an:05346973
Zbl 1152.65121
Mori, Masatake; Nurmuhammad, Ahniyaz; Murai, Takefumi
Numerical solution of Volterra integral equations with weakly singular kernel based on the DE-sinc method
EN
Japan J. Ind. Appl. Math. 25, No. 2, 165-183 (2008).
0916-7005 1868-937X
2008
j
65R20 45E10
numerical examples; double exponential transformation; sinc method; weakly singular kernel
A method for numerical solution of Volterra integral equations of the second kind with a weakly singular kernel based on the double exponential (DE) transformation is proposed. In this method we first express the approximate solution in the form of a Sinc expansion based on the double exponential transformation by \textit{H. Takahasi} and \textit{M. Mori} [Publ. Res. Inst. Math. Sci., Kyoto Univ. 9, 721--741 (1974; Zbl 0293.65011)] followed by collocation at the Sinc points. We also apply the DE formula to the kernel integration. In every sample equation a numerical solution with very high accuracy is obtained and a nearly exponential convergence rate \(\exp(-cM/\log M)\), \(c > 0\) in the error is observed where \(M\) is a parameter representing the number of terms in the Sinc expansion. We compare the result with the one based on the single exponential (SE) transformation by \textit{B. V. Riley} [Appl. Numer. Math. 9, No.~3--5, 249--257 (1992; Zbl 0757.65148)] which made us confirm the high efficiency of the present method.
0293.65011; 0757.65148