High Accuracy Algorithm Of The Numerical Solution Of The Hypersingular Integral Equation

This paper investigates the numerical solution of the hyper-singular integral equations. Hyper-singular integral equations have been widely used in various practical engineering problems(mathematics and physics problems), such as theory of close and long-distance interaction, soliton’s theory, theory of elasticity and thermoelasticity, aerodynamics, electrodynamics and so on. However, this kind of equations is divergence in the common sense and principal value sense, which increases the difficulty of the research. The contents of this paper include: the numerical solution of the Fredholm equation of the second kind, the numerical solution of the hyper-singular equations of the first kind.Firstly, the problem of the numerical solution of the hyper-singular equations of the first kind is investigated. Traditional collection method with the linear interpolation function cannot be used to solve the hyper-singular equations because of this method unable to solve the singularity on interpolation nodes. The modified linear collection method is introduced by improving the linear basic function. Through the study of discrete matrix, we can demonstrate the uniqueness of the approximate solution and give the posteriori error estimation between analytical solution and approximate solution. Finally, the feasibility and effectiveness of the algorithm is shown by the numerical simulation.Secondly, the problem of the numerical solution of the general fractional order hyper-singular equations of the first kind is studied. By enhancement to the traditional constant collection method, the singularity of the interpolation node is eliminated. By using the Hadamard theorem to analysis the discrete matrix, the existence and uniqueness of the solution of the system is testified. And we get the posteriori error estimation between the analytic solution and numerical solution. At last, a numerical example is given to illustrate this theorem.Thirdly, the numerical solution of the hyper-singular Fredholm equation of the first kind is considered. By the contrast of const collection method and linear collection method, we choose linear collection method to solve this problem, which proposed in the first part. Through the theoretical proof and numerical example, the high accuracy of numerical solution is verified.Finally, the numerical algorithm of hyper-singular integral is studied in this paper, which lay the foundation of follow-up work to study the approximate solution of the hyper-singular equations with logarithmic singular.