Analytical and Numerical Solution for Fractional Gas Dynamic Equations Using Residual Power Series Method

Abuteen, Eman; Freihet, Asad

doi:10.2139/ssrn.3270460

Cited by 4 publications

(4 citation statements)

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“…The 3D plot of SHAM, PITM and Exact for Eqn. (18) were presented and the shape of the figure are the same for the three methods which demonstrated the behaviour of the classical equation of motion for a free massive scalar field Klein Gordon equation.Table 2 shows the results of the non linear gas dynamic partial differential together with absolute error which was obtained by subtracting each result obtained at each values of t from the exact solution ,it was observed that the Mean Absolute Error obtained from SHAM was the same with the solution of [10] which shows that SHAM compared favourably with the referenced solution. the graph agrees with each other as it describes the behaviour of gases in motion under the influence of various forces.…”

Section: Applicationmentioning

confidence: 88%

“…Several integral transforms such as Laplace [16,29], Elzaki [2], Aboodh [27], Sumudu, Mohand [31], Sawi [9,13,14,23] have been used extensively to find analytical solution of linear problems of differential equations be it classical (ordinary differential equation, partial differential equation,integral equations )and fractional order differential equations. These integral transforms were combined with other numerical methods such as homotopy pertubation method [32], homotopy analysis method [16,18,29], variational iteration method [7], adomian decomposition method [13,] to find approximate solution of integro-differential equation, ordinary differential equation, partial differential equations of classical type as well as fractional order differential equation [1,3,8,10,12,16,18,19,22]. Klein-Gordon equation has important applications in plasma physics together with Zakharov equations describing the interaction of Langmuir wave also known as a plasma oscillation and the ion acoustic wave in plasma, It is a relativistic wave equation that describes the behaviour of scalar particles with spin zero.…”

Section: Introductionmentioning

confidence: 99%

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Exponential Attractors for the Viscous Cahn-Hilliard System

Ossebi,

Mangoubi,

Mahamat

et al. 2024

Adv. Math., Sci. J.

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In this article, we explore a viscous Cahn-Hilliard system, which has applications in biology. By imposing appropriate boundary and initial conditions, we examine the asymptotic behavior of its solutions. First, we show that the problem of initial and limit values generates by a continuous semigroup on an appropriate phase space, which has a global attractor denoted $\mathcal{A}$. Subsequently, we establish the existence of an exponential attractor $\mathcal{M}$. Therefore, the global attractor $\mathcal{A}$ has a finite fractal dimension.

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Section: Applicationmentioning

confidence: 88%

Section: Introductionmentioning

confidence: 99%

Exponential Attractors for the Viscous Cahn-Hilliard System

Ossebi,

Mangoubi,

Mahamat

et al. 2024

Adv. Math., Sci. J.

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“…The solutions to the fractional order Schrodinger equations were determined using RPSM in [50]. The RPSM has been used to solve a variety of problems, including the gas dynamic equation [51], the Emden-Fowler equation, Berger-Fisher equation, and the Benney-Lin equation in its fractional format, solved by RPSM in [52]. Similarly, RPSM was used by Al-Smadi [53] to overcome the solutions to initial value problems.…”

Section: Introductionmentioning

confidence: 99%

Numerical and Analytical Simulations of Nonlinear Time Fractional Advection and Burger’s Equations

Khan

Tchier

et al. 2022

Journal of Function Spaces

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In this paper, the higher nonlinear problems of fractional advection-diffusion equations and systems of nonlinear fractional Burger’s equations are solved by using two sophisticated procedures, namely, the q-homotopy analysis transform method and the residual power series method. The proposed methods are implemented with the Caputo operator. The present techniques are utilised in a very comprehensive and effective manner to obtain the solutions to the suggested fractional-order problems. The nonlinearity of the problem was controlled tactfully. The numerical results of a few examples are calculated and analyzed. The tables and graphs are constructed to understand the higher accuracy and applicability of the current method. The obtained results that are in good contact with the actual dynamics of the given problem, which is verified by the graphs and tables. The present techniques require fewer calculations and are associated with a higher degree of accuracy, and therefore can be extended to solve other high nonlinear fractional problems.

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“…In [13], an approach was applied to nd the exact solutions of fractional-order time-dependent Schrödinger equations. e residual power series method was also used for many other problems, such as the gas dynamic equation [14], the fractional initial Emden-Fowler equation [15], the generalized Berger-Fisher equation [16], and the nonlinear time-space-fractional Benney-Lin equation [17].…”

Section: Introductionmentioning

confidence: 99%

Least‐Squares Residual Power Series Method for the Time‐Fractional Differential Equations

Zhang

Wei

et al. 2019

Complexity

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In this study, an applicable and effective method, which is based on a least-squares residual power series method (LSRPSM), is proposed to solve the time-fractional differential equations. The least-squares residual power series method combines the residual power series method with the least-squares method. These calculations depend on the sense of Caputo. Firstly, using the classic residual power series method, the analytical solution can be solved. Secondly, the concept of fractional Wronskian is introduced, which is applied to validate the linear independence of the functions. Thirdly, a linear combination of the first few terms as an approximate solution is used, which contains unknown coefficients. Finally, the least-squares method is proposed to obtain the unknown coefficients. The approximate solutions are solved by the least-squares residual power series method with the fewer expansion terms than the classic residual power series method. The examples are shown in datum and images.The examples show that the new method has an accelerate convergence than the classic residual power series method.

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Analytical and Numerical Solution for Fractional Gas Dynamic Equations Using Residual Power Series Method

Cited by 4 publications

References 0 publications

Exponential Attractors for the Viscous Cahn-Hilliard System

Exponential Attractors for the Viscous Cahn-Hilliard System

Numerical and Analytical Simulations of Nonlinear Time Fractional Advection and Burger’s Equations

Least‐Squares Residual Power Series Method for the Time‐Fractional Differential Equations

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