A M\"untz-Collocation spectral method for weakly singular volterra integral equations

In this paper we propose and analyze a fractional Jacobi-collocation spectral method for the second kind Volterra integral equations (VIEs) with weakly singular kernel $(x-s)^{-\mu},0<\mu<1$. First we develop a family of fractional Jacobi polynomials, along with basic approximation results for some weighted projection and interpolation operators defined in suitable weighted Sobolev spaces. Then we construct an efficient fractional Jacobi-collocation spectral method for the VIEs using the zeros of the new developed fractional Jacobi polynomial. A detailed convergence analysis is carried out to derive error estimates of the numerical solution in both $L^{\infty}$- and weighted $L^{2}$-norms. The main novelty of the paper is that the proposed method is highly efficient for typical solutions that VIEs usually possess. Precisely, it is proved that the exponential convergence rate can be achieved for solutions which are smooth after the variable change $x\rightarrow x^{1/\lambda}$ for a suitable real number $\lambda$. Finally a series of numerical examples are presented to demonstrate the efficiency of the method

High-Order Multivariate Spectral Algorithms for High-Dimensional Nonlinear Weakly Singular Integral Equations with Delay

One of the open problems in the numerical analysis of solutions to high-dimensional nonlinear integral equations with memory kernel and proportional delay is how to preserve the high-order accuracy for nonsmooth solutions. It is well-known that the solutions to these equations display a typical weak singularity at the initial time, which causes challenges in developing high-order and efficient numerical algorithms. The key idea of the proposed approach is to adopt a smoothing transformation for the multivariate spectral collocation method to circumvent the curse of singularity at the beginning of time. Therefore, the singularity of the approximate solution can be tailored to that of the exact one, resulting in high-order spectral collocation algorithms. Moreover, we provide a framework for studying the rate of convergence of the proposed algorithm. Finally, we give a numerical test example to show that the approach can preserve the nonsmooth solution to the underlying problems. © 2022 by the authors.King Saud University, KSUM. A. Zaky and A. Aldraiweesh extend their appreciation to Distinguished Scientist Fellowship Program (DSFP) at King Saud University (Saudi Arabia)

Variational Iteration Method for Volterra Functional Integrodifferential Equations with Vanishing Linear Delays

Application process of variational iteration method is presented in order to solve the Volterra functional integrodifferential equations which have multi terms and vanishing delays where the delay function θ(t) vanishes inside the integral limits such that θ(t)=qt for 0<q<1, t≥0. Either the approximate solutions that are converging to the exact solutions or the exact solutions of three test problems are obtained by using this presented process. The numerical solutions and the absolute errors are shown in figures and tables

A SPECTRAL METHOD FOR PANTOGRAPH-TYPE DELAY DIFFERENTIAL EQUATIONS AND ITS CONVERGENCE ANALYSIS

Abstract We propose a novel numerical approach for delay differential equations with vanishing proportional delays based on spectral methods. A Legendre-collocation method is employed to obtain highly accurate numerical approximations to the exact solution. It is proved theoretically and demonstrated numerically that the proposed method converges exponentially provided that the data in the given pantograph delay differential equation are smooth. Mathematics subject classification: 65M06, 65N12

An hp hp -version spectral collocation method for fractional Volterra integro-differential equations with weakly singular kernels

We present a multi-step spectral collocation method to solve Caputo-type fractional integro-differential equations (FIDEs) involving weakly singular kernels. We reformulate the problem as the second type Volterra integral equation (VIE) with two different weakly singular kernels. Based on these integral equations, we construct a multi-step Legendre-Gauss spectral collocation scheme for the problem. The $hp$-version convergence is established rigorously. To demonstrate the effectiveness of the suggested method and the validity of the theoretical results, the results of some numerical experiments are presented

Equations with infinite delay : Numerical bifurcation analysis via pseudospectral discretization

We address the problem of the numerical bifurcation analysis of general nonlinear delay equations, including integral and integro-differential equations, for which no software is currently available. Pseudospectral discretization is applied to the abstract reformulation of equations with infinite delay to obtain a finite dimensional system of ordinary differential equations, whose properties can be numerically studied with well-developed software. We explore the applicability of the method on some test problems and provide some numerical evidence of the convergence of the approximations.Peer reviewe