The exact solution of a linear integral equation with weakly singular kernel

A space W₂¹ [a, b], which is proved to be a reproducing kernel space with simple reproducing kernel, is defined. The expression of its reproducing kernel function is given. Subsequently, a class of linear Volterra integral equation (VIE) with weakly singular kernel is discussed in the new reproducing kernel space. The reproducing kernel method of linear operator equation A u = f, which request the image space of operator A is W₂¹ [a, b] and operator A is bounded, is improved. Namely, the request for the image space is weakened to be L² [a, b], and the boundedness of operator A is also not required. As a result, the exact solution of the equation is obtained. The numerical experiments show the efficiency of our method.