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A Fractional spectral method for weakly singular Volterra integro-differential equations with delays of the third-kind

Borui Zhao

Abstract

In this paper, we present a fractional spectral collocation method for solving a class of weakly singular Volterra integro-differential equations (VDIEs) with proportional delays and cordial operators. Assuming the underlying solutions are in a specific function space, we derive error estimates in the $L^2_{ω^{α,β,λ}}$ and $L^{\infty}$-norms. A rigorous proof reveals that the numerical errors decay exponentially with the appropriate selections of parameters $λ$. Subsequently, numerical experiments are conducted to validate the effectiveness of the method.

A Fractional spectral method for weakly singular Volterra integro-differential equations with delays of the third-kind

Abstract

In this paper, we present a fractional spectral collocation method for solving a class of weakly singular Volterra integro-differential equations (VDIEs) with proportional delays and cordial operators. Assuming the underlying solutions are in a specific function space, we derive error estimates in the and -norms. A rigorous proof reveals that the numerical errors decay exponentially with the appropriate selections of parameters . Subsequently, numerical experiments are conducted to validate the effectiveness of the method.
Paper Structure (12 sections, 15 theorems, 122 equations, 10 figures, 10 tables)

This paper contains 12 sections, 15 theorems, 122 equations, 10 figures, 10 tables.

Table of Contents

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

Key Result

Theorem 1

Let $D = {\lbrace(t,s),(t,\tau):0 \leq s,\tau \leq t \leq T\rbrace}$, $K_1(t, s),K_2(t, \tau) \in C(D)$ and $p_1{(t)}$,$q_1{(t)}$,$g_1{(t)}$$\in C[0, T]$, theneq_prime has a unique solution $y \in C^1{[0, T]}$ on the interval $[0, T]$. Furthermore, when $p_1{(t)},q_1{(t)},g_1{(t)} \in C^{m}[0, T], K

Figures (10)

  • Figure 1: Example \ref{['example_third_NO1']} with $\lambda=\frac{1}{2}$
  • Figure 2: Example \ref{['example_third_NO1']} with $\lambda=1$
  • Figure 3: Example \ref{['example_third_NO2']} with $\lambda=\frac{1}{3}$
  • Figure 4: Example \ref{['example_third_NO2']} with $\lambda=1$
  • Figure 5: Example \ref{['example_third_NO3']} with $\lambda=\frac{1}{2}$
  • ...and 5 more figures

Theorems & Definitions (27)

  • Theorem 1
  • proof
  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • proof
  • Lemma 5
  • proof
  • Lemma 6
  • ...and 17 more