Numerical Solution of Fredholm and Volterra Integral Equations of the First Kind Using Wavelets Bases
Authors
Maryam Bahmanpour
 Department of Mathematics, Sama Technical and Vocational Training College, Islamic Azad University, Khorasgan (Isfahan) Branch, Isfahan, Iran.
Mohammad Ali Fariborzi Araghi
 Department of Mathematics, Central Tehran Branch, Islamic Azad University, P.O.Box 13185.768, Tehran, Iran.
Abstract
The Fredholm and Volterra types of integral equations are appeared in many engineering fields. In this paper, we suggest a method for solving Fredholm and Volterra integral equations of the first kind based on the wavelet bases. The Haar, continuous Legendre, CAS, Chebyshev wavelets of the first kind (CFK) and of the second kind (CSK) are used on [0,1] and are utilized as a basis in Galerkin or collocation method to approximate the solution of the integral equations. In this case, the integral equation converts to the system of linear equations. Then, in some examples the mentioned wavelets are compared with each other.
Keywords
 First kind Volterra and Fredholm integral equation
 Galerkin method
 Collocation method
 Haar
 Legendre
 CAS
 CFK
 CSK wavelets.
MSC
References

[1]
E. Babolian, F. Fattahzadeh, Numerical computation method in solving integral equations by using Chebyshev wavelet operational matrix of integration, Appl. Math. Compt., (2007), 10161022.

[2]
L. M. Delves, J. L. Mohamed, Computational Methods for integral equations, Cambridge University Press, (1985)

[3]
M. A. Fariborzi Araghi, M. Bahmanpour, Numerical Solution of Fredholm Integral Equation of the First kind using Legendre, Chebyshev and CAS wavelets, International J. of Math. Sci. & Engg. Appls. (IJMSEA), 2 (2008), 19.

[4]
M. A. Fariborzi Araghi, M. Bahmanpour, Numerical Solution of Fredholm Integral Equation of the First Kind by Using Chebyshev Wavelets, Proceeding of the International Conference of Approximation on Scientific Computing (ICASC'08), ISCAS, Beijing, China (2008)

[5]
J. S. Gu, W. S. Jiang, The Haar wavelets operational matrix of integration, Int. J. Sys. Sci., 27 (1996), 623628.

[6]
U. Lepik, Solving integral and differential equations by the aid of nonuniform Haar wavelets , Appl. Math. Compt., 198 (2008), 326332.

[7]
K. Maleknejad, R. Mollapourasl, M. Alizadeh, Numerical solution of Volterra type integral equation of the first kind with wavelet basis, Appl. Math. Compt. , 194 (2007), 400405.

[8]
M. Maleknejad, S. Sohrabi, Numerical solution of Fredholm integral equation of the first kind by using Legendre wavelets, Appl. Math. Compt., (2007), 836843.

[9]
M. Rabbani, R. Jamali , Solving Nonlinear System of Mixed VolterraFredholm Integral Equations by Using Variational Iteration Method, The Journal of Mathematics and Computer Science, 5 (2012), 280287.

[10]
T. J. Rivlin, Chebyshev Polynomials, John Wiley and Sons, Second Edition (1990)

[11]
X. Shang, D. Han, Numerical solution of Fredholm integral equation of the first kind by using linear Legendre multiwavelets, Appl. Math. Compt. , 191 (2007), 440444.

[12]
S. Yousefi, A. Banifatemi., Numerical solution of Fredholm integral equations by using CAS wavelets, Appl. Math. Compt., (2006), 458463.