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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.

\[ u(t)+\int_a^t\frac{H(t,s)}{(t-s)^\alpha }u(s) \,ds=f(t),\quad 0<\alpha <1,\;a\leq t\leq b. \]

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.

Reviewer: Adrian Carabineanu (Bucureşti)

### MSC:

45E10 | Integral equations of the convolution type (Abel, Picard, Toeplitz and Wiener-Hopf type) |

65R20 | Numerical methods for integral equations |

### Keywords:

weakly singular kernel; Volterra linear integral equation; reproducing kernel; exact solution; linear operator equation; numerical experiments
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\textit{Z. Chen} and \textit{Y. Lin}, J. Math. Anal. Appl. 344, No. 2, 726--734 (2008; Zbl 1144.45002)

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### 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 |

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