Evaluating the solution for second kind nonlinear Volterra Fredholm integral equations using hybrid method

Subject Areas : Applied Mathematics

Ahmad Shahsavaran ¹ (Islamic Azad University, Boroujerd Branch, Boroujerd, Iran.)
Akbar Shahsavaran ² (Islamic Azad University, Boroujerd Branch, Boroujerd, Iran.)

Received: 2015-09-21 Accepted : 2015-09-21 Published : 2015-12-01

Keywords: Nonlinear Fredholm Volterra integral equation, Haar wavelet, Haar coecient matrix, Block-Pulse Function, Collocation points,

Abstract :

In this work, we present a computational method for solving second kindnonlinear Fredholm Volterra integral equations which is based on the use ofHaar wavelets. These functions together with the collocation method are thenutilized to reduce the Fredholm Volterra integral equations to the solution ofalgebraic equations. Finally, we also give some numerical examples that showsvalidity and applicability of the technique.

References:

[1] E. Babolian, A. Shahsavaran, Numerical solution of nonlinear Fredholm
integral equations of the second kind using Haar wavelets, J. Comput. Appl.
Math. 225 (2009) 87{95.
[2] C. F. Chen, C. H. Hsiao, Haar wavelet method for solving lumped and
distributed-parameter systems, IEE Proc. Control Theory Appl. 144 (1997)
87{94.
[3] M. Ghasemi, A. Azizi and M. Fardi, Numerical solution of seven order
Sawada Kotara equations by homotopy perturbation method, Mathematics
Scientic Journal 1 (2011) 69{77.
[4] L. Hooshangian, D. Mirzaei, A Legendre spectral scheme for solution
of nonlinear system of Volterra Fredholm integral equations, Mathematics
Scientic Journal 1 (2012) 1{14.
[5] U. Lepik, E. Tamme, Application of the Haar wavelets for solution of linear
integral equations, in:Dynamical systems and applications 1 (2005) 395{407.
[6] K. Maleknejad, T. Lot, Numerical expansion method for solving integral
equation by interpolation and Gauss quadrature rules, Appl. Math. Comput.
168 (2005) 111{124.
[7] K. Maleknejad, M. T. Kajani, Y. Mahmoudi, Numerical solution of the
second kind integral equation by using Legendre wavelets, J. Kybornet In
press.
[8] K. Maleknejad, M. Karami, Numerical solution of non-linear Fredholm
integral Equations by using multiwavelet in Petrov-Galerkin method, Appl.
Math. Comput. 168 (2005) 102{110.
[9] K. Maleknejad, Y. Mahmoudi, Numerical solution of linear Fredholm
integral equation by using hybrid Taylor and Block-Pulse functions, Appl.
Math. Comput. 149 (2004) 799{806.
[10] K. Maleknejad, M. Shahrezaee, Using Runge-Kutta method for numerical
solution of the system of Volterra integral equation, Appl. Math. Comput.
149 (2004) 399-410.
[11] K. Maleknejad, M. T. Kajani, Solving second kind integral equations by
Galerkin method with hybrid Legendre and Block-Pulse functions, Appl.
Math. Comput. 145 (2003) 623{629.

[12] Y. Mahmoudi, Wavelet Galerkin method for numerical solution of
nonlinear integral equation, Appl. Math. Comput. 167 (2005) 1119{1129.
[13] K. Maleknejad, F. Mirzaee, Using rationalized Haar wavelet for solving
linear integral equations, Appl. Math. Comput. 160 (2005) 579{587.
[14] K. Maleknejad, F. Mirzaee, Numerical solution of linear Fredholm integral
equations system by rationalized Haar functions method, Int. J. Comput.
Math. 8 (2003) 1397{1405.
[15] A. Shahsavaran, Numerical solution of linear Volterra and Fredholm
integro dierential equations using Haar wavelets, Mathematics Scientic
Journal 1 (2010) 85{96.
[16] C. P. Rao, Piecewise constant orthogonal functions and their applications
to system and control, Springer, Berlin 1983.
[17] P. Wojtaszczyk, A mathematical introduction to wavelets, Cambridge
University Press 1997.