Abstract
In this study, an effective technique upon compactly supported semi orthogonal cubic B-spline wavelets for solving nonlinear Volterra-Fredholm integral equations is proposed. Properties of B-spline wavelets and function approximation by them are first presented and the exponential convergence rate of the approximation, Ο(2-4j ), is proved. For solving the nonlinear Volterra-Fredholm integral equation, the unknown function of problem is approximated by cubic B-spline wavelets. Then Properties of these functions are used to reduce nonlinear mixed integral equation to some algebraic system. For solving the mentioned system, Galerkin and collocation methods are applied. In the both methods, Cubic B-spline wavelets are used as testing and weighting functions. Convergence and error analysis of the method is described through some proved theorems. Because of having vanishing moments, compact support and semi orthogonality properties of these wavelets, operational matrices of the Galerkin and collocation methods are very sparse. In fact the entries with significant magnitude are in the diagonal of operational matrices, and other entries are very small and hence can be set to zero without significantly affecting the solution. Because of having low memory requirement, high speed and accuracy of the method, the presented procedure is more practical with respect to many of other methods for solving this class of integral equations. The method is computationally attractive and applications are demonstrated through illustrative examples. As is shown in the reported tables of examples, compare the error of three methods, we can find that the presented method get better approximate solution.

Keywords
Fredholm-Volterra-Hammerstein integral equations, collocation method, Galerkin method, Cubic B-spline wavelets, error analysis

Cite this paper
Monireh Nosrati Sahlan, Hamid Reza Marasi, Convergence of Approximate Solution of Nonlinear Volterra-Fredholm Integral Equations , SCIREA Journal of Physics. Volume 1, Issue 2, December 2016 | PP. 125-143.

References

[ 1 ]	Pathpatte, B.G., On mixed Volterra–Fredholm integral equations. Indian J. Pure. Appl. Math., 17 (1986), pp. 488–496.
[ 2 ]	Diekmann, O., Thresholds and traveling for the geographical spread of infection. J Math Biol., 6 (1978), pp. 109-130.
[ 3 ]	Thieme, H.R., A model for the spatial spread of an epidemic. J. Math. Biol., 4 (1977), pp. 337–351.
[ 4 ]	Brunner, H., Implicitly linear collocation method for nonlinear Volterra equations. J Appl. Num. Math., 9 (1982), pp. 235-247.
[ 5 ]	Guoqiang, H., Asymptotic error expansion variation of a collocation method for Volterra Hammerstein equations. J Appl. Num. Math., 13 (1993), pp. 357-369.
[ 6 ]	Kumar, S and I.H. Sloan., A new collocation-type method for Hammerstein integral equations. J Math. Comput., 48 (1987), pp. 123-129.
[ 7 ]	Lardy, L.J., A variation of Nystrom’s method for Hammerstein equations. Journal of Integral Equations, 3 (1982), pp. 123-129.
[ 8 ]	Yalcinbas, S., Taylor polynomial solution of nonlinear Volterra–Fredholm integral equations. Appl Math Comput, 127 (2002), pp. 195–206.
[ 9 ]	Kauthen, J.P., Continuous time collocation method for Volterra-Fredholm integral equations. Numer. Math., 56 (1989), pp. 409-424.
[ 10 ]	Hacia, L., On approximate solution for integral equations of mixed type. ZAMM Z. Angew. Math. Mech., 76 (1996), pp. 415-416.
[ 11 ]	Hacia, L., Projection methods for integral equations in epidemic. J Math Model Anal, 7 (2) (2002), pp. 229-240.
[ 12 ]	Maleknejad, K and M, Hadizadeh., A new computational method for Volterra–Fredholm integral equations. J Comput. Appl Math, 37 (1999), pp. 1–8.
[ 13 ]	Wazwaz, A.M., A reliable treatment for mixed Volterra–Fredholm integral equations. Appl Math Comput, 127 (2002), pp. 405–414.
[ 14 ]	Yildirim, A., Homotopy perturbation method for the mixed Volterra–Fredholm integral equations. Chaos Solitons Fractals, 42 (2009), 2760–2764.
[ 15 ]	Banifatemi, E., M, Razzaghi and S, Yousefi., Two-dimensional Legendre wavelets method for the mixed Volterra-Fredholm integral equations. J Vibr Control, 13 (2007), pp. 1667-1675.
[ 16 ]	Hadizadeh, M and M, Asgari., An effective numerical approximation for the linear class of mixed integral equations. Appl Math Comput, 167 (2005), pp. 1090–1100.
[ 17 ]	Tricomi, F.G., Integral Equations. Dover, 1982.
[ 18 ]	Ala, G., M.L. Silvestre., E, Francomano and A, Tortorici., An advanced numerical model in solving thin-wire integral equations by using semi orthogonal compactly supported spline wavelets. IEEE Trans Electromagn Compact, 45 (2003), pp. 218–228.
[ 19 ]	Nevels, R.D., J.C. Goswami and H, Tehrani., Semi orthogonal versus orthogonal wavelet basis sets for solving integral equations. IEEE Trans Antennas Propagat, 45 (1997), pp. 1332–1339.
[ 20 ]	Chui, C., An introduction to wavelets. New York, Academic press, 1992.
[ 21 ]	Chui, C., Wavelets: a mathematical tool for signal analysis. Philadelphia, PA: SIAM, 1997.
[ 22 ]	Mallat, S.G., A theory for multiresolution signal decomposition: The wavelet representation. IEEE Trans, Pattern Anal Mach Intell, 11 (1989), pp. 674-693.
[ 23 ]	Daubechies, I., Ten lectures on wavelets. Philadelphia, PA: SIAM, 1992.
[ 24 ]	Maleknejad, K., K, Nouri and M, Nosrati Sahlan., Convergence of approximate solution of nonlinear Fredholm Hammerstein integral equations. Commun Nonlinear Sci Num Simul, 15(6) (2010), pp. 1432-1443.
[ 25 ]	Strang G and G, Fix., A Fourier analysis of the finite element variational method, in: Constructive Aspect of Functional Analysis, Edizioni Cremonese, Rome, 1971, pp. 796-830.
[ 26 ]	Strang G., Wavelets and dilation equations: a brief introduction. SIAM Review, 31 (1989), pp. 614-627.
[ 27 ]	Unser, M., 1996. Approximation power of biorthogonal wavelet expansions. IEEE Trans Signal Processing, 44(3) (1996), pp. 519-527.0
[ 28 ]	Kalman, R.E. and R.E. Kalaba., Quasi linearization and Nonlinear Boundary-Value Problems. Elsevier, New York, 1969.
[ 29 ]	Ordokhani, Y. and M, Razzaghi., Solution of nonlinear Volterra Fredholm Hammerstein integral equations via a collocation method and rationalized Haar functions. Applied Mathematics Letters, 21 (2008), pp. 4-9.