The spectral method for solving systems of Volterra integral equations

This paper presents a high accurate and stable Legendre-collocation method for solving systems of Volterra integral equations (SVIEs) of the second kind. The method transforms the linear SVIEs into the associated matrix equation. In the nonlinear case, after applying our method we solve a system of nonlinear algebraic equations. Also, sufficient conditions for the existence and uniqueness of the Linear SVIEs, in which the coefficient of the main term is a singular (or nonsingular) matrix, have been formulated. Several examples are included to illustrate the efficiency and accuracy of the proposed technique and also the results are compared with the different methods. All of the numerical computations have been performed on a PC using several programs written in MAPLE 13.

[1]  L. Delves,et al.  Computational methods for integral equations: Frontmatter , 1985 .

[2]  Ivan P. Gavrilyuk,et al.  Collocation methods for Volterra integral and related functional equations , 2006, Math. Comput..

[3]  M. V. Bulatov.  Regularization of singular systems of Volterra integral equations , 2002 .

[4]  Jan S. Hesthaven,et al.  Spectral Methods for Time-Dependent Problems: Contents , 2007 .

[5]  Jalil Rashidinia,et al.  Approximate solution of systems of Volterra integral equations with error analysis , 2010, Int. J. Comput. Math..

[6]  Jafar Biazar,et al.  He's homotopy perturbation method for solving systems of Volterra integral equations of the second kind , 2009 .

[7]  Jafar Biazar,et al.  Solution of a system of Volterra integral equations of the first kind by Adomian method , 2003, Appl. Math. Comput..

[8]  Jafar Biazar,et al.  Chebyshev wavelets approach for nonlinear systems of Volterra integral equations , 2012, Comput. Math. Appl..

[9]  Suayip Yüzbasi,et al.  A collocation approach for solving systems of linear Volterra integral equations with variable coefficients , 2011, Comput. Math. Appl..

[10]  Khosrow Maleknejad,et al.  Numerical solution of singular Volterra integral equations system of convolution type by using operational matrices , 2008, Appl. Math. Comput..

[11]  Emran Tohidi.  Legendre Approximation for Solving Linear HPDEs and Comparison with Taylor and Bernoulli Matrix Methods , 2012 .

[12]  M. V. Bulatov.  Transformations of differential-algebraic systems of equations , 1994 .

[13]  R. Kress.  Linear Integral Equations , 1989 .

[14]  M. Hadizadeh,et al.  Jacobi spectral solution for integral algebraic equations of index-2 , 2011 .

[15]  M. V. Bulatov,et al.  Two-dimensional integral-algebraic systems: Analysis and computational methods , 2011, J. Comput. Appl. Math..

[16]  Tang,et al.  ON SPECTRAL METHODS FOR VOLTERRA INTEGRAL EQUATIONS AND THE CONVERGENCE ANALYSIS , 2008 .

[17]  T. A. Zang,et al.  Spectral methods for fluid dynamics , 1987 .

[18]  Mohsen Rabbani,et al.  Numerical computational solution of the Volterra integral equations system of the second kind by using an expansion method , 2007, Appl. Math. Comput..

[19]  J.-P. Kauthen.  The numerical solution of integral-algebraic equations of index 1 by polynomial spline collocation methods , 2001, Math. Comput..

[20]  Jafar Biazar,et al.  A new analytical method for solving systems of Volterra integral equations , 2010, Int. J. Comput. Math..

[21]  Sedaghat Shahmorad,et al.  Block by block method for the systems of nonlinear Volterra integral equations , 2010 .

[22]  Peter Linz,et al.  Analytical and numerical methods for Volterra equations , 1985, SIAM studies in applied and numerical mathematics.

[23]  Yingjun Jiang,et al.  On spectral methods for Volterra-type integro-differential equations , 2009 .

[24]  Zorica Uzelac,et al.  Family of quadratic spline difference schemes for a convection–diffusion problem , 2007, Int. J. Comput. Math..

[25]  T. A. Zang,et al.  Spectral Methods: Fundamentals in Single Domains , 2010 .

[26]  Gamal N. Elnagar,et al.  Chebyshev spectral solution of nonlinear Volterra-Hammerstein integral equations , 1996 .

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

[28]  Salih Yalçınbaş,et al.  Approximate solutions of linear Volterra integral equation systems with variable coefficients , 2010 .

[29]  M. I. Berenguer,et al.  Biorthogonal systems for solving Volterra integral equation systems of the second kind , 2011, J. Comput. Appl. Math..

[30]  Emran Tohidi,et al.  Optimal control of nonlinear Volterra integral equations via Legendre polynomials , 2013, IMA J. Math. Control. Inf..

[31]  Emran Tohidi,et al.  Legendre Approximation for Solving a Class of Nonlinear Optimal Control Problems , 2011 .

[32]  Yunqing Huang,et al.  Legendre spectral Galerkin method for second-kind Volterra integral equations , 2009 .

[33]  Elçin Yusufoglu.  Numerical expansion methods for solving systems of linear integral equations using interpolation and quadrature rules , 2007, Int. J. Comput. Math..

[34]  Yanping Chen,et al.  Spectral methods for weakly singular Volterra integral equations with smooth solutions , 2009, J. Comput. Appl. Math..

[35]  Jafar Saberi-Nadjafi,et al.  The variational iteration method: A highly promising method for solving the system of integro-differential equations , 2008, Comput. Math. Appl..