Journal of Scientific Computing

, Volume 69, Issue 2, pp 673–695 | Cite as

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

  • Sonia Seyed Allaei
  • Teresa Diogo
  • Magda Rebelo


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.


Jacobi spectral collocation method Nonlinear Volterra integral equation Weakly singular kernel Convergence analysis 

Mathematics Subject Classification

65R20 45J05 



This work was partially supported by Fundação para a Ciência e a Tecnologia (Portuguese Foundation for Science and Technology), through the projects Pest-OE/MAT/UI0822/2014 and PTDC/MAT/101867/2008. The research of the first author (S. Seyed Allaei) was also co-financed by the Hong Kong Research Grants Council (RGC Project HKBU 200113 and 1369648). The work of the third author was also partially supported by the FCT Project UID/MAT/00297/2013 (Centro de Matemática e Aplicações). The first author would like to thank Professor Hermann Brunner for his valuable suggestions and constructive discussions.


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Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Department of MathematicsHong Kong Baptist UniversityKowloon TongChina
  2. 2.Center for Computational and Stochastic Mathematics (CEMAT), Department of Mathematics, Instituto Superior TécnicoUniversidade de LisboaLisbonPortugal
  3. 3.Centro de Matemática e Aplicações (CMA) and Departamento de Matemática, Faculdade de Ciências e TecnologiaUniversidade NOVA de LisboaCaparicaPortugal

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