The Jacobi Collocation Method for a Class of Nonlinear Volterra Integral Equations with Weakly Singular Kernel

详细信息    查看全文

• 作者：Sonia Seyed Allaei ; Teresa Diogo ; Magda Rebelo
• 关键词：Jacobi spectral collocation method ; Nonlinear Volterra integral equation ; Weakly singular kernel ; Convergence analysis
• 刊名：Journal of Scientific Computing
• 出版年：2016
• 出版时间：November 2016
• 年：2016
• 卷：69
• 期：2
• 页码：673-695
• 全文大小：666 KB
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Algorithms
  Computational Mathematics and Numerical Analysis
  Applied Mathematics and Computational Methods of Engineering
  Mathematical and Computational Physics
  
• 出版者：Springer Netherlands
• ISSN：1573-7691
• 卷排序：69

文摘

A Jacobi spectral collocation method is proposed for the solution of a class of nonlinear Volterra integral equations with a kernel of the general form \( x^{\beta }\, (z-x)^{-\alpha } \, g(y(x))\), where \(\alpha \in (0,1), \beta >0\) and g(y) is a nonlinear function. Typically, the kernel will contain both an Abel-type and an end point singularity. The solution to these equations will in general have a nonsmooth behaviour which causes a drop in the global convergence orders of numerical methods with uniform meshes. In the considered approach a transformation of the independent variable is first introduced in order to obtain a new equation with a smoother solution. The Jacobi collocation method is then applied to the transformed equation and a complete convergence analysis of the method is carried out for the \(\displaystyle L^{\infty }\) and the \(L^2\) norms. Some numerical examples are presented to illustrate the exponential decay of the errors in the spectral approximation.