数学
分数阶微积分
离散化
非线性系统
格朗沃尔不等式
独特性
应用数学
核(代数)
数学分析
理论(学习稳定性)
纯数学
物理
量子力学
机器学习
计算机科学
不平等
作者
Anatoly A. Alikhanov,Mohammad Shahbazi Asl,Chengming Huang,Aslanbek Khibiev
标识
DOI:10.1016/j.cam.2023.115515
摘要
This paper investigates a class of the time-fractional diffusion-wave equation (TFDWE), which incorporates a fractional derivative in the Caputo sense of order α+1 where 0<α<1. Initially, the original problem is transformed into a new model that incorporates the fractional Riemann–Liouville integral. Subsequently, a more generalized model is considered and investigated, which includes the generalized fractional integral with memory kernel. The paper establishes the existence and uniqueness of a numerical solution for the problem. We introduce a discretization technique for approximating the generalized fractional Riemann–Liouville integral and study its fundamental properties. Based on this technique, we propose a second-order numerical scheme for approximating the generalized model. The proof of the stability of the proposed difference scheme when considered in terms of the L2 norm is established. Furthermore, a difference scheme is devised for solving a nonlinear generalized model with time delay. The convergence and stability analysis of this particular case are established using the discrete Gronwall inequality. Numerical examples are provided to evaluate the effectiveness and reliability of the proposed methods and to support our theoretical analysis.
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