Analytic technique for solving temporal time-fractional gas dynamics equations with Caputo fractional derivative

Abstract

<abstract><p>Constructing mathematical models of fractional order for real-world problems and developing numeric-analytic solutions are extremely significant subjects in diverse fields of physics, applied mathematics and engineering problems. In this work, a novel analytical treatment technique called the Laplace residual power series (LRPS) technique is performed to produce approximate solutions for a non-linear time-fractional gas dynamics equation (FGDE) in a multiple fractional power series (MFPS) formula. The LRPS technique is a coupling of the RPS approach with the Laplace transform operator. The implementation of the proposed technique to handle time-FGDE models is introduced in detail. The MFPS solution for the target model is produced by solving it in the Laplace space by utilizing the limit concept with fewer computations and more accuracy. The applicability and performance of the technique have been validated via testing three attractive initial value problems for non-linear FGDEs. The impact of the fractional order <italic>β</italic> on the behavior of the MFPS approximate solutions is numerically and graphically described. The <italic>j</italic>th MFPS approximate solutions were found to be in full harmony with the exact solutions. The solutions obtained by the LRPS technique indicate and emphasize that the technique is easy to perform with computational efficiency for different kinds of time-fractional models in physical phenomena.</p></abstract>

New Bright and Kink Soliton Solutions for Fractional Complex Ginzburg–Landau Equation with Non-Local Nonlinearity Term

“…We have classified the gained equilibrium point E * of the system (17) using the planar dynamic system theory as follows: if J > 0, then the point is center. The point is said to be saddle if J < 0 and the point is cusp if J = 0, provided the poincare index of the equilibrium point is zero [44,45], see also [29][30][31][32][33][34][46][47][48] for further details. The phase portrait for the system ( 17) is shown in Figures 1-4.…”

“…The complex-valued function ψ = ψ(t, x) constitutes the wave profile, while ψ * (t, x) is assigned to its conjugate. However, the novelty of this paper is to investigate bright and kink soliton solutions using the ansatz method for the governing model (1) and study its dynamic behavior using the theory of the dynamic planner system [27][28][29][30][31][32][33][34]. In this paper, the ansatz method is utilized for the first time to establish wave solutions for the fractional complex Ginzburg-Landau equation with non-local nonlinearity term.…”

Usage of the Fuzzy Adomian Decomposition Method for Solving Some Fuzzy Fractional Partial Differential Equations

“…Te frequent use of FPDEs in engineering and scientifc applications has led many researchers in this feld to develop new results in theoretical and applied research methods [6][7][8][9][10][11][12][13][14]. Recently, several authors have solved linear and nonlinear FPDEs by using diferent methods, such as the homotopy perturbation method, Adomian decomposition method, variational iteration method, and homotopy analysis method, as mentioned in [15][16][17][18][19][20][21][22].…”

A Novel Solution Approach for Time-Fractional Hyperbolic Telegraph Differential Equation with Caputo Time Differentiation

“…The Laplace FPSM provides a solution in the form of a fast convergence Maclaurin series, either exact or with ASs. It has been used to study a wide range of physical issues, including generating analytical data and for various systems of linear and nonlinear fractional problems [34], studying a non-linear time-fractional generalized biological population model [35], solving temporal time-fractional gas dynamics equations [36], solving non-linear time-fractional Kolmogorov and Rosenau-Hyman models with suitable initial data [37] and investigating the ASs for a nonlinear fractional reaction-diffusion for a bacteria growth model [38].…”