Solutions of fractional gas dynamics equation by a new technique

Abstract

In this paper, a novel technique is formed to obtain the solution of a fractional gas dynamics equation. Some reproducing kernel Hilbert spaces are defined.Reproducing kernel functions of these spaces have been found. Some numerical examples are shown to confirm the efficiency of the reproducing kernel Hilbert space method. The accurate pulchritude of the paper is arisen in its strong implementation of Caputo fractional order time derivative on the classical equations with the success of the highly accurate solutions by the series solutions. Reproducing kernel Hilbert space method is actually capable of reducing the size of the numerical work. Numerical results for different particular cases of the equations are given in the numerical section. KEYWORDS fractional gas dynamics equation, Hilbert space, operators MSC CLASSIFICATION46E22; 34A08 Recently, many numerical methods have been implemented to solve fractional differential equations. These techniques are Adomian decomposition method, 17 homotopy analysis method, 18 homotopy perturbation method, 19 and variational iteration method. 20 Singh et al 21 investigated the fractional partial differential equations showing up in biological populations. Generalized differential transform method has been implemented to search the solution of time-fractional reaction diffusion equations. 22 Singh et al 23 have investigated many nonlinear fractional differential equations. Bhrawy et al 24 have applied the operational matrix concept for solving the fractional diffusion equations. Machado et al 25 and Yang et al 26 have implemented local fractional series method to investigate fractional differential equations.

Fox H-functions as exact solutions for Caputo type mass spring damper system under Sumudu transform

“…Various other mathematical systems can be obtained in research areas including computational biology, medicine, microbiology, biophysics, physiology, environmental science, and many more. Classical systems modeled with ordinary derivatives have been investigated under operators from fractional calculus [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21]. These fractional operators called Riemann--Liouville, Caputo, Caputo-Fabrizio, Atangana-Baleanu, Atangana-Gomez and a few others have characteristics of retaining memory unlike classical operators.…”

Implementing Reproducing Kernel Method to Solve Singularly Perturbed Convection-Diffusion Parabolic Problems

“…Some applications of the RKM and singularly perturbed problems are introduced in [14,16,26,28]. In addition, we can observe applications of the RKM method to solve fractional problems in [1,2,3,4,5,9,27,29].…”

A numerical study on time fractional Fisher equation using an extended cubic B-spline approximation

“…Veeresha et al [32] have obtained a numerical technique for solving TFFE via q-homotopy analysis transform method. Akgül et al [5] derived solutions of fractional gas dynamics equation by a new technique using reproducing kernel method. Akgül [3] investigated boundary layer flow of a Powell-Eyring non-Newtonian fluid by reproducing kernel Hilbert space method.…”