Mechanical quadrature methods and their extrapolation for solving first kind Abel integral equations

作者：Liu Ya ping^*; Tao Lu

来源：Journal of Computational and Applied Mathematics, 2007, 201(1): 300-313.

DOI：10.1016/j.cam.2006.02.021

• the first kind Abel integral equation
• quadrature method
• asymptotic expansion
• extrapolation

This paper presents high accuracy mechanical quadrature methods for solving first kind Abel integral equations. To avoid the ill-posedness of problem, the first kind Abel integral equation is transformed to the second kind Volterra integral equation with a continuous kernel and a smooth right-hand side term expressed by weakly singular integrals. By using periodization method and modified trapezoidal integration rule, not only high accuracy approximation of the kernel and the right-hand side term can be easily computed, but also two quadrature algorithms for solving first kind Abel integral equations are proposed, which have the high accuracy O(h(2)) and asymptotic expansion of the errors. Then by means of Richardson extrapolation, an approximation with higher accuracy order O(h(3)) is obtained. Moreover, an a posteriori error estimate for the algorithms is derived. Some numerical results show the efficiency of our methods.

• 出版日期2007-4-1
• 单位四川大学