Fully discretized collocation methods for nonlinear singular Volterra integral equations

We consider a nonlinear weakly singular Volterra integral equation arising from a problem studied by Lighthill (1950) [1]. A series expansion for the solution is obtained and shown to be convergent in a neighbourhood of the origin. Owing to the singularity of the solution at the origin, the global convergence order of product integration and collocation methods is not optimal. However, the optimal orders can be recovered if we use the fully discretized collocation methods based on graded meshes. A theoretical proof is given and we present some numerical results which illustrate the performance of the methods.

[1]  Richard K Miller,et al.  Nonlinear Volterra Integral Equations , 1970 .

[2]  M. Lighthill.  Contributions to the theory of heat transfer through a laminar boundary layer , 1950, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

[3]  I. J. Kumar,et al.  On a Mellin transform technique for the asymptotic solution of a nonlinear Volterra integral equation , 1977, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.

[4]  Hermann Brunner,et al.  The numerical solution of weakly singular Volterra integral equations by collocation on graded meshes , 1985 .

[5]  Numerical solution of a nonlinear Abel type Volterra integral equation , 2006 .

[6]  Gennadi Vainikko,et al.  Cordial Volterra Integral Equations 1 , 2009 .

[7]  On the asymptotic solution of a nonlinear Volterra integral equation , 1971, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.

[8]  Hermann Brunner,et al.  The piecewise polynomial collocation method for nonlinear weakly singular Volterra equations , 1999, Math. Comput..

[9]  S. McKee,et al.  Weakly Singular Discrete Gronwall Inequalities , 1986 .

[10]  Arvet Pedas,et al.  Integral Equations with Diagonal and Boundary Singularities of the Kernel , 2006 .

[11]  Ivan P. Gavrilyuk,et al.  Collocation methods for Volterra integral and related functional equations , 2006, Math. Comput..

[12]  G. Vainikko,et al.  High-Order Methods for Volterra Integral Equations with General Weak Singularities , 2009 .

[13]  Gennadi Vainikko,et al.  Spline Collocation-Interpolation Method for Linear and Nonlinear Cordial Volterra Integral Equations , 2010 .

[14]  Teresa Diogo,et al.  A hybrid collocation method for a nonlinear Volterra integral equation with weakly singular kernel , 2010, J. Comput. Appl. Math..