Some Modifications Of Sinc Collocation Method For Integral Equations

ABSTRACT

 The research is concerned with the implementation of sinc collocation method for the solution of Volterra and Volterra-FredhoIm integral equations of the second kind, using a variable transformation method. By composition of functions, a single exponential transformation formula that relates the real line R to the interval (a, b) is modified to obtain a new one, within the same interval of relationship, which possesses double exponential decay characteristic. We show the derivation of Haber's formula for indefinite integration with improved error bound after the replacement of the single exponential formula with the modified one. This formula and the properties of sinc spaces of approximation are employed in the construction of a modified collocation scheme which assists in the conversion of the Volterra integral equation into algebraic equations. The theoretical convergence rates for the collocation schemes based on single exponential formula and double exponential formula are established as  (_!-exp -cIAi) (logN —C1N  , respectively, where C, C1 and C2 are independent of the number of terms N. The requirement for optimal convergence is discussed with respect to the mesh size, h. A region for optimal convergence is also proposed based on the ratio, dia , of the parameters d and a, where d and a are infinite strip width and a positive constant, respectively. This proposal is verified with numerical experiments carried out with MATLAB computational software for comparison of maximum absolute error values corresponding to different ratios, d/a, with the proposed size for optimal convergence. It is also shown that the collocation scheme based on the modified formula produced improved results over the one obtained using single exponential formula.