zbMATH — the first resource for mathematics

Numerical solution of Volterra integral equations with weakly singular kernel based on the DE-sinc method. (English) Zbl 1152.65121
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 H. Takahasi and 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 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.

MSC:
 65R20 Numerical methods for integral equations 45E10 Integral equations of the convolution type (Abel, Picard, Toeplitz and Wiener-Hopf type)
Full Text:
References:
 [1] H. Brunner and P.J.van der Houwen, The Numerical Solution of Volterra Equations. North-Holland, Amsterdam, 1986. · Zbl 0611.65092 [2] G.E. Forsythe, M.A. Malcolm and C.B. Moler, Computer Methods for Mathematical Computations. Prentice-Hall, Inc., Englewood Cliffs, New Jersey, 1977. · Zbl 0361.65002 [3] M. Muhammad and M. Mori, Double exponential formulas for numerical indefinite integration. J. Comput. Appl. Math.,161 (2003), 431–448. · Zbl 1038.65018 [4] M. Muhammad, A. Nurmuhammad, M. Mori and M. Sugihara, Numerical solution of integral equations by means of the Sinc collocation method based on the double exponential transformation. J. Comput. Appl. Math.,177 (2005), 269–286. · Zbl 1072.65168 [5] B.V. Riley, The numerical solution of Volterra integral equations with nonsmooth solutions based on sinc approximation. J. Comput. Appl. Math.,9 (1992), 249–257. · Zbl 0757.65148 [6] F. Stenger, Numerical Methods Based on Sinc and Analytic Functions. Springer-Verlag, New York, 1993. · Zbl 0803.65141 [7] M. Sugihara, Near optimality of the Sinc approximation. Math. Comput.,72 (2003), 767–786. · Zbl 1013.41009 [8] H. Takahasi, Complex function theory and numerical analysis (in Japanese). Kokyuroku, RIMS Kyoto Univ.,253 (1975), 24–37. [9] H. Takahasi, Complex function theory and numerical analysis. Publ. RIMS Kyoto Univ.,41 (2005), 979–988. · Zbl 1100.65020 [10] H. Takahasi and M. Mori, Double exponential formulas for numerical integration. Publ. RIMS Kyoto Univ.,9 (1974), 721–741. · Zbl 0293.65011
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.