The Reproducing Kernel Method For Solving The Volterra Integral Equation Systems With Variable Coefficients

Volterra integral equation systems are used in many applied sciences to model dynamical systems and social systems. It is important for us to study the Volterra integral equation systems.In this paper, the new numerical method is given to obtain the approximate solution. Based on the theory of the reproducing kernel space, the exact solution and the approximate solution are constructed by the reproducing property.Firstly, according to the form of the integral equation systems, the new Hilbert space is constructed by the reproducing kernel space. In the reproducing kernel space, the reproducing kernel function is expressed with piecewise polynomial. In the same, the variable coefficients of equation systems are assigned to the space. So, the computation of systems is simplified. We provide the good space frame.Secondly, making good use of the reproducing kernel techniques, a complete system of the Hilbert space is obtained. Employing the Gram-Schmidt process, the orthonormal system is given. We expand the function and get the exact solution and approximate solution.Finally, the validity check of the algorithm is done by the practical examples given in the paper. Compared with the references, our algorithm is of rapid convergence and computational accuracy.