×

The exact solution of a linear integral equation with weakly singular kernel. (English) Zbl 1144.45002

The paper deals with the weakly-singular linear Volterra integral equation of the second kind
\[ u(t)+\int_a^t\frac{H(t,s)}{(t-s)^\alpha }u(s) \,ds=f(t),\quad 0<\alpha <1,\;a\leq t\leq b. \]
The unique solution of the equation belongs to \(W_2^1[a,b]\) which is proved to be a reproducing kernel space with simple reproducing kernel. The expression of the reproducing kernel is given. The reproducing kernel method of a linear operator equation \(Au=f\) , which requests the image space of the operator \(A\) is \(W_2^1[a,b]\) and the operator \(A\) is bounded, is improved. Namely, the request for the image space is weakened to be \(L^2[a,b],\) and the boundedness of the operator \(A\) is also not required. The authors give the exact solution of the equation, denoted by a series in the reproducing kernel space \(W_2^1[a,b].\) After truncating the series, the approximate solution (which converges uniformly to the exact solution) is obtained. The effectiveness of the method is shown by the numerical experiments.

MSC:

45E10 Integral equations of the convolution type (Abel, Picard, Toeplitz and Wiener-Hopf type)
65R20 Numerical methods for integral equations
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] de Hong, F.R.; Weiss, R., High order methods for a class of Volterra integral equation with weakly singular kernels, SIAM J. numer. anal., 11, 1166-1180, (1974) · Zbl 0292.65067
[2] Brunner, H., The numerical solution of weakly singular Volterra integral equations by collocation on graded meshes, Math. comp., 45, 417-437, (1985) · Zbl 0584.65093
[3] Galperin, E.A.; Kansa, E.J.; Makroglou, A.; Nelson, S.A., Mathematical programming methods in the numerical solution of Volterra integral and integral-differential equations with weakly-singular kernel, Nonlinear anal., 30, 3, 1505-1513, (1997) · Zbl 0891.65141
[4] Galperin, E.A.; Kansa, E.J.; Makroglon, A.; Nelson, S.A., Variable transformations in the numerical solution of second kind Volterra integral equations with continuous and weakly singular extension to Fredholm integral equation, J. comput. appl. math., 115, 193-211, (2000) · Zbl 0958.65144
[5] Cui, M.G.; Wu, B.Y., Numerical analysis in reproducing kernel space, (2004), Scientific Press Peking
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.