A Fractional Order Collocation Method for Second Kind Volterra Integral Equations with Weakly Singular Kernels

• Published in: Journal of scientific computing Vol. 75; no. 2; pp. 970 - 992
• Main Authors: Cai, Haotao, Chen, Yanping
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.05.2018; Springer Nature B.V
• Subjects: Accuracy; Algorithms; Chebyshev approximation; Collocation methods; Computational Mathematics and Numerical Analysis; Integral equations; Interpolation; Mathematical analysis; Mathematical and Computational Engineering; Mathematical and Computational Physics; Mathematics; Mathematics and Statistics; Methods; Numerical analysis; Quadratures; Theoretical; Volterra integral equations; Second kind Volterra integral equations with weakly singular kernels; 65R20; Condition number; A fractional order collocations spectral method; Stability analysis; 45E05; Convergence analysis
• ISSN: 0885-7474; 1573-7691
• DOI: 10.1007/s10915-017-0568-7

In this paper, we develop a fractional order spectral collocation method for solving second kind Volterra integral equations with weakly singular kernels. It is well known that the original solution of second kind Volterra integral equations with weakly singular kernels usually can be split into two parts, the first is the singular part and the second is the smooth part with the assumption that the integer m being its smooth order. On the basis of this characteristic of the solution, we first choose the fractional order Lagrange interpolation function of Chebyshev type as the basis of the approximate space in the collocation method, and then construct a simple quadrature rule to obtain a fully discrete linear system. Consequently, with the help of the Lagrange interpolation approximate theory we establish that the fully discrete approximate equation has a unique solution for sufficiently large n , where n + 1 denotes the dimension of the approximate space. Moreover, we prove that the approximate solution arrives at an optimal convergence order O ( n - m log n ) in the infinite norm and O ( n - m ) in the weighted square norm. In addition, we prove that for sufficiently large n , the infinity-norm condition number of the coefficient matrix corresponding to the linear system is O ( log 2 n ) and its spectral condition number is O ( 1 ) . Numerical examples are presented to demonstrate the effectiveness of the proposed method.