On the numerical solution of Volterra-Fredholm integral equations with logarithmic kernel using smoothing transformation
Keywords:Volterra-Fredholm Integral Equations, Logarithmic Kernels, Integral Equation, Collocation Matrix Method, Legendre and Chebychev Polynomials.
A smoothing transformation, Legendre and Chebyshev collocation method areÂ presented to solve numerically the Voltterra-Fredholm Integral Equations withÂ Logarithmic Kernel. We transform the Volterra Fredholm integral equationsÂ to a system of Fredholm integral equations of the second kind, using a smoothing transformation to cancel the singularities in the kernel, a system Fredholm integral equation with smooth kernel is obtained and will be solved using Legendre and Chebyshev polynomials. This lead to a system of algebraic equations with Legendre or Chebychev coefficients. Thus, by solving the matrix equation, Legendre and Chebychev coefficients are obtained. Some numerical examples are included to demonstrate the validity and applicability of the proposed technique.
 C.Chniti, Numerical Approximations of Fredholm Integral equations with Abel Kernel using Legendre and Chebychev Polynomials, J. Math. Comput. Sci. 3 (2013), No. 2, 655-667.
 C. Chniti, On The Numerical Solution of Volterra-Fredholm Integral Equations With Abel Kernel Using Legendre Polynomials, IJAST , Issue 3 volume 1,(2013), 404-412.
 L. Fox, Chebyshev Methods for Ordinary Differential Equations, Comput Journal , 4, 318-331 (1962).
 R. Ezzati, F. Mokhtari, Numerical solution of Fredholm integral equations of the second kind by using fuzzy transforms, International Journal of Physical Science, 7 (2012) 1578- 1583.
 R. Ezzati, S. Najafalizadeh, Numerical solution of nonlinear Voltra-Fredholm integral equation by using Chebyshev polynomials, Mathematical Sciences Quarterly Journal, 5 (2011) 14-22.
 H. Guoqiang, Z. Liqing, Asymptotic expantion for the trapezoidal Nystrom method of linear Voltra-Fredholm integral equations, J Comput Math. Appl, 51 (1994) 339-348.
 PG. Kauthen, Continuous time collocation methods for Voltra-Fredholm integral equations, Numer Math 56 (1989) 409-424.
 K. Maleknejad, MR. Fadaei Yami, A computational method for system of Volterra- Fredholm integral equations, Appl. Math. Comput, 188 (2006) 589-595.
 K. Maleknejad, M. Hadizadeh, A new computational method for Volterra-Fredholm integral equation, J. Comput. Math. Appl, 37 (1999) 37-48.
 K. Maleknejad, Y. Mahmodi, Taylor polynomial solution of high-order nonlinear Volterra-Fredholm integro-differential equations, Appl. Math. Comput, 145 (2003) 641-653.
 K. Maleknejad, S. Sohrabi. Y. Rostami, Numerical solution of nonlinear Voltra integral equation of the second kind by using Chebyshev polynomials, Appl. Math. Comput, 188 (2007) 123-128.
 K. Maleknejad, M. Tavassoli, Y. Mahmoudi, Numerical solution of linear Fredholm and Volterra integral equations of the second kind by using Legendre wavelet, J. Syst. Math, 32 (2003) 1530-1539.
 S. Yousefi, S. M. Razzaghi, Legendre wavelets method for the nonlinear Volterra-Fredholm integral equations, Math. Comput. Simul, 70 (2005) 1-8.
 S. Yalsinbas, Taylor polynomial solution of nonlinear Volterra-Fredholm integral equations, Appl. Math. Comput, 127 (2002) 195-206.
 David Elliott, Chebyshev series method for the numerical solution of Fredholm integral equations. Comput. J, 6 (1963/1964), 102-111.
 W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in FORTRAN: the art of scientific computing, Cambridge University Press, New York (1992).