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Yalçinbaş, Salih; Sezer, Mehmet; Sorkun, Hüseyin Hilmi

Legendre polynomial solutions of high-order linear Fredholm integro-differential equations. (English) Zbl 1162.65420

Appl. Math. Comput. 210, No. 2, 334-349 (2009).
Summary: A Legendre collocation matrix method is presented to solve high-order Linear Fredholm integro-differential equations under the mixed conditions in terms of Legendre polynomials. The proposed method converts the equation and conditions to matrix equations, by means of collocation points on the interval [-1,1], which corresponding to systems of linear algebraic equations with Legendre coefficients. Thus, by solving the matrix equation, Legendre coefficients and polynomial approach are obtained. Also examples that illustrate the pertinent features of the method are presented and by using the error analysis, the results are discussed.

Cited in 61 Documents

MSC:

65R20 Numerical methods for integral equations

Keywords:

Legendre polynomials and series; Fredholm integro-differential equations; collocation points
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References:

[1] Yalçınbaş, S.; Sezer, M., The approximate solution of high-order linear Volterra-Fredholm integro-differential equations in terms of Taylor polynomials, Appl. Math. Comput., 112, 291-308 (2000) · Zbl 1023.65147
[2] Yalçınbaş, S., Taylor polynomial solutions of nonlinear Volterra-Fredholm integral equations, Appl. Math. Comput., 127, 195-206 (2002) · Zbl 1025.45003
[3] Ren, Y.; Zhang, B.; Qiao, H., A simple Taylor-series expansion method for a class of second kind integral equations, J. Comput. Appl. Math., 110, 15-24 (1999) · Zbl 0936.65146
[4] Maleknejad, K.; Mahmoudi, Y., Numerical solution of linear Fredholm integral equation by using hybrid Taylor and Block-Pulse functions, Appl. Math. Comput., 149, 799-806 (2004) · Zbl 1038.65147
[5] Maleknejad, K.; Aghazadeh, N., Numerical solutions of Volterra integral equations of the second kind with convolution kernel by using Taylor-series expansion method, Appl. Math. Comput., 161, 3, 915-922 (2005) · Zbl 1061.65145
[6] Delves, L. M.; Mohamed, J. L., Computational Methods for Integral Equations (1985), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0592.65093
[7] Ortiz, E. L.; Samara, L., An operational approach to the tau method for the numerical solution of nonlinear differential equations, Computing, 27, 15-25 (1981) · Zbl 0449.65053
[8] Pour-Mahmoud, J.; Rahimi-Ardabili, M. Y.; Shahmorad, S., Numerical solution of the system of Fredholm integro-differential equations by the tau method, Appl. Math. Comput., 168, 465-478 (2005) · Zbl 1082.65600
[9] Hosseini, S. M.; Shahmorad, S., Numerical solution of a class of integro-differential equations by the tau method with an error estimation, Appl. Math. Comput., 136, 559-570 (2003) · Zbl 1027.65182
[10] Razzaghi, M.; Yousefi, S., Legendre wavelets method for the nonlinear Volterra Fredholm integral equations, Math. Comput. Simul., 70, 1-8 (2005) · Zbl 1205.65342
[11] El-Mikkawy, M. E.A.; Cheon, G. S., Combinatorial and hypergeometric identities via the Legendre polynomials – a computational approach, Appl. Math. Comput., 166, 181-195 (2005) · Zbl 1073.65019
[12] Marzban, H. R.; Razzaghi, M., Optimal control of linear delay systems via hybrid of block-pulse and Legendre polynomials, J. Franklin Inst., 341, 279-293 (2004) · Zbl 1070.93028
[13] Maleknejad, K.; Tavassoli Kajani, M., Solving second kind integral equation by Galerkin methods with hybrid Legendre and Block-Pulse functions, Appl. Math. Comput., 145, 623-629 (2003) · Zbl 1101.65323
[14] Everitt, W. N.; Litlejohn, L. L.; Wellman, R., Legendre polynomials, Legendre-Stirling numbers and the left-definite spectral analysis of the Legendre differential expressions, J. Comput. Appl. Math., 148 (2002) · Zbl 1014.33003
[15] Spiegel, M. R., Theory and Problems of Fourier Analysis, Schaum’s Outline Series (1994), McGraw-Hill Inc.: McGraw-Hill Inc. New York
[16] Fox, L.; Parker, I., Chebyshev Polynomials in Numerical Analysis (1968), Clarendon Press: Clarendon Press Oxford · Zbl 0153.17502
[17] Morris, A. G.; Horner, T. S., Chebyshev Polynomials in Numerical Analysis (1968), Clarendon Press: Clarendon Press Oxford · Zbl 0386.65040
[18] Elbarbary, E. M.E., Legendre expansion method for the solution of the second-and fourth-order elliptic equations, Math. Comput. Simul., 59, 389-399 (2002) · Zbl 1004.65120
[19] Streltsov, I. P., Approximation of Chebyshev and Legendre polynomials on discrete point set to function interpolation and solving Fredholm integral equations, Comput. Phys. Commun., 126, 178-181 (2000) · Zbl 0963.65143
[20] Sezer, M., Taylor polynomial solutions of Volterra integral equations, Int. J. Math. Educ. Sci. Technol., 25, 5, 625-633 (1994) · Zbl 0823.45005
[21] Sezer, M., A method for approximate solution of the second order linear differential equations in terms of Taylor polynomials, Int. J. Math. Educ. Sci. Technol., 27, 6, 821-834 (1996) · Zbl 0887.65084
[22] Nas, S.; Yalçınbaş, S.; Sezer, M., A Taylor polynomial approach for solving high-order linear Fredholm integro-differential equations, Int. J. Math. Educ. Sci. Technol., 31, 2, 213-225 (2000) · Zbl 1018.65152
[23] Karamete, A.; Sezer, M., A Taylor collocation method for the solution of linear integro-differential equations, Int. J. Comput. Math., 79, 9, 987-1000 (2002) · Zbl 1006.65144
[24] Gulsu, M.; Sezer, M., A method for the approximate solution of the high-order linear difference equations in terms of Taylor polynomials, Int. J. Comput. Math., 82, 5, 629-642 (2005) · Zbl 1072.65164
[25] Gülsu, M.; Sezer, M.; Tanay, B., A matrix method for solving high-order linear difference equations with mixed argument using hybrid Legendre and Taylor polynomials, J. Franklin Inst., 343, 647-659 (2006) · Zbl 1106.65110
[26] A. Akyüz, M. Sezer, A Taylor polynomial approach for solving high-order linear Fredholm integro-differential equations in the most general form, Int. J. Comp. Math. 527-539 (2007).; A. Akyüz, M. Sezer, A Taylor polynomial approach for solving high-order linear Fredholm integro-differential equations in the most general form, Int. J. Comp. Math. 527-539 (2007). · Zbl 1118.65129
[27] Maleknejad, K.; Nouri, K.; Yousefi, M., Discussion on convergence of Legendre polynomial for numerical solution of integral equations, Appl. Math. Comput., 193, 335-339 (2007) · Zbl 1193.65223
[28] Hosseini, S. M.; Shahmorad, S., Numerical piecewise approximate solution of Fredholm integro-differential equations by the Tau method, Appl. Math. Model, 29, 1005-1021 (2005) · Zbl 1099.65136
[29] Canuto, C.; Hussaini, M. Y.; Quarteroni, A.; Zang, T. A., Spectral Methods on Fluid Dynamics (1988), Springer-Verlag · Zbl 0658.76001
[30] Atkinson, Kendall E., An Introduction to Numerical Analysis (1978), Jon Wiley and Sons · Zbl 0402.65001
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