Solving Fredholm integral equations of the first kind by using wavelet bases

Abstract

In this paper, we used a project technique for solving integral equation of the first kind by wavelet families via regularization approach and we proved the convergence for the numerical method and error consideration. Semi-orthogonal B-spline scaling functions and wavelets of degree 4 and their dual functions are presented to approximate the solutions to integral equations. Sparse matrix will product of semi-orthoganality and vanishing moment properties of B-spline wavelets.

Keywords

• Fredholm integral equations,projection method,regularization,quartic B-spline wavele

References

1. [1] H. Adibi and P. Assari, Chebyshev wavelet method for numerical solution of Fredholm integral equation of the first kind, Math. Prob. Eng, 2010, Article ID 138408, 17 pp., 2010.
2. [2] K. Atkinson, The Numerical Solution of Integral Equations of the Second Kind, Cambridge University Press 2, 1997.
3. [3] C.K. Chui, L. Montefusco and L. Puccio, Wavelets, Theory algorithm and applications, Academic press, 1994.
4. [4] A. Cohen, Numerical Analysis of Wavelet Methods, New York, Academic Press, 2003.
5. [5] P.K. Kythe and P. Puri, Computational Methods for Linear Integral Equations, Springer Science, 2002.
6. [6] K. Maleknejad, T. Lotfi and K. Mahdiani, Numerical solution of first kind Fredholm integral equation with wavelets-Galerkin method (WGM) and wavelets precondition, Appl. Math. Comput. 186, 794-800, 2007.
7. [7] K. Maleknejad, T. Lotfi and Y. Rostami, Numerical Computational Method in Solving Fredholm Integral Equations of the Second Kind by Using Coifman Wavelet, Appl. Math. Comput. 186, 212-218, 2007.
8. [8] K. Maleknejad, H. Mesgarani and T. Nikazad, Wavelet-Galerkin Solutions For Fredholm Integral Equations of The Second Kind, Internat. J. Engng. Sci. 13, 75-80, 2002.

9. [9] K. Maleknejad and S. Sohrabi,Numerical solution of Fredholm integral equation of the first kind by using Legendre wavelets, Appl. Math. Comput. 186, 836-843, 2007.
10. [10] M.T. Rashed, Numerical solutions of the integral equations of the first kind, Appl. Math. Comput. 2-3 , 413-420, 2003.
11. [11] P.K. Sahu and S. Saha Ray, Numerical solutions for the system of Fredholm integral equations of second kind by a new approach involving semiorthogonal B-spline wavelet collocation method. Appl. Math. Comput. 234, 368-379, 2014.
12. [12] M. Shamsi and M. Razzaghi, Solution of Hallens integral equation using multiwavelets, Comput. Phys. Comm. C 168, 187-197, 2015.
13. [13] A.M. Wazwaz, The regularization method for Fredholm integral equations of the first kind, Comput. Math. Appl. 61 (10), 2981-2986, 2011.
14. [14] G.A. Zakeri and M. Navab, Sinc collocation approximation of non-smooth solution of a nonlinear weakly singular Volterra integral equation, J. Comput. Phys. 229, 6548-6557, 2010.