A semi-analytical approach to solve integro-differential equations

In this work, we present a semi-analytical method for solving integro-differential equations under multi-point or mixed boundary conditions. The proposed method solves linear and nonlinear FredholmVolterra integro-differential equations. A convergence analysis of the proposed method is directly examined. Numerical examples are worked out to demonstrate the main results. Moreover, proper graphs are provided to confirm the efficiency and the accuracy of the proposed scheme. We show that with a few number of obtained approximating terms, we achieve a high accuracy level of the obtained results. However, increasing the number of approximating terms, yields a significant decrease of the error of the approximation. The proposed method is very useful, reliable, and flexible for solving different kinds of integro-differential equations.

[1]  Mehmet Sezer,et al.  The approximate solution of high-order linear Volterra-Fredholm integro-differential equations in terms of Taylor polynomials , 2000, Appl. Math. Comput..

[2]  E. Yusufoglu,et al.  IMPROVED HOMOTOPY PERTURBATION METHOD FOR SOLVING FREDHOLM TYPE INTEGRO-DIFFERENTIAL EQUATIONS , 2009 .

[3]  M. H. Aliabadi,et al.  A matrix formulation of the Tau Method for Fredholm and Volterra linear integro-differential equations , 2002 .

[4]  Mehmet Sezer,et al.  A Bessel collocation method for numerical solution of generalized pantograph equations , 2012 .

[5]  Mehmet Sezer,et al.  Chelyshkov collocation method for a class of mixed functional integro-differential equations , 2015, Appl. Math. Comput..

[6]  Khadijeh Sadri,et al.  A new operational approach for numerical solution of generalized functional integro-differential equations , 2015, J. Comput. Appl. Math..

[7]  Bo Zhang,et al.  A simple Taylor-series expansion method for a class of second kind integral equations , 1999 .

[8]  Xufeng Shang,et al.  Numerical solution of integro-differential equations by using CAS wavelet operational matrix of integration , 2007, Appl. Math. Comput..

[9]  Mohsen Razzaghi,et al.  Legendre wavelets method for the nonlinear Volterra-Fredholm integral equations , 2005, Math. Comput. Simul..

[10]  Abdul-Majid Wazwaz,et al.  The combined Laplace transform-Adomian decomposition method for handling nonlinear Volterra integro-differential equations , 2010, Appl. Math. Comput..

[11]  Han Dan-fu,et al.  Numerical solution of integro-differential equations by using CAS wavelet operational matrix of integration , 2007 .

[12]  S. Shahmorad,et al.  Numerical solution of a class of Integro-Differential equations by the Tau Method with an error estimation , 2003, Appl. Math. Comput..

[13]  Robert M. Corless,et al.  Compact finite difference method for integro-differential equations , 2006, Appl. Math. Comput..

[14]  Farshid Mirzaee,et al.  Numerical solution of integro‐differential equations by using rationalized Haar functions method , 2006 .

[15]  Weiming Wang,et al.  A new algorithm for integral of trigonometric functions with mechanization , 2005, Appl. Math. Comput..

[16]  Chun-Hui Hsiao,et al.  Hybrid function method for solving Fredholm and Volterra integral equations of the second kind , 2009 .

[17]  Sergiy Yu. Reutskiy,et al.  The backward substitution method for multipoint problems with linear Volterra-Fredholm integro-differential equations of the neutral type , 2016, J. Comput. Appl. Math..

[18]  Kyunghoon Kim,et al.  A novel semi-analytical approach for solving nonlinear Volterra integro-differential equations , 2015, Appl. Math. Comput..

[19]  M. T. Rashed.  Numerical solution of functional differential, integral and integro-differential equations , 2004, Appl. Math. Comput..

[20]  Weigao Ge,et al.  Existence of solutions of boundary value problems with integral boundary conditions for second-order impulsive integro-differential equations in Banach spaces , 2010, J. Comput. Appl. Math..

[21]  S. Shahmorad,et al.  Numerical solution of the system of Fredholm integro-differential equations by the Tau method , 2005, Appl. Math. Comput..

[22]  Ibrahim Özkol,et al.  Solution of boundary value problems for integro-differential equations by using differential transform method , 2005, Appl. Math. Comput..

[23]  Suayip Yüzbasi,et al.  A collocation approach for solving high-order linear Fredholm-Volterra integro-differential equations , 2012, Math. Comput. Model..

[24]  Esmail Babolian,et al.  Comparison between the homotopy perturbation method and the sine-cosine wavelet method for solving linear integro-differential equations , 2007, Comput. Math. Appl..

[25]  Khosrow Maleknejad,et al.  Numerical solution of linear Fredholm integral equation by using hybrid Taylor and Block-Pulse functions , 2004, Appl. Math. Comput..

[26]  Ali R. Ansari,et al.  A new iterative technique for solving nonlinear second order multi-point boundary value problems , 2011, Appl. Math. Comput..

[27]  Mehmet Sezer,et al.  Chebyshev polynomial solutions of systems of higher-order linear Fredholm-Volterra integro-differential equations , 2005, J. Frankl. Inst..

[28]  Esmail Babolian,et al.  Numerical solution of linear integro-differential equation by using sine-cosine wavelets , 2006, Appl. Math. Comput..

[29]  Willi Törnig,et al.  Numerical Treatment of Differential Equations in Applications , 1978 .

[30]  M. Reihani,et al.  Rationalized Haar functions method for solving Fredholm and Volterra integral equations , 2007 .

[31]  Eduardo L. Ortiz,et al.  An operational approach to the Tau method for the numerical solution of non-linear differential equations , 1981, Computing.

[32]  Weiming Wang,et al.  An algorithm for solving the high-order nonlinear Volterra-Fredholm integro-differential equation with mechanization , 2006, Appl. Math. Comput..

[33]  E. Ortiz,et al.  On the numerical solution of nonlinear and functional differential equations with the tau method , 1978 .