Numerical Solution for Integro-Differential Equations by Using Quartic B-Spline Wavelet and Operational Matrices

Authors: Khosrow Maleknejad, Yaser Rostami

Abstract:

In this paper, Semi-orthogonal B-spline scaling functions and wavelets and their dual functions are presented to approximate the solutions of integro-differential equations.The B-spline scaling functions and wavelets, their properties and the operational matrices of derivative for this function are presented to reduce the solution of integro-differential equations to the solution of algebraic equations. Here we compute B-spline scaling functions of degree 4 and their dual, then we will show that by using them we have better approximation results for the solution of integro-differential equations in comparison with less degrees of scaling functions

Keywords: Integro-differential equations, Quartic B-spline wavelet, Operational matrices.

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1093706

References:

[1] K. Atkinson The Numerical Solution of Integral Equations of the Second Kind. Cambridge University Press 2, 1997.
[2] K. Maleknejad,S. Sohrabi and Y. Rostami Application of Wavelet Transform Analysis in Medical Frames Compression. Kybernetes, Int. J. Sys. Math, 37, 2, 343-351, 2008.
[3] A.M. Wazwaz Linear and nonlinear integral equations: Methods and applications. Higher Education, Springer, 2011.
[4] K. Maleknejad,R. Mollapourasl and M. Alizadeh, Numerical Solution of the Volterra Type Integral Equation of the First Kind with Wavelet Basis. Applied Mathematics and Computation , 194, 400-405, 2007.
[5] K. Maleknejad and M. Rabbani, A modification for solving Fredholm-Hammerstein integral equation by using wavelet basis. Kybernetes, Int. J. Sys. Math, 38, 615-620, 2009.
[6] K. Maleknejad and M. Nosrati Sahlan, The Method of Moments for Solution of Second Kind Fredholm Integral Equations Based on B-Spline Wavelets. International Journal of Computer Mathematics, Vol 87, No 7, 1602-1616, 2010.
[7] K. Maleknejad,R. Mollapourasl and M. shahabi, On solution of nonlinear integral equation based on fixed point technique and cubic B-spline scaling functions. Journal of Computational and Applied Mathematics, Volume 239, 346-358, 2013.
[8] K. Maleknejad,R. Mollapourasl and P. Mirzaei, Numerical solution of Volterra functional integral equation by using cubic B-spline scaling functions. Journal of Numerical Methods for Partial Differential Equations, 2013.
[9] K. Maleknejad,T. Lotfi and Y. Rostami, Numerical Computational Method in Solving Fredholm Integral Equations of the Second Kind by Using Coifman Wavelet. Applied Mathematics and Computation , 186, 212-218, 2007.
[10] C.K. Chui, An introduction to wavelets,Wavelet analysis and its applications. New york. Academic press, 1992.
[11] S.G. Mallat, A theory for multiresolution signal decomposition. The wavelet representation, IEEE Trans, Pattern Anal. Mach. Intell. vol 11, pp 674-693, 1989.
[12] E.A. Rawashdeh, Numerical solution of fractional integro-differential equations by collocation method. Applied Mathematics and Computation, vol 176, pp 1-6, 2006.
[13] N. Aghazadeh and K. Maleknejad Using quadratic B-spline scaling functions for solving integral equations. International Journal: Mathematical Manuscripts, No 1, 1-6, 2007.
[14] K. Koro and K. Abe Non-orthogonal spline wavelets for boundary element analysis. Engineering Analysis with Boundary Elements, 25, 149-164, 2001.
[15] M. Lakestani,M. Razzaghi and M. Dehghan Solution of nonlinear Fredholm- Hammerstein integral equations by using semiorthogonal spline wavelets. Mathematical Problems in Engineering, 113-121,
[16] M. Lakestani,M. Razzaghi and M. Dehghan Semiorthogonal spline wavelets approximation for Fredholm integro-differential equations. Mathematical Problems in Engineering, Article ID 96184, 12 pages, 2006.
[17] G. Ala,M.L. Di 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 Transactions on Electromagnetic Compatibility 45 , No 2, 218-228, 2003.
[18] K. Maleknejad and M. Tavassoli Kajani Solving integro-differential equation by using hybrid Legendre and Block-Pulse functions. International Journal of Applied Mathematics, 11(1), 67-76, 2002.
[19] K. Maleknejad and Y. Mahmoudi Numerical solution of Integro-Differential Equation By Using Hybrid Taylor And Block-Pulse Functions. Far East Journal of Mathematical science , 9(2) 203-213, 2003.
[20] K. Maleknejad and T. Lotfi Method for Linear Integral Equations by Cardinal B-Spline Wavelet and Shannon Wavelet as Bases for Obtain Galerkin System. Applied Mathematics and Computation , 175, 347-355, 2006.
[21] K. Maleknejad,S. Sohrabi and Y. Rostami Numerical Solution of Nonlinear Volterra Integral Equations of the Second Kind by Using Chebyshev Polynomials. Applied Mathematics and Computation, 188, 123-128, 2007.
[22] K. Maleknejad,B. Basirat and E. Hashemizadeh Hybrid Legendre polynomials and Block-Pulse functions approach for nonlinear Volterra-Fredholm integro-differential equations. Computer and Mathematics with Application, Vol 61, 2821-2828, 2011.
[23] K. Maleknejad and M. Attary A Chebyshev collocation method for the solution of higher-order Fredholm-Volterra integro-differential equation system. U.P.B. Sci. Bull., Series A, Vol 74, Iss 4, 2012.
[24] A. Ayad Spline approximation for first order Fredholm integro-differential equations. Universitatis Babes-Bolyai. Studia. Mathematica 41 , No 3, 18 , 1996.
[25] G. Micula and G. Fairweather Direct numerical spline methods for first-order Fredholm integrodifferential equations. Revue dAnalyse Numerique et de Theorie de lApproximation 22 , No 1, 59-66, 1993.
[26] R.D. Nevels, J.C. Goswami and H. Tehrani Semi-orthogonal versus orthogonal wavelet basis sets for solving integral equations. IEEE Transactions on Antennas and Propagation 45 , No 9, 1332-1339,1997.
[27] M.A. Fariborzi Araghi,S. Daliri and M. Bahmanpour numerical solution of integro-differential equation by using chebyshev wavelet operational matrix of integration. international journal of mathematical modelling and computations, 02, 127-136, 2012.
[28] G. Ala,N.L. D 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, Compat,1995
[29] B.K. Alpert Wavelets and other bases for fast numerical linear algebra, Wavelets: A Tutorial Theory and Applications. Wavelet Anal, Appl, vol 2, Academic Press, Massachusetts,181-216,1992.
[30] K. Mustapha A Petrov-Galerkin method for integro-differential equations with a memory term. Int. J. Open Problems Compt, 2008.
[31] J.C. Goswami,A.K. Chan and C.K. Chui On Solving First-Kind Integral Equations Using Wavelets on a Bounded Interval. IEEE Trance.Antennas propaget,43(6),614-622, 1995.