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An \(hp\)-version Legendre-Jacobi spectral collocation method for Volterra integro-differential equations with smooth and weakly singular kernels. (English) Zbl 1364.65299

Summary: In this paper, we present an \( hp\)-version Legendre-Jacobi spectral collocation method for Volterra integro-differential equations with smooth and weakly singular kernels. We establish several new approximation results of the Legendre/Jacobi polynomial interpolations for both smooth and singular functions. As applications of these approximation results, we derive \( hp\)-version error bounds of the Legendre-Jacobi collocation method under the \( H^1\)-norm for the Volterra integro-differential equations with smooth solutions on arbitrary meshes and singular solutions on quasi-uniform meshes. We also show the exponential rates of convergence for singular solutions by using geometric time partitions and linearly increasing polynomial degrees. Numerical experiments are included to illustrate the theoretical results.

MSC:

65R20 Numerical methods for integral equations
45D05 Volterra integral equations
45J05 Integro-ordinary differential equations
45E10 Integral equations of the convolution type (Abel, Picard, Toeplitz and Wiener-Hopf type)
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