Modified Three-Point Fractional Formulas with Richardson Extrapolation

Abstract

In this paper, we introduce new three-point fractional formulas which represent three generalizations for the well-known classical three-point formulas; central, forward and backward formulas. This has enabled us to study the function’s behavior according to different fractional-order values of α numerically. Accordingly, we then introduce a new methodology for Richardson extrapolation depending on the fractional central formula in order to obtain a high accuracy for the gained approximations. We compare the efficiency of the proposed methods by using tables and figures to show their reliability.

Mathematical analysis of a randomized method for fractional Carathéodory type equation with time‐irregular coefficients

“…Complex modeling problems in physics, mechanics, biology, and engineering are the driving force for the theoretical and applied study of fractional calculus [2]. Since most fractional models cannot be solved analytically, many researchers resort to developing efficient and reliable numerical methods to solve fractional equations [3,4]. This paper is devoted to consider randomized estimates for fractional Carathéodory type differential equations in the following form, { 𝐶 D 𝛼 0,𝑡 𝑢(𝑡) = 𝑓(𝑡, 𝑢(𝑡)), 𝑡 ∈ [0, 𝑇], 𝛼 ∈ (0, 1) 𝑢(0) = 𝑢 0 , (1.1) where 𝑇 ∈ (0, ∞), 𝑢 ∶ [0, 𝑇] → ℝ 𝑑 , 𝑑 ∈ ℕ, and the initial condition 𝑢 0 ∈ ℝ 𝑑 .…”

“…Complex modeling problems in physics, mechanics, biology, and engineering are the driving force for the theoretical and applied study of fractional calculus [2]. Since most fractional models cannot be solved analytically, many researchers resort to developing efficient and reliable numerical methods to solve fractional equations [3, 4].…”

Uncertain Stochastic Hybrid Age-Dependent Population Equation Based on Subadditive Measure: Existence, Uniqueness and Exponential Stability

“…For more models on using the Euler-Maruyama approximation method to deal with stochastic differential systems driven by Wiener process, please refer to Refs. [6,7], Moreover, in recent decades, a significant and expanding body of literature has emerged, investigating models and numerical algorithms for stochastic age-dependent population equations, such as [8,9].…”

The FitzHugh–Nagumo Model Described by Fractional Difference Equations: Stability and Numerical Simulation

“…With the nabla operator, Atici and Eloe [13] investigated the structure of a discrete fractional calculus. For additional information on recent advances in fractional discrete calculus, see [14][15][16][17][18][19].…”