Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Euler-Maruyama approximation for stochastic fractional neutral integro-differential equations with weakly singular kernel

Javad A. Asadzade, Nazim I. Mahmudov

TL;DR

This work addresses nonlinear stochastic fractional neutral integro-differential equations with Abel-type weakly singular kernels by proving well-posedness under local Lipschitz and linear growth conditions. It develops an Euler–Maruyama discretization, proving strong convergence and, under global Lipschitz assumptions, a quantified convergence rate of $\big(E[|Z(t)-z(t)|^2]\big)^{1/2}\le C h^{\alpha}$. The analysis leverages a stochastic Volterra integral equation reformulation, moment bounds, and weakly singular Gronwall-type inequalities, providing both theoretical guarantees and a numerical validation in a one-dimensional example. The results advance reliable numerical approximation for SFNIDEs with memory and noise, relevant to applications in physics, finance, and engineering where fractional dynamics and stochasticity interact with weakly singular kernels.

Abstract

This manuscript examines the problem of nonlinear stochastic fractional neutral integro-differential equations with weakly singular kernels. Our focus is on obtaining precise estimates to cover all possible cases of Abel-type singular kernels. Initially, we establish the existence, uniqueness, and continuous dependence on the initial value of the true solution, assuming a local Lipschitz condition and linear growth condition. Additionally, we develop the Euler-Maruyama method for the numerical solution of the equation and prove its strong convergence under the same conditions as the well-posedness. Moreover, we determine the accurate convergence rate of this method under global Lipschitz conditions and linear growth conditions.

Euler-Maruyama approximation for stochastic fractional neutral integro-differential equations with weakly singular kernel

TL;DR

This work addresses nonlinear stochastic fractional neutral integro-differential equations with Abel-type weakly singular kernels by proving well-posedness under local Lipschitz and linear growth conditions. It develops an Euler–Maruyama discretization, proving strong convergence and, under global Lipschitz assumptions, a quantified convergence rate of . The analysis leverages a stochastic Volterra integral equation reformulation, moment bounds, and weakly singular Gronwall-type inequalities, providing both theoretical guarantees and a numerical validation in a one-dimensional example. The results advance reliable numerical approximation for SFNIDEs with memory and noise, relevant to applications in physics, finance, and engineering where fractional dynamics and stochasticity interact with weakly singular kernels.

Abstract

This manuscript examines the problem of nonlinear stochastic fractional neutral integro-differential equations with weakly singular kernels. Our focus is on obtaining precise estimates to cover all possible cases of Abel-type singular kernels. Initially, we establish the existence, uniqueness, and continuous dependence on the initial value of the true solution, assuming a local Lipschitz condition and linear growth condition. Additionally, we develop the Euler-Maruyama method for the numerical solution of the equation and prove its strong convergence under the same conditions as the well-posedness. Moreover, we determine the accurate convergence rate of this method under global Lipschitz conditions and linear growth conditions.
Paper Structure (14 sections, 12 theorems, 84 equations, 1 figure)

This paper contains 14 sections, 12 theorems, 84 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Assumptions
  4. Some auxiliary results.
  5. Well-posedness of stochastic fractional neutral integro-differentional equation
  6. Existence and uniqueness of solution of stochastic fractional neutral integro-differentional equation
  7. Proof of the Theorem 1.
  8. Continious dependence of solutions on the initial value.
  9. Proof of the Theorem 2.
  10. Strong convergence and convergence rate of the Euler-Maruyama method.
  11. Strong convergence.
  12. Convergence rate.
  13. Numerical example
  14. Conclusion

Key Result

Theorem 1

The Stochastic Fractional Neutral Integro-differential Equation (A) has a unique solution $z(t)$ under Assumptions (1), (3), and (4). Furthermore, for each positive $p\geq 2$,

Figures (1)

  • Figure 1: The mean square errors of the EM scheme (\ref{['18']}) for Example 1. $\color{blue}\boxdot$$\alpha=0.4, \alpha_{1}=0.5, \beta_{1}=0.6, \beta_{2}=0.4$, $\color{red}\boxdot$$\alpha=0.4, \alpha_{1}=0.5, \beta_{1}=0.8, \beta_{2}=0.3$, $\color{yellow}\boxdot$$\alpha=0.8, \alpha_{1}=0.9, \beta_{1}=0.6, \beta_{2}=0.4$, $\color{purple}\boxdot$$\alpha=0.8, \alpha_{1}=0.9, \beta_{1}=0.8, \beta_{2}=0.3$.

Theorems & Definitions (25)

  • Theorem 1
  • Theorem 2
  • Remark 1
  • Proposition 3
  • Definition 1
  • Theorem 4
  • proof
  • Lemma 1
  • Lemma 2
  • Lemma 3
  • ...and 15 more