scholarly journals New Explicit Solutions to the Fractional-Order Burgers’ Equation

2021 ◽  
Vol 2021 ◽  
pp. 1-11
Author(s):  
M. Hafiz Uddin ◽  
Mohammad Asif Arefin ◽  
M. Ali Akbar ◽  
Mustafa Inc
Keyword(s):  
Differential Equations ◽  
Rational Function ◽  
Acoustic Waves ◽  
Burgers Equation ◽  
Three Dimensional ◽  
Wall Friction ◽  
Weakly Nonlinear ◽  
Bubbly Liquids ◽  
Fractional Differential ◽  
Accuracy Of Results

The closed-form wave solutions to the time-fractional Burgers’ equation have been investigated by the use of the two variables G ′ / G , 1 / G -expansion, the extended tanh function, and the exp-function methods translating the nonlinear fractional differential equations (NLFDEs) into ordinary differential equations. In this article, we ascertain the solutions in terms of tanh , sech , sinh , rational function, hyperbolic rational function, exponential function, and their integration with parameters. Advanced and standard solutions can be found by setting definite values of the parameters in the general solutions. Mathematical analysis of the solutions confirms the existence of different soliton forms, namely, kink, single soliton, periodic soliton, singular kink soliton, and some other types of solitons which are shown in three-dimensional plots. The attained solutions may be functional to examine unidirectional propagation of weakly nonlinear acoustic waves, the memory effect of the wall friction through the boundary layer, bubbly liquids, etc. The methods suggested are direct, compatible, and speedy to simulate using algebraic computation schemes, such as Maple, and can be used to verify the accuracy of results.

Application of the Exponential Rational Function Method to Some Fractional Soliton Equations

2020 ◽  
pp. 13-32
Author(s):  
Mustafa Ekici ◽  
Metin Ünal
Keyword(s):  
Differential Equations ◽  
Rational Function ◽  
Function Method ◽  
Space Time ◽  
Fractional Equations ◽  
New Exact Solutions ◽  
Kdv Equations ◽  
Fractional Differential ◽  
Exponential Rational Function Method ◽  
Fifth Order

In this chapter, the authors study the exponential rational function method to find new exact solutions for the time-fractional fifth-order Sawada-Kotera equation, the space-time fractional Whitham-Broer-Kaup equations, and the space-time fractional generalized Hirota-Satsuma coupled KdV equations. These fractional differential equations are converted into ordinary differential equations by using the fractional complex transform. The fractional derivatives are defined in the sense of Jumarie's modified Riemann-Liouville. The proposed method is direct and effective for solving different kind of nonlinear fractional equations in mathematical physics.

Waves of Acoustically Induced Transparency in Bubbly Liquids: Theoretical Prediction and Experimental Validation

2013 ◽  
Author(s):  
Nail A. Gumerov ◽  
Iskander S. Akhatov ◽  
Claus-Dieter Ohl ◽  
Sergei P. Sametov ◽  
Maxim V. Khasimulin ◽  
...  
Keyword(s):  
Acoustic Field ◽  
Acoustic Waves ◽  
Three Dimensional ◽  
Self Organization ◽  
Nonlinear Phenomenon ◽  
Three Dimensional Model ◽  
Bubbly Liquids ◽  
Strongly Nonlinear ◽  
Induced Transparency ◽  
Good Agreement

Self-organization of bubbles in acoustic fields, or self-action of the acoustic waves in bubbly liquids is a strongly nonlinear phenomenon due to two-way interaction of the bubbles and the acoustic field. Theoretical model and preliminary computations predict that waves of self-induced acoustic transparency may exist. Such effect is confirmed in the experiments presented in this paper. Formation of a wave of void fraction which rapidly propagates through the bubbly medium leaving a region almost free of bubbles behind its front is observed in the experiments. Measurements of the dynamics of such a wave at different acoustic frequencies and amplitudes are carried out. A three dimensional model of self-organization of a polydisperse bubble continuum in acoustic field is developed and the results of simulations are compared with experiments. A good agreement of the theory and experiment is found.

On the Cole–Hopf transformation and integration by parts formulae in computational methods within fractional differential equations and fractional optimal control theory

2021 ◽  
pp. 107754632110310
Author(s):  
Ahmed S Hendy ◽  
Mahmoud A Zaky ◽  
José A Tenreiro Machado
Keyword(s):  
Optimal Control ◽  
Differential Equations ◽  
Heat Equation ◽  
Fractional Differential Equations ◽  
Burgers Equation ◽  
Variable Order ◽  
Integer Order ◽  
Order Variable ◽  
Fractional Differential ◽  
Distributed Order

The treatment of fractional differential equations and fractional optimal control problems is more difficult to tackle than the standard integer-order counterpart and may pose problems to non-specialists. Due to this reason, the analytical and numerical methods proposed in the literature may be applied incorrectly. Often, such methods were established for the classical integer-order operators and are then applied directly without having in mind the restrictions posed by their fractional-order versions. It was recently reported that the Cole–Hopf transformation can be used to convert the time-fractional nonlinear Burgers’ equation into the time-fractional linear heat equation. In this article, we show that, unlike integer-order differential equations, employing the Cole–Hopf transformation for reducing the nonlinear time-fractional Burgers’ equation into the time-fractional heat equation is wrong from two different perspectives. Indeed, such a reduction is accomplished using incorrect transcripts of the Leibniz or chain rules. Hence, providing numerical or analytical schemes based on the Cole–Hopf transformation leads to erroneous results for the nonlinear time-fractional Burgers’ equation. Regarding constant-order, variable-order, and distributed-order Caputo fractional optimal problems, we note an inconsistency in the necessary optimality conditions derived in the literature. The transversality conditions were introduced as identical to those for the integer-order case, with a vanishing multiplier at the terminal of the interval. The correct condition should involve a constant-order, variable-order, or distributed-order fractional integral operator. We also deduce that if the control system is defined with a Caputo derivative, then the adjoint equations should be expressed in the Riemann–Liouville sense and vice versa. In fact, neglecting some terms in the integration by parts formulae, during the derivation of the optimality conditions, causes some confusion in the literature.

scholarly journals A modified exp-function method for fractional partial differential equations

2021 ◽  
pp. 17-17
Author(s):  
Yi Tian ◽  
Jun Liu
Keyword(s):  
Differential Equations ◽  
Exact Solutions ◽  
Rational Function ◽  
Fractional Differential Equations ◽  
Present Method ◽  
Function Method ◽  
Solution Procedure ◽  
Algebraic Equations ◽  
Fractional Partial Differential Equations ◽  
Fractional Differential

This paper proposes a novel exponential rational function method, a modification of the well-known exp-function method, to find exact solutions of the time fractional Cahn-Allen equation and the time fractional Phi-4 equation. The solution procedure is reduced to solve a system of algebraic equations, which is then solved by Wu?s method. The results show that the present method is effective, and can be applied to other fractional differential equations.

Generalized exponential rational function method for extended Zakharov–Kuzetsov equation with conformable derivative

2019 ◽  
Vol 34 (20) ◽  
pp. 1950155 ◽  
Author(s):  
Behzad Ghanbari ◽  
M. S. Osman ◽  
Dumitru Baleanu
Keyword(s):  
Differential Equations ◽  
Rational Function ◽  
Function Method ◽  
Three Dimensional ◽  
Conformable Derivative ◽  
Reduced Equations ◽  
Wave Solutions ◽  
Contour Plots ◽  
Free Parameters ◽  
Exponential Rational Function Method

In this paper, new analytical obliquely propagating wave solutions for the time fractional extended Zakharov–Kuzetsov (FEZK) equation of conformable derivative are investigated. By using the main properties of the conformable derivative, the FEZK equation is transformed into integer-order differential equations, and the reduced equations are solved via the generalized exponential rational function method (GERFM). The shape and features for the resulting solutions are illustrated through three-dimensional (3D) plots and corresponding contour plots for various values of the free parameters.

scholarly journals Laplace homotopy perturbation method for Burgers equation with space- and time-fractional order

Open Physics ◽  
2016 ◽  
Vol 14 (1) ◽  
pp. 247-252 ◽  
Author(s):  
S. J. Johnston ◽  
H. Jafari ◽  
S. P. Moshokoa ◽  
V. M. Ariyan ◽  
D. Baleanu
Keyword(s):  
Perturbation Method ◽  
Acoustic Waves ◽  
Homotopy Perturbation Method ◽  
Burgers Equation ◽  
Physical Processes ◽  
Weakly Nonlinear ◽  
Homotopy Perturbation ◽  
The Laplace Transform ◽  
Nonlinear Acoustic ◽  
Fractional Orders

AbstractThe fractional Burgers equation describes the physical processes of unidirectional propagation of weakly nonlinear acoustic waves through a gas-filled pipe. The Laplace homotopy perturbation method is discussed to obtain the approximate analytical solution of space-fractional and time-fractional Burgers equations. The method used combines the Laplace transform and the homotopy perturbation method. Numerical results show that the approach is easy to implement and accurate when applied to partial differential equations of fractional orders.

scholarly journals Hyperbolic rational solutions to a variety of conformable fractional Boussinesq-Like equations

2019 ◽  
Vol 8 (1) ◽  
pp. 224-230 ◽  
Author(s):  
Hadi Rezazadeh ◽  
M.S. Osman ◽  
Mostafa Eslami ◽  
Mohammad Mirzazadeh ◽  
Qin Zhou ◽  
...  
Keyword(s):  
Differential Equations ◽  
Exact Solutions ◽  
Rational Function ◽  
Fractional Differential Equations ◽  
Present Method ◽  
Physical Phenomenon ◽  
Ocean Engineering ◽  
Rational Solutions ◽  
Fractional Differential

Abstract The aim of this paper is to investigate hyperbolic rational solutions of four conformable fractional Boussinesq-like equations using the method of exponential rational function (ERF). The present method is a good scheme, reveal distinct exact solutions and convenient for solving other types of nonlinear conformable fractional differential equations. These solutions are of significant importance in coastal and ocean engineering where the fractional Boussinesq-like equations modeled for some special physical phenomenon.

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