A Legendre-spectral scheme for solution of nonlinear system of Volterra-Fredholm integral equations

Subject Areas : Applied Mathematics

L. Hooshangian ¹ (Department of Mathematics, Dezful Branch, Islamic Azad University, Dezful, Ira)
D. Mirzaie ² (Department of Mathematics, University of Isfahan, Isfahan, Iran)

Received: 2011-03-10 Accepted : 2012-01-11 Published : 2014-05-01

Keywords: Volterra-Fredholm integral equations, Nonlinear integral equations, Legendre-spectral method, Gauss Legendre points, Lagrange interpolatio,

Abstract :

This paper gives an ecient numerical method for solving the nonlinear systemof Volterra-Fredholm integral equations. A Legendre-spectral method based onthe Legendre integration Gauss points and Lagrange interpolation is proposedto convert the nonlinear integral equations to a nonlinear system of equationswhere the solution leads to the values of unknown functions at collocationpoints.

References:

[1] O. Diekman, Thresholds and traveling waves for the geographical spread
of infection, J. Math. Biol. 6 (1978) 109{130
[2] B.G. Pachpatte, On mixed Volterra-Fredholm type integral equations,
Indian J. Pure Appl. Math. 17 (1986), pp. 488{496.
[3] L. Hacia, On approximate solution for integral equations of mixed type,
ZAMM. Z. Angew. Math. Mech. 76 (1996) 415{416.
[4] P.J. Kauthen, Continuous time collocation methods for Volterra-Fredholm
integral equations, Numer. Math. 56, (1989) 409{424.
[5] L. Hacia, On approximate solving of the Fourier problems, Demonstratio
Math. 12 (1979) 913{922.
[6] H.R. Thieme, A model for the spatio spread of an epidemic, J. Math. Biol.
4 (1977) 337{351.
[7] H. Guoqiang and Z. Liqing, Asymptotic expansion for the trapezoidal
Nystrom method of linear Volterra- Fredholm equations, J. Comput. Appl.
Math. 51 (3) (1994) 339{348.
[8] H. Brunner, On the numerical solution of nonlinear Volterra-Fredholm
integral equation by collocation methods, SIAM J. Numer. Anal. 27
(4)(1990) 987-1000.
[9] H. Guoqiang, Asymptotic error expansion for the Nystrom method for a
nonlinear Volterra-Fredholm integral equations, J. Comput. Appl. Math. 59
(1995) 490{59.

[10] K. Maleknejad and M. Hadizadeh, A new computational method for
Volterra-Fredholm integral iquations, Comput. Math. Appl. 37 (1999) 1-8.
[11] M. Hadizadeh, Posteriori error estimates for the nonlinear Volterra-
Fredholm integral equations, Comput. Math. Appl. 45 (2003) 677{687.
[12] K. Maleknejad , M.R. Fadaei Yami, A computational method for system
of Volterra-Fredholm integral equations, Appl. Math. Comput. 183 (2006)
589{595.
[13] Gamal N. Elnagar, Mohammad A. Kazemi, A cell-averaging chebyshev
spectral method for nonlinear fredholm-hammerstein integral equations, Int.
J. Comput. Math. 60 (1996) 91{104.
[14] Y.J.Jiang, On spectral methods for Volterra-type integro-dierential
equations, J. Comput. Appl. Math. 230 (2009) 333{340.
[15] M. Lakestani, B. N. Saray, M. Dehghan, Numerical solution for the
weakly singular Fredholm integro-dierential equations using Legendre
multiwavelets, J. Comput. Appl. Math. 235 (2011) 3291{3303.
[16] F. Fakhar-Izadi, M. Dehghan, The spectral methods for parabolic Volterra
integro-dierential equations, J. Comput. Appl. Math. 235 (2011) 4032{
4046.
[17] M. Tavassoli Kajani, S. Mahdavi, Numerical solution of nonlinear integral
equations by Galerkin methods with hybrid Legendre and Block-Pulse
functions, Mathematics Scientic Journal, 7 (2011) 97{105.
[18] Y. Liu, Solving abel integral equation by using Chebyshev wavelets,
Mathematics Scientic Journal, 6 (2010) 51{57.
[19] Y. Mahmoudi, Chebyshev collocation method for integral equations with
singular kernels, Mathematics Scientic Journal, 5 (2010) 41{49.
[20] A. Shahsavaran, Numerical solution of linear Volterra and Fredholm
integro dierential equations using Haar wavelets, Mathematics Scientic
Journal, 6 (2010) 85{96.
[21] T. Tang, Xiang Xu, Jin Cheng, On spectral methods for Volterra type
integral equations and the convergence analysis, J. Comput. Math. 26 (2008)
825{837.
13

[22] Y. Chen, T. Tang, Convergence analysis of the Jacobi spectral-collocation
methods for Volterra integral equations with a weakly singular kernel, Math.
Comput. 79 (2010) 147{167.