The Research And Its Related Applications On Modified Collocation Method For Integral Equations Of Second Kind With Weakly Singular Kernel

The numerical solution of the integral equations is one of the general interests among the researchers in the field of Computational Mathematics including the creation,the modifying,and the applications of the numerical methods.In recent years,the Fredholm integral equations of the second kind with weakly singular kernel and the fractional Integro-Differential equations,whose numerical solutions can not be solved by the traditional numerical methods directly because of their singularity,make it essential for us to develop new numerical methods or modify the traditional numerical methods.The main work of this paper is modifying the traditional collocation method with Bernstein collocation method by iterating the kernel,selecting proper configuration points,discrete the integration and choosing the appropriate orthogonal polynomial functions such that the equations are transformed into linear algebraic equations,which make it suitable for resolving the integral equations with weakly singular kernel,and the modified collocation method has obtained higher accuracy,less calculation time and wider range of applications compared with the traditional one.Further more,numerical experiments with the integral equations with weakly singular kernel shows that the subdividing approach of the integration nodes and the configuration nodes should not be the same.Three kinds of approaches of subdividing the intervals had been presented and different approaches corresponding to different numerical accuracy,and the best subdividing approach has been obtained in this paper.Finally,the Simulated Annealing Algorithm has been introduced as the auxiliary algorithm for seeking the optimizing configuration nodes by iterations to improve the numerical accuracy of the numerical method,which has achieved the optimal results.