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.


45E10 Integral equations of the convolution type (Abel, Picard, Toeplitz and Wiener-Hopf type)
65R20 Numerical methods for integral equations
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