Spectral collocation methods for nonlinear weakly singular Volterra integro-differential equations. (English) Zbl 1416.65236
Summary: In this paper, a spectral collocation approximation is proposed for neutral and nonlinear weakly singular Volterra integro-differential equations (VIDEs) with non-smooth solutions. We use some suitable variable transformations to change the original equation into a new equation, so that the solution of the resulting equation possesses better regularity, and the the Jacobi orthogonal polynomial theory can be applied conveniently. Under reasonable assumptions on the nonlinearity, we carry out a rigorous error analysis in \({L}^\infty\) norm and weighted \({L}^2\) norm. To perform the numerical simulations, some test examples (linear and nonlinear) are considered with nonsmooth solutions, and numerical results are presented. Further more, the comparative study of the proposed methods with some existing numerical methods is provided.
MSC:
| 65L60 | Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations |
| 45J05 | Integro-ordinary differential equations |
| 45D05 | Volterra integral equations |
Keywords:
exponential rate of convergence; nonlinear Volterra integro-differential equations; spectral collocation methodReferences:
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