## 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
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### References:

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