Mathematics and computation, 151, 869-878.
[15] Yalcinbas, S., Sezer, M. (2000). The approximate solution of high-order linear Volterra-Fredholm integro-differential equations in terms of Taylor polynomials, Appl. Math. Comput. 112 , 291-308.
[16] Dascioghlu, A. A.,  Sezer, M. (2005). Chebyshev polynomial solutions of systems of higher order linear Fredholm-Volterra integro-differential equations, Journal of the Franklin Institute, 342, 688-701.
[17] Kajani, M. T., Ghasemi, M., Babolian, E. (2006). Numerical solution of linear integro-differential equation by using sine-cosine wavelets, Applied Mathematics and Computation, 180, 569-574.
[18] Aminikhah, H., Salahi, M. (2010). A new analytical method for solving systems of Volterra integral equations, International Journal of Computer Mathematics, 87, 1142–1157.
[19] Aminikhah, H., Salahi, M. (2009). A New HPM for Integral Equations, Applications and Applied Mathematics: An International Journal, 4, 122-133.
[20] Vahidi, A. R.,  Babolian, E.,  Cordshooli, G. A.,  Azimzadeh, Z. (2009).  Numerical solution of Fredholm integro-differential equation by Adomian decomposition method, Int. Journal of Math. Analysis, 36, 1769-1773.
[21] Hashim, I. (2006). Adomian decomposition method for solving BVPs for fourth-order integro-differential equations, Journal of Computational and Applied Mathematics, 193, 658-664.
[22] Ghorbani, A. (2009). Beyond Adomian polynomials: He polynomials. Chaos, Solitons & Fractals, 39, 1486-1492.