In this paper we present a computational technique for solving the second-order Fredholm integro-differential equations. The method is based on a non-classical pseudospectral method. The differential matrices are computed and are utilized to reduce the second-order Fredholm integro-differential equations to algebraic equations. Numerical examples are presented to demonstrate the validity and applicability of the new method.
AlaGSilvestreMLDFrancomanoETortoriciA (2003) An advanced numerical model in solving thin-wire integral equations by using semiorthogonal compactly supported spline wavelet. IEEE Transactions on Electromagnetic Compatibility45: 216–228.
2.
AyadA (1996) Spline approximation for first-order Fredholm integro-differential equations. Studia universitatis babes-Bolyai Mathematica41: 1–8.
3.
AyadA (2002) Wavelet methods for the numerical solution of Fredholm integro-differential equations. International Journal of Applied Mathematics11: 27–35.
4.
BehirySHHashishH (2002) Wavelet methods for the numerical solution of Fredholm integro-differential equations. International Journal of Applied Mathematics11: 27–35.
5.
ChenHShizgalBD (2001) A spectral solution of the Sturm–Liouville equation: Comparison of classical and non-classical basis sets. Journal of Computation and Applied Mathematics136: 17–35.
6.
CostaBDonWS (2000) On the computation of high order pseudospectral derivatives. Applied Numerical Mathematics33: 151–159.
7.
DehghanM (2006) Solution of a partial integro–differential equation arising from viscoelasticity. International Journal of Computer Mathematics83: 123–129.
8.
DehghanMSaadatmandiA (2008) Chebyshev finite difference method for Fredholm integro–differential equation. International Journal of Computer Mathematics85: 123–130.
9.
DehghanMShakeriF (2010) Solution of parabolic integro–differential equations arising in heat conduction in materials with memory via He s variational iteration technique. International Journal for Numerical Methods in Engineering26: 705–715.
10.
DehghanMShakourifarMHamidiA (2009) The solution of linear and nonlinear systems of Volterra functional equations using Adomian–Pade technique. Chaos Solitons and Fractals39: 2509–2521.
11.
ElnagarGNKazemiMA (1998) Pseudospectral Chebyshev optimal control of constrained non-linear dynamical systems. Computational Optimization and Applications11: 195–217.
12.
ElnagarGNKazemiMARazzaghiM (1995) The pseudospectral Legendre method for discretizing optimal control problems. IEEE Transactions on Automatic Control40: 1793–1796.
13.
GautschiW (1985) Orthogonal polynomials-constructive: Theory and applications. Journal of Computation and Applied Mathematics12/13: 61–75.
14.
GhasemiMTavassoli KajaniMBabolianE (2007) Application of He's homotopy perturbation method to non-linear integro-differential equations. Applied Mathematics and Computation188: 538–548.
15.
GolubGHWelschJH (1969) Calculation of Gauss quadrature rules. Mathematics of Computation23: 221–230.
16.
GoswamiJCChanAKChuiCK (1995) On solving first-kind integral equations using wavelets on a bounded interval. IEEE Transactions on Antennas and Propagation43: 614–622.
17.
HosseiniSMShahmoradS (2002) A matrix formulation of the Tau for Fredholm and Volterra linear integro-differential equations. Journal of Computational and Applied Mathematics9: 497–507.
18.
HosseiniSMShahmoradS (2005) Numerical piecewise approximate solution of Fredholm integro-differential equations by the Tau method. Applied Mathematical Modelling29: 1005–1021.
19.
IvazKShahmoradSMostahkanBS (2009) Newton–Tau numerical solution of one-diemensional non-linear integro-differential equations. Southeast Asian Bulletin of Mathematics33: 733–740.
20.
LakestaniMDehghanM (2010) Numerical solution of fourth-order integro-differential equations using Chebyshev cardinal functions. International Journal of Computer Mathematics87: 1389–1394.
21.
LinzP (1974) A method for the approximate solution of linear integro-differential equations. SIAM Journal of Numerical Analysis11: 137–144.
22.
MirzaeiDDehghanM (2010) A meshless based method for solution of integral equations. Applied Numerical Mathematics60: 245–262.
23.
NevaiP (1990) Orthogonal Polynomials: Theory and Practice. Norwell, MA: Kluwer Academic/Cambridge University Press.
24.
SaadatmandiADehghanM (2010) Numerical solution of the higher-order linear Fredholm integro–differential difference equation with variable coefficients. Computers and Mathematics with Applications59: 2996–3004.
25.
ShakourifarMDehghanM (2008) On the numerical solution of nonlinear systems of Volterra integro–differential equations with delay arguments. Computing82: 241–260.
26.
ShamsiMRazzaghiM (2005) Solution of Hallen's integral equation using multi-wavelets. Computer Physics Communications168: 187–197.
27.
ShizgalBD (1981) A Gaussian quadrature procedure for use in the solution of the Boltzmann equation and related problems. Journal of Computational Physics41: 309–328.
28.
ShizgalBDBlackmoreR (1984) A discrete ordinate method of solution of linear boundary value and eigenvalue problems. Journal of Computational Physics55: 313–327.
29.
WazwazAM (1997) A First Course in Integral Equations. Singapore: World Scientific.
30.
WeidemanJAC (1999) Spectral method based on non-classical orthogonal polynomials. International Series in Numerical Mathematics131: 239–251.
31.
WelfertBD (1997) Generation of pseudospectral differentiation matrices. SIAM Journal of Numerical Analysis24: 1640–1657.
32.
WilliamsP (2004) A quadrature discretization method for solving optimal control problems. Advances in the Astronautical Sciences, Spaceflight19: 703–722.
33.
YousefiSRazzaghiM (2005) Legendre wavelets method for the non-linear Volterra–Fredholm integral equations. Mathematics and Computers in Simulation70: 1–8.