Study On Numerical Methods For Singular Integral Equations Derived From Two Kinds Of Mechanical Problems

In recent years,many scholars have made great achievements in the study of singular integral equations.Singular integral equations also play an important role in solving mathematical physics problems such as elastic-ity theory and fracture mechanics.In this thesis,a novel numerical method is proposed to find the approximate solution of the singular integral equa-tions derived from two kinds of mechanical problems,and the convergence and error estimation of the numerical method are analysed.Some numerical examples are carried out to show the feasibility and effectiveness of the pro-posed method by comparing with the existing ones.The main contents are given as follows.(1)A weakly singular integral equation derived from the fluid mechan-ics is investigated.Firstly,the fractional differential Bagley-Torvik equation with variable coefficients is integrated and transformed into the second kind of Volterra integral equation with weakly singular kernel function.Then the contraction operator theorem in Banach spaces is further used to address the uniqueness of the solution for the obtained Volterra integral equation,and the corresponding sufficient conditions are given.An approximate solution of the second kind Volterra integral equation with weakly singular kernel is constructed and its convergence and error estimate are made.Finally,the ap-proximate solution of the fractional differential Bagley-Torvik equation with variable coefficients is calculated and compared with other methods to verify the proposed method.In particular,for the integral equation with the exact solution as a polynomial function,the method of this paper can obtain the exact solution.If the exact solution of the integral equation to be solved is unknown,the comparison with the classical difference method shows that the proposed method is still valid.(2)The singular integral equation derived from the cruciform crack prob-lem in fracture mechanics is investigated.The main difference between the singular integral equation and the general integral equation is that it has sin-gularity.A modified numerical method is proposed to construct the approx-imate solution of the derived equation.By introducing the weight parameter?,the accuracy of the approximate solution can be improved.The error and convergence analysis are further made,and some numerical examples are of-fered to verify the modified method.The above research fully reflects the application of singular integral e-quation in practical problems,and provides a certain theoretical basis for solving similar issues such as mechanics and engineering.