Conformable Time-fractional Derivative Research Articles

Dynamical analysis of optical soliton solutions for CGL equation with Kerr law nonlinearity in classical, truncated [formula omitted]-fractional derivative, beta fractional derivative, and conformable fractional derivative types

The study of optical soliton solutions plays a vital role in nonlinear optics. The foremost area of optical solitons research encompasses around optical fiber, telecommunication, meta-surfaces and others related technologies. The aim of this work is to integrate optical soliton solutions of the complex Ginzburg-Landau (CGL) model with Kerr law nonlinearity, also showing the effect of diverse fraction derivative and comparing it with the classical form. Here, the local derivative is used as the conformable wisdom known as the truncated M-fractional derivative, beta fractional derivative, and conformable fraction derivative. We also deliberated on some assets satisfied by the derivative. The CGL model is useful to describe the light propagation in optical communications, optical transmission, and nonlinear optical fiber. Under the right circumstances, the affectionate unified scheme is implemented for the complex Ginzburg-Landau model to generate the optical wave pattern. For α1=2α2, the unified scheme generates the solution of CGL model in terms of hyperbolic, trigonometric, and rational function solutions. This scheme provides some novel optical solitons such as periodic waves, periodic with rogue waves, breather waves, different types of periodic rogue waves, a singular soliton solution, and rogue with the periodic wave for the special value of the free parameters. For α2=-ρ+6α18, the unified scheme generates the solutions of CGL model in terms of hyperbolic, trigonometric, and rational function solutions. This scheme offers some fresh optical solitons such as periodic rogue waves, multi-rogue waves, double periodic waves, and periodic waves. In numerical argument, the wave patterns are offered with 3-D and density plots. To test the stability of the obtained solutions, we show diverse fractional forms such as beta time fractional, and conformable time fractional derivative and compare these fractional derivatives with their classical form in 2D plots. The investigation reveals innovative and explicit solutions, providing insight into the dynamics of the related physical processes. This paper provides a comprehensive examination of the obtained solutions, emphasizing their distinct features and depictions using unified technique. These findings are especially advantageous for specialists in the fields of nonlinear science and mathematical physics, providing significant insights into the behavior and development of nonlinear waves in various physical situations.

On exact solutions of some important nonlinear conformable time-fractional differential equations

The nonlinear fractional Boussinesq equations are known as the fractional differential equation class that has an important place in mathematical physics. In this study, a method called $$\Big (\frac{G'}{G^2}\Big )$$ -extension method which works well and reveals exact solutions is used to examine nonlinear Boussinesq equations with conformable time-fractional derivative. This method is a very useful approach and extremely utility compared to other analytical methods. With the proposed method, there are three unique types of solutions such as hyperbolic, trigonometric and rational solutions. This approach can similarly be applied to other nonlinear fractional models.

Analytic solutions for a modified fractional three wave interaction equations with conformable derivative by unified method

The present work implements the unified method to a class of fractional partial differential systems corresponding to modified fractional three wave interaction equations (FTWIEs). The fractional derivative of the three waves envelopes is considered under the conformable sense. The conformable FTWIEs is derived in the (3 + 1) dimensions under the conformable time-fractional derivative of order α∈(0,1] based on a modification of the Lax-pair created by Zakharov-Manakov which includes three-dimensional velocity vector dotted with the usual del operator. Then, the unified method for creating solutions for nonlinear evolution FTWIEs is applied. Subsequently, a systematic algorithm is used to obtain an infinite set of exact rational solutions to the novel constructed system. By randomly selecting the special values for the parameters, three-dimensional graphs are also given for different patterns. The obtained solutions might play an essential role in many other nonlinear evolution equations that occur in the fields of engineering and mathematical physics.

Asymptotic behaviour of solution and non-existence of global solution to a class of conformable time-fractional stochastic equation

Consider the following class of conformable time-fractional stochastic equation, for any x∈R fixed, Tα,tau(x,t)=λσ(u(x,t))Ẇt,t∈[a,∞),0<α<1, with a non-random initial condition u(x,0)=u0(x),x∈R assumed to be non-negative and bounded, Tα,ta is a conformable time-fractional derivative, σ:R→R is globally Lipschitz continuous, Ẇt a generalized derivative of Wiener process and λ>0 is the noise level. Given some precise and suitable conditions on the non-random initial function, we study the asymptotic behaviour of the solution with respect to the time parameter t and the noise level parameter λ. We also show that when the non-linear term σ grows faster than linear, the energy of the solution blows-up at finite time for all α∈(0,1).

Solitons in conformable time-fractional Wu–Zhang system arising in coastal design

The modified exp $$\left( {-\Omega \left( \zeta \right) } \right) $$ -expansion function method is applied to the Wu–Zhang system with conformable time-fractional derivative to construct new analytical solutions. We have obtained some soliton-type solutions such as dark, singular and combined soliton solutions. We have seen that all solutions have provided the mentioned equation system. For the suitable value of the solutions, the 2D–3D and contour surfaces have been plotted.

Moment Bound of Solution to a Class of Conformable Time-Fractional Stochastic Equation

We study a class of conformable time-fractional stochastic equation T α , t a u ( x , t ) = σ ( u ( x , t ) ) W ˙ t , x ∈ R , t ∈ [ a , T ] , T &lt; ∞ , 0 &lt; α &lt; 1 . The initial condition u ( x , 0 ) = u 0 ( x ) , x ∈ R is a non-random function assumed to be non-negative and bounded, T α , t a is a conformable time-fractional derivative, σ : R → R is Lipschitz continuous and W ˙ t a generalized derivative of Wiener process. Some precise condition for the existence and uniqueness of a solution of the class of equation is given and we also give an upper bound estimate on the growth moment of the solution. Unlike the growth moment of stochastic fractional heat equation with Riemann–Liouville or Caputo–Dzhrbashyan fractional derivative which grows in time like t c 1 exp ( c 2 t ) , c 1 , c 2 &gt; 0 ; our result also shows that the energy of the solution (the second moment) grows exponentially in time for t ∈ [ a , T ] , T &lt; ∞ but with at most c 1 exp ( c 2 ( t − a ) 2 α − 1 ) for some constants c 1 , and c 2 .

Nonlocal Telegraph Equation in Frame of the Conformable Time-Fractional Derivative

We use the cosine family of linear operators to prove the existence, uniqueness, and stability of the integral solution of a nonlocal telegraph equation in frame of the conformable time-fractional derivative. Moreover, we give its implicit fundamental solution in terms of the classical trigonometric functions.

ANALYTICAL STUDY OF THE TEO-DIMENSIONAL TIME-FRTACTIONAL NAVIER-STOKES EQUATIONS

In this paper, the two-dimensional (2D) Holf-Cole transformation with mass conservation in the frame of conformable derivative is developed, and then by introducing some exact solutions that satisfy linear differential equations and using the symbolic computation method, four exact solutions of 2D-nonlinear Navier-Stokes equations (NSEs) with the conformable time-fractional derivative are established. Some physical properties of the exact solutions are described preliminarily. Our results are the first ones on analytical study for the 2D time-fractional NSEs.

The $$\phi ^{6}$$ ϕ 6 -model expansion method for solving the nonlinear conformable time-fractional Schrödinger equation with fourth-order dispersion and parabolic law nonlinearity

The $$\phi ^{6}$$ -model expansion method combined with the conformable time-fractional derivative is applied in this paper for finding many new exact solutions including Jacobi elliptic function solutions, solitary wave solutions, trigonometric function solutions and other solutions to the nonlinear conformable time-fractional Schrodinger equation with fourth-order dispersion and parabolic law nonlinearity. This method presents a wider applicability for handling the nonlinear partial differential equations. Comparing our results with the well-known results are given.

Some new exact solutions of (2+1)-dimensional nonlinear Heisenberg ferromagnetic spin chain with the conformable time fractional derivative

There are various types of fractional derivatives in literature e.g. Caputo, Riemann–Liouville, conformable and so on. In the present paper, a new integrable nonlinear Schrodinger type equation in (2+1)-dimensions is investigated, which describes the spin dynamics of Heisenberg ferromagnetic spin chains. Some bright and dark solitons and other exact solutions of this equation with conformable time fractional derivative are reported. Then, fractionalize effects of this model are graphically investigated. Moreover, the simplest equation method and Nucci’s method are utilized for differential equations with conformable fractional derivatives.

Two reliable techniques for the analytical study of conformable time-fractional Phi-4 equation

In this article, two newly developed methods, namely the exp $$(-~\Phi (\xi ))$$ method and the modified Kudryashov method are used to extract the exact solutions of the nonlinear fractional Phi-4 equation in the sense of conformable time-fractional derivative. The obtained solutions confirm the significant features of the proposed methods in handling the conformable time fractional Phi-4 equation.

Analytical treatment of nonlinear conformable time-fractional Boussinesq equations by three integration methods

Under investigation in this paper is a nonlinear conformable time-fractional Boussinesq equations as an important class of fractional differential equations in mathematical physics. The extended trial equation method, the $$\exp (-\Omega (\eta ))$$ -expansion method and the $$\tan (\phi (\eta )/2)$$ -expansion method are used in examining the analytical solution of the nonlinear fractional equations. The proposed methods are based on the integration method and a wave transformation. The fractional derivative in the sense of conformable time-fractional derivative is defined. Fractional complex transform is implemented to change fractional differential equations into ordinary differential equations in this paper. In addition, explicit new exact solutions are derived in different form such as dark solitons, bright solitons, solitary wave, periodic solitary wave, rational function, and elliptic function solutions of nonlinear conformable time-fractional Boussinesq equations.

Exact solutions of nonlinear conformable time-fractional Boussinesq equations using the $$\exp \left( { - \phi \left( \varepsilon \right)} \right)$$ exp - ϕ ε -expansion method

Nonlinear fractional Boussinesq equations are considered as an important class of fractional differential equations in mathematical physics. In this article, a newly developed method called the $$\exp \left( { - \phi \left( \varepsilon \right)} \right)$$ -expansion method is utilized to study the nonlinear Boussinesq equations with the conformable time-fractional derivative. Different forms of solutions, including the hyperbolic, trigonometric and rational function solutions are formally extracted. The method suggests a useful and efficient technique to look for the exact solutions of a wide range of nonlinear fractional differential equations.

New exact solutions of nonlinear conformable time-fractional Boussinesq equations using the modified Kudryashov method

In this paper, the nonlinear Boussinesq equations with the conformable time-fractional derivative are solved analytically using the well-established modified Kudryashov method. As a consequence, a number of new exact solutions for this type of equations are formally derived. It is believed that the method is one of the most effective techniques for extracting new exact solutions of nonlinear fractional differential equations.

The first integral method for Wu–Zhang system with conformable time-fractional derivative

In this paper, the first integral method is used to construct exact solutions of the time-fractional Wu---Zhang system. Fractional derivatives are described by conformable fractional derivative. This method is based on the ring theory of commutative algebra. The results obtained confirm that the proposed method is an efficient technique for analytic treatment of a wide variety of nonlinear conformable time-fractional partial differential equations.