The Reproducing Kernel Method For The Singular Integral Equations With Abel Kernel

Singular integral equations and system of integral equations with singularity arise in a variety of applied subjects such as dynamics, fracture mechanics, solid mechanics and so on. Due to its extensive applications, there is considerable interest for many scholars to solve singular integral equations. Because the problem of the singularity of the equations, generally numerical methods fail to produce good approximations to the solutions of the equations. Hence, it is a practical significance to form an efficient numerical method for solving the equations.The paper aims at finding a new method for solving singular integral equation (equations) in reproducing kernel space. Based on reproducing kernel theory, and employing the reproducing property of reproducing kernel function, we give the exact solution of singular integral equation (equations) in the reproducing kernel space.Firstly, the fundamental theories of the reproducing kernel space and the reproducing kernel function are introduced simplify. According to the definition of the inner product, the expression of the reproducing kernel function is given. The reproducing kernel function is constructed by the form of piecewise polynomial, which can simplify the calculation process. In this paper, improving the requirements of the operator, we construct a complete system of the reproducing kernel space directly by employing the property of the reproducing kernel function. Then the exact solution of the singular integral equation is given in the form of series. Finally the approximate solution is obtained by truncating series.In this paper, the system of integral equations with Abel kernel is discussed in a similar method. Firstly, according to the form of the system of integral equations, the new Hilbert space is constructed. Secondly, definition of the vector is applied and a complete system is constructed, then the exact solution and approximate solution of the equations are given.The method introduced in this paper overcome the singularity and obtain better results. Finally, the numerical examples demonstrate the feasibility and validity of the method. Compared with the references, the reproducing kernel method solving the singular integral equations has higher precision and computational speed.