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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)+ a t H(t,s) (t-s) α u(s)ds=f(t),0<α<1,atb·

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:
45E10Integral equations of the convolution type
65R20Integral equations (numerical methods)
References:
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[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 · doi:10.2307/2008134
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[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 · doi:10.1016/S0377-0427(99)00297-6
[5]Cui, M. G.; Wu, B. Y.: Numerical analysis in reproducing kernel space, (2004)