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A Müntz-collocation spectral method for weakly singular Volterra delay-integro-differential equations

Borui Zhao

Abstract

A Müntz spectral collocation method is implemented for solving weakly singular Volterra integro-differential equations (VDIEs) with proportional delays. After constructing the numerical scheme to seek an approximate solution, we derive error estimates in a weighted $L^2$ and $L^{\infty}$-norms. A rigorous proof reveals that the proposed method can handle the weak singularity of the exact solution at the initial point $t=0$, with the numerical errors decaying exponentially in certain cases. Moreover, several examples will illustrate our convergence analysis.

A Müntz-collocation spectral method for weakly singular Volterra delay-integro-differential equations

Abstract

A Müntz spectral collocation method is implemented for solving weakly singular Volterra integro-differential equations (VDIEs) with proportional delays. After constructing the numerical scheme to seek an approximate solution, we derive error estimates in a weighted and -norms. A rigorous proof reveals that the proposed method can handle the weak singularity of the exact solution at the initial point , with the numerical errors decaying exponentially in certain cases. Moreover, several examples will illustrate our convergence analysis.
Paper Structure (11 sections, 14 theorems, 106 equations, 8 figures, 8 tables)

This paper contains 11 sections, 14 theorems, 106 equations, 8 figures, 8 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Function spaces
  4. Fractional Jaboci polynomials
  5. Elementary lemmas
  6. Numerical Scheme
  7. Convergence Analysis
  8. Error estimate in $L^{\infty}$-norm
  9. Error estimate in $L_{\omega^{\alpha,\beta,\lambda}}^{2}$-norm
  10. Numerical experiments
  11. Conclusions

Key Result

Lemma 1

$($Gronwall inequality$)$ Assume that If $f(\theta), g(\theta)$ are non-negative integrable functions on $[0, 1]$ and $C > 0$, then there exists $L>0$ such that:

Figures (8)

  • Figure 1: Example \ref{['example_second_NO1']} with $\lambda=\frac{1}{2}$
  • Figure 2: Example \ref{['example_second_NO1']} with $\lambda=1$
  • Figure 3: Example \ref{['example_second_NO2']} with $\lambda=\frac{1}{3}$
  • Figure 4: Example \ref{['example_second_NO2']} with $\lambda=1$
  • Figure 5: Example \ref{['example_second_NO3']} with $\lambda=\frac{1}{2}$
  • ...and 3 more figures

Theorems & Definitions (25)

  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • proof
  • Lemma 5
  • proof
  • Lemma 6
  • Lemma 7
  • Lemma 8
  • ...and 15 more