Spline Collocation for Solving System of Fredholm and Volterra Integral Equations
In this paper, numerical solution of system of Fredholm and Volterra integral equations by means of the Spline collocation method is considered. This approximation reduces the system of integral equations to an explicit system of algebraic equations. The solution is collocated by cubic B-spline and the integrand is approximated by the Newton-Cotes formula. The error analysis of proposed numerical method is studied theoretically. The results are compared with the results obtained by other methods to illustrate the accuracy and the implementation of our method.
Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1097100Procedia APA BibTeX Chicago EndNote Harvard JSON MLA RIS XML ISO 690 PDF Downloads 1686
 Atkinson,K. E: The Numerical Solution of Integral Equations of the Second Kind, Cambridge University Press, New York, (1997).
 Delves,L. M , Mohammed,J. L: Computational Methods for Integral Equations , Cambridge University Press, Cambridge, (1985).
 Saeed,R. K , Ahmed,C. S: Approximate Solution for the System of Nonlinear Volterra Integral Equations of the Second Kind by using Blockby- block Method,Australian Journal of Basic and Applied Sciences, 2(1), ,114-124(2008).
 Babolian,E , Masouri,Z, Hatamzadeh-Varmazyar,S: A Direct Method for Numerically Solving Integral Equations System Using Orthogonal Triangular Functions, Int. J. Industrial Mathematics.,1(2), 135- 145(2009).
 Maleknejad,K, Shahrezaee,M: Using RungeKutta method for numerical solution of the system of Volterra integral equation, Applied Mathematics and Computation, 149 ,399-410(2004).
 Babolian,E , Mordad,M: A numerical method for solving systems of linear and nonlinear integral equations of the second kind by hat basis functions,Computers and Mathematics with Applications ,62 ,187- 198(2011).
 Rabbani,M, Maleknejad,K , Aghazadeh,N: Numerical computational solution of the Volterra integral equations system of the second kind by using an expansion method,Applied Mathematics and Computation ,187, 1143-1146(2007).
 Bakodah,H. O: Some Modifications of Adomian Decomposition Method Applied to Nonlinear System of Fredholm Integral Equations of the Second Kind, Int. J. Contemp. Math. Sciences., 7(19),929 - 942(2012)
 Prenter,P.M: Spline and Variational Methods, Wiley & Sons, New- York, (1975).
 Mahmoodi,Z,Rashidinia ,J, Babolian,E: B-Spline collocation method for linear and nonlinear Fredholm and Volterra integro-differential equations, Applicable Analysis ,1-16(2012).
 Rashidinia ,J, Babolian ,E, Mahmoodi,Z : Spline Collocation for Fredholm Integral Equations, Mathematical Sciences, 5(2),147- 158(2011).
 Rashidinia ,J,Babolian,E,Mahmoodi,Z: Spline Collocation for nonlinear Fredholm Integral Equations, International Journal of Mathematical Modelling&Computations.,1(1) ,69-75(2011).
 Babolian,E, Biazar,J, Vahidi,A.R: The decomposition method applied to systems of Fredholm integral equations of the second kind, Appl. Math. Comput., 148 ,443-452(2004).
 Maleknejad,K , Mirzaee,F: Numerical solution of linear Fredholm integral equations system by rationalized Haar functions method, Int. J. Comput. Math., 80(11),1397-1405(2003).
 Maleknejad,K, Shahrezaee,M,Khatami,H: Numerical solution of integral equations system of the second kind by Block-Pulse functions, Appl. Math. Comput. ,166,15-24(2005).
 Maleknejad,K,Aghazadeh,N, Rabbani,M: Numerical solution of second kind Fredholm integral equations system by using a Taylor-series expansion method, Appl. Math. Comput.,175,1229-1234(2006).
 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).
 Babolian,E, Fattahzadeh,F: Numerical computation method in solving integral equations by using Chebyshev wavelet operational matrix of integration, Appl. Math. Comput.,188,1016-1022(2007).
 E. Yusufo Glu,E: A homotopy perturbation algorithm to solve a system of Fredholm-Volterra type integral equations, Mathematical and Computer Modelling,47, 1099-1107(2008).
 Matinfar,M,Saeidy,M,Vahidi,J: Application of Homotopy Analysis Method for Solving Systems of Volterra Integral Equations , Adv. Appl. Math. Mech., 4 ,36-45(2012).
 Javidi,M,Golbabai,A: A numerical solution for solving system of Fredholm integral equations by using homotopy perturbation method, Appl. Math. Comput.,189,1921-1928(2007).
 Biazar,J, Eslami,M: Modified HPM for solving systems of Volterra integral equations of the second kind ,Journal of King Saud University , 23,35-39(2011).
 Muhammad,M,Nurmuhammad,A,Mori,M, M. Sugihara,M: Numerical solution of integral equations by means of the Sinc collocation method based on the double exponential transformation, J. Comput. Appl. Math.,177 ,269-286(2005).
 Rashidinia,J,Zarebnia,M: Convergence of approximate solution of system of Fredholm integral equations, J. Math. Anal. Appl.,333,1216- 1227(2007).
 Wazwaz,A. M: A First Course in Integral Equations, WSPC, New Jersey, (1997).
 Biazar,J: Solution of systems of integral-differential equations by Adomian decomposition method, Appl. Math. Comput.,168,1232- 1238(2005).
 Jafari,H, Daftardar-Gejji,V: Solving a system of nonlinear fractional differential equations using Adomian decomposition, J. Comput. Appl. Math.,196 ,644-651(2006).
 Vahidi,A.R,Mokhtar,M: On the decomposition method for system of linear Fredholm integral equationsof the second kind, Appl. Math. Sci., 2,57-62(2008).