Tau approximation method for the weakly singular Volterra-Hammerstein integral equations

In this paper, we propose a useful method based on the Tau method with arbitrary bases to find the numerical solution of weakly singular Voletrra-Hammerstein integral equations. In this scheme an operational approach using the orthogonal polynomial bases is presented for converting the problem under consideration to its matrix-vector representation. Error analysis of this method is also presented. Some numerical examples are provided to illustrate the accuracy and computational efficiency of the method.

[1]  Eduardo L. Ortiz,et al.  An operational approach to the Tau method for the numerical solution of non-linear differential equations , 1981, Computing.

[2]  H. Srivastava,et al.  A new Neumann series method for solving a family of local fractional Fredholm and Volterra integral equations , 2013 .

[3]  Fazlollah Soleymani,et al.  Tau approximate solution of weakly singular Volterra integral equations , 2013, Math. Comput. Model..

[4]  Donal O'Regan,et al.  Existence Results for Nonlinear Integral Equations , 1995 .

[5]  Józef Banaś,et al.  On existence and asymptotic stability of solutions of a nonlinear integral equation , 2003 .

[6]  Tao Tang,et al.  Convergence analysis of Jacobi spectral collocation methods for Abel-Volterra integral equations of second kind , 2012 .

[7]  Sedaghat Shahmorad,et al.  NUMERICAL PIECEWISE APPROXIMATE SOLUTION OF FREDHOLM INTEGRO-DIFFERENTIAL EQUATIONS BY THE TAU METHOD , 2005 .

[8]  M. E. Froes Bunchaft.  Some extensions of the Lanczos-Ortiz theory of canonical polynomials in the Tau Method , 1997, Math. Comput..

[9]  R. Malarkodi,et al.  Fractal Boundary Value Problems for Integral and Differential Equations with Local Fractional Operators , 2014 .

[10]  Farideh Ghoreishi,et al.  Numerical computation of the Tau approximation for the Volterra-Hammerstein integral equations , 2009, Numerical Algorithms.

[11]  S. Shahmorad,et al.  Numerical solution of the general form linear Fredholm-Volterra integro-differential equations by the Tau method with an error estimation , 2005, Appl. Math. Comput..

[12]  Hosseini Seyed Mohammad,et al.  Tau numerical solution of Fredholm integro-differential equations with arbitrary polynomial bases , 2003 .

[13]  Yong-sheng Ding,et al.  A generalized Gronwall inequality and its application to a fractional differential equation , 2007 .