Legendre Spectral Collocation Methods for Pantograph Volterra Delay-Integro-Differential Equations

Convergence Analysis of Generalized Jacobi-Galerkin Methods for Second Kind Volterra Integral Equations with Weakly Singular Kernels

“…There are many numerical attempts based on the spline approximation to overcome the difficulty caused by the singularity of the solution of (2) (see [1][2][3][4][5][6][7][8]). Recently, spectral methods using Jacobi polynomial basis have received considerable attention to approximating the solution of integral equations due to their high accuracy and easy implementation (see [9][10][11][12][13][14][15][16][17]). In particular, Chen and Tang in [11] proposed a Jacobi-collocation spectral method for second kind Volterra integral equations with weakly singular kernels.…”

Spectral-Collocation Method for Volterra Delay Integro-Differential Equations with Weakly Singular Kernels

“…Wan, Chen and Huang [21] used the spectral Galerkin method to solve the nonlinear Volterra integral equations of the second kind. The authors in [1,9,10,19,22,23,26,31] proposed the spectral Legendre-collocation method for Volterra integral or integro-differential equations with smooth kernels. In [6-8, 12, 24, 25, 29, 30, 32] the spectral Jacobi-collocation method was successfully applied to solve Volterra integral or integro-differential equations with weakly kernels and fractional integro-differential equations.…”

“…In this work, we will consider the special case that the exact solutions of (1.1a)-(1.1b) are smooth (see [2]). The main purpose of this paper is to develop a spectral Jacobi-collocation method for Volterra integro-differential equations with weakly singular kernels and pantograph delays based on the works of [23] and [32]. This paper is arranged as follows.…”

Legendre Spectral Collocation Methods for Volterra Delay-Integro-Differential Equations

“…They proposed the spectral collocation method after the results on existence, uniqueness and regularity of the solutions were established. The analytical and numerical techniques used in these works can be extended to delay differential and delay integral equations [2,3,27]. Wei and Chen [27] introduced Legendre spectral collocation methods to approximate smooth solutions of Volterra integro-differential equations with proportional delays.…”

“…The analytical and numerical techniques used in these works can be extended to delay differential and delay integral equations [2,3,27]. Wei and Chen [27] introduced Legendre spectral collocation methods to approximate smooth solutions of Volterra integro-differential equations with proportional delays. Meanwhile, they provided a vigorous error analysis and proved that both the errors of approximate solutions and the errors of approximate derivatives decayed exponentially in L 2 -norm and L ∞ -norm.…”