The Rayleigh–Stokes Problem for a Heated Generalized Second-Grade Fluid with Fractional Derivative: An Implicit Scheme via Riemann–Liouville Integral

Abstract

The goal of this study is to use the fast algorithm to solve the Rayleigh–Stokes problem for heated generalized second-grade fluid (RSP-HGSGF) with Riemann–Liouville time fractional derivative using the fast algorithm. The modified implicit scheme, which is formulated by the Riemann–Liouville integral formula and applied to the fractional RSP-HGSGF, is proposed. Numerical experiments will be carried out to demonstrate that the scheme is simple to implement, and the results will reveal the best way to implement the suggested technique. The proposed scheme’s stability and convergence will be examined using the Fourier series. The method is stable, and the approximation solution approaches the exact solution. A numerical demonstration will be provided to demonstrate the applicability and viability of the suggested strategy.

Coupled system of three sequential Caputo fractional differential equations: Existence and stability analysis

“…Many important and interesting areas concerning research for fractional differential equations and fractional systems are devoted to the existence theory and stability analysis of the solutions. In recent times, researchers have given the existence, uniqueness, and Ulam stability of solutions for differential equations and systems of arbitrary order; see [17][18][19][20][21][22][23][24][25][26][27][28][29][30][31][32][33][34][35][36]. In recent years, many scientific researchers have considered differential equations and systems containing sequential fractional derivatives of different types; for instance, see [37][38][39][40][41][42][43][44].…”

Solvability and Mittag–Leffler–Ulam stability for fractional Duffing problem with three sequential fractional derivatives

“…In differential equations, the fractional derivative operators have shown its wings in the modeling of several problems in science, engineering, and technology as can be seen in earlier studies [1][2][3][4][5][6][7][8][9] and the references therein. In quite recent times, many researchers have given the existence, uniqueness, and various structures of Ulam stability and Mittag-Leffler-Ulam stability of solutions for differential equations of fractional order as in previous research [10][11][12][13][14][15][16][17][18][19][20][21][22][23] and the references therein. In recent years, many researchers have exposed the attention in the field of theory of nonlinear fractional differential equations, which will be used to describe the phenomena of the present day problems; for example, see earlier studies [24][25][26][27][28][29][30][31][32][33][34].…”

“…for 0 < 𝜂, 𝜗, 𝛿 < 1, 0 < 𝛽 < 𝛿 where R.L D 𝜗 , C D 𝜍 , 𝜍 ∈ {𝜂, 𝛿, 𝛽} represent the Riemann-Liouville and Caputo fractional derivatives, I 𝛼 represents the Riemann-Liouville integral of order 𝛼, 𝜑, 𝜓 ∶ J × R × R → R, and 𝜙 ∶ J → R are given continuous functions; R.L D 𝜗 denotes the fractional derivative in the notion of Riemann-Liouville [15,27] as…”

Fractional Order Environmental and Economic Model Investigations Using Artificial Neural Network

“…The integer order shows remembrance, whereas, the memory function represents the form of FO. The FO form is applied to the application of the real-world systems [26][27][28][29][30][31][32][33][34]. Few novel features of the ANNs-LMBM for the FO-NEE model are provided as:…”