Numerical solution of high-order linear integro-differential equations with variable coefficients using two proposed schemes for rational Chebyshev functions

Abstract

In this paper, a rational Chebyshev (RC) collocation method is presented to solve high-order linear Fredholm integro-differential equations with variable coefficients under the mixed conditions, in terms of RC functions by two proposed schemes. The proposed method converts the integral equation and its conditions to matrix equations, by means of collocation points on the semi–infinite interval, which corresponding to systems of linear algebraic equations in RC coefficients unknowns. Thus, by solving the matrix equation, RC coefficients are obtained and hence the approximate solution is expressed in terms of RC functions. Numerical examples are given to illustrate the validity and applicability of the method. The proposed method numerically compared with others existing methods as well as the exact solutions where it maintains better accuracy.

Keywords

• Rational Chebyshev functions,Collocation method

References

1. A. Avudainayagam, C. Vani, Wavelet-Galerkin method for integro-differential equations, Appl. Numer. Math. 32 (2000) 247–254.
2. M. T. Rashed, Lagrange interpolation to compute the numerical solutions of differential, integral and integro differential equations, Appl. Math. Comput., 151(2004) 869- 878.
3. M. Sezer, M. Gülsu, A new polynomial approach for solving difference and Fredholm integro-difference equations with mixed argument, Appl. Math. Comput. 171(2005) 332–344.
4. A. Karamete, M. Sezer, A. Taylor, A collocation method for the solution of linear integro-differential equations, Intern. J. Computer Math., 79(9) (2002) 987–1000.
5. M. Dehghan, A. Saadatmandi, Chebyshev finite difference method for Fredholm integro-differential equation, Inter. J. Comp. Math. , 85(1) (2008)123–130.
6. M. Sezer, M. Gülsu, Y. Öztürk, A new collocation method for solution of mixed linear integro-differential-difference equations, Appl. Math. Comput. 216(2010) 2183–2198.
7. M. Sezer, M. Gülsu, G. Yüksel, A Chebyshev polynomial approach for high-order linear Fredholm-Volterra integro-differential equations, Gazi University J. Sci., 25(2)(2012)393-401.
8. A. R. Vahidi, E. Babolian, Gh. Cordshooli , Z. Azimzadeh , Numerical Solution of Fredholm Integro-Differential Equation by Adomian’s Decomposition Method, Int. J. Math. Anal., 3(36) (2009)1769 – 1773.

9. S. M. El-Sayed, M. R. Abdel-Aziz, A comparison of Adomian’s decomposition method and Wavelet-Galerkin method for solving integro-differential equations, Appl. Math. Comput. 136(2003)151–159.
10. P. Darania, A. Ebadian, A method for the numerical solution of the integro-differential equations, Appl. Math. Comput. 188 (2007) 657–668.
11. S. H. Behiry, S. I. Mohamed, Solving high-order nonlinear Volterra-Fredholm integro-differential equations by differential transform method, Natural Science, 4(8) (2012)581-587.
12. S. Yalcinbas, M. Sezer and Hüseyin Hilmi Sorkun, Legendre polynomial solutions of high-order linear Fredholm integro-differential equations, Appl. Math. Comput. 210(2009) 334–349.
13. H. Danfu, Sh. Xufeng, Numerical solution of integro-differential equations by applying CAS Wavelet operational matrix of integration, Appl. Math. Comput. 194(2007) 460-466.
14. K. Yildirim, A Solution Method For Solving Systems Of Nonlinear Pdes, World Appl. Science J, 18 (11) (2012) 1527-1532.
15. K. Yildirim, M. Bayram, Approximates Method for Solving an Elasticity Problem of Settled of the Elastic Ground with Variable Coefficients, Appl. Math. & Info. Science, 7(4) (2013) 1351-1357.
16. M. Sezer, M. Gulsu B. Tanay, Rational Chebyshev collocation method for solving higher-order linear ordinary differential equations, Wiley Online Library, DOI 10.1002/num.20573, 2010.
17. [17] M. A. Ramadan, K. R. Raslan, M. A. Nassar, An approximate analytical solution of higher order linear differential equations with variable coefficients using improved rational Chebyshev collocation method, Appl. Comput. Math. 3(6) (2014) 315-322.