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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 $$u(t)+\int_a^t\frac{H(t,s)}{(t-s)^\alpha }u(s) \,ds=f(t),\quad 0<\alpha <1,\ a\leq t\leq b.$$ 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 65R20 Integral equations (numerical methods)
Full Text:
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 · doi:10.1137/0711088 [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 [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, No. 3, 1505-1513 (1997) · Zbl 0891.65141 · doi:10.1016/S0362-546X(96)00340-9 [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)