A new collocation approach for solving systems of high-order linear Volterra integro-differential equations with variable coefficients

This paper contributes an efficient numerical approach for solving the systems of high-order linear Volterra integro-differential equations with variable coefficients under the mixed conditions. The method we have used consists of reducing the problem to a matrix equation which corresponds to a system of linear algebraic equations. The obtained matrix equation is based on the matrix forms of Fibonacci polynomials and their derivatives by means of collocations. In addition, the method is presented with error. Numerical results with comparisons are given to demonstrate the applicability, efficiency and accuracy of the proposed method. The results of the examples indicated that the method is simple and effective, and could provide an approximate solution with high accuracy or exact solution of the system of high-order linear Volterra integro-differential equations.

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

[2]  M. Javidi,et al.  Modified homotopy perturbation method for solving system of linear Fredholm integral equations , 2009, Math. Comput. Model..

[3]  Ángel Plaza,et al.  On k-Fibonacci sequences and polynomials and their derivatives , 2009 .

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

[5]  Ibrahim Özkol,et al.  Solutions of integral and integro-differential equation systems by using differential transform method , 2008, Comput. Math. Appl..

[6]  M. Roodaki,et al.  Delta basis functions and their applications to systems of integral equations , 2012, Comput. Math. Appl..

[7]  Khosrow Maleknejad,et al.  Solving linear integro-differential equation system by Galerkin methods with hybrid functions , 2004, Appl. Math. Comput..

[8]  Farshid Mirzaee,et al.  Application of Fibonacci collocation method for solving Volterra-Fredholm integral equations , 2016, Appl. Math. Comput..

[9]  Christopher T. H. Baker,et al.  Stability analysis of continuous implicit Runge-Kutta methods for Volterra integro-differential systems with unbounded delays , 1997 .

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

[11]  M. I. Berenguer,et al.  An approximation method for solving systems of Volterra integro-differential equations , 2013 .

[12]  Li Xie,et al.  Numerical Approach to Wind Ripple in Desert , 2007 .

[13]  Sana’a A. Zarea,et al.  The Decomposition Method Applied to Solve High-order Linear Volterra-Fredholm Integro-differential Equations , 2004 .

[14]  Nicola Bellomo,et al.  BIFURCATION ANALYSIS FOR A NONLINEAR SYSTEM OF INTEGRO-DIFFERENTIAL EQUATIONS MODELLING TUMOR-IMMUNE CELLS COMPETITION , 1999 .

[15]  Lan Xu,et al.  Electrospun Nanoporous Spheres with Chinese Drug , 2007 .

[16]  K. Maleknejad,et al.  Numerical solution of an integral equations system of the first kind by using an operational matrix with block pulse functions , 2011, Int. J. Syst. Sci..

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

[18]  Hao Wang,et al.  A Practical Thermodynamic Method to Calculate the Best Glass-forming Composition for Bulk Metallic Glasses , 2007 .

[19]  Ian H. Sloan,et al.  Optimal order spline methods for nonlinear differential and integro-differential equations , 1999 .

[20]  M. El-shahed.  Application of He's Homotopy Perturbation Method to Volterra's Integro-differential Equation , 2005 .

[21]  Temur Jangveladze,et al.  Finite difference approximation of a nonlinear integro-differential system , 2009, Appl. Math. Comput..

[22]  Mehmet Sezer,et al.  Polynomial solution of high-order linear Fredholm integro-differential equations with constant coefficients , 2008, J. Frankl. Inst..

[23]  Abdul-Majid Wazwaz,et al.  The existence of noise terms for systems of inhomogeneous differential and integral equations , 2003, Appl. Math. Comput..

[24]  Farshid Mirzaee,et al.  Numerical solution of linear Fredholm integral equations system by rationalized Haar functions method , 2003, Int. J. Comput. Math..

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

[26]  F. Sun,et al.  The Fractal Dimension of the Fractal Model of Dropwise Condensation and Its Experimental Study , 2007 .

[27]  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..

[29]  Farshid Mirzaee,et al.  Solving linear integro-differential equations system by using rationalized Haar functions method , 2004, Appl. Math. Comput..

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

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

[32]  M. Golberg,et al.  Discrete projection methods for integral equations , 1996 .

[33]  Esmail Babolian,et al.  Application of He's homotopy perturbation method to nonlinear integro-differential equations , 2007, Appl. Math. Comput..