The functional variable method for finding exact solutions of some nonlinear time-fractional differential equations

In this paper, we implemented the functional variable method and the modified Riemann–Liouville derivative for the exact solitary wave solutions and periodic wave solutions of the time-fractional Klein–Gordon equation, and the time-fractional Hirota–Satsuma coupled KdV system. This method is extremely simple but effective for handling nonlinear time-fractional differential equations.     

Scaling behavior of the time-fractional equations for molecular-beam epitaxy growth: scaling analysis versus numerical stimulations

The scaling behavior of the time-fractional molecular-beam epitaxy (TFMBE) equations in 1+1 dimensions is investigated by numerical simulations and scaling analysis, respectively. The growth equations studied include the time-fractional linear molecular-beam epitaxy (TFLMBE) and the time-fractional Lai-Das Sarma-Villain (TFLDV). Growth exponents are obtained using the two methods. The analytical results are consistent with the corresponding numerical solutions based on Caputo-type fractional derivative.     

Analytical and approximate solutions of(2+1)-dimensional time-fractional Burgers-Kadomtsev-Petviashvili equation

In this paper, we applied the sub-equation method to obtain a new exact solution set for the extended version of the time-fractional Kadomtsev-Petviashvili equation, namely BurgersKadomtsev-Petviashvili equation(Burgers-K-P) that arises in shallow water waves.Furthermore, using the residual power series method(RPSM), approximate solutions of the equation were obtained with the help of the Mathematica symbolic computation package. We also presented a few graphical illustrations for some surfaces. The fractional derivatives were considered in the conformable sense. All of the obtained solutions were replaced back in the governing equation to check and ensure the reliability of the method. The numerical outcomes confirmed that both methods are simple, robust and effective to achieve exact and approximate solutions of nonlinear fractional differential equations.     

Exact solutions of a nonlinear conformable time-fractional parabolic equation with exponential nonlinearity using reliable methods

A nonlinear conformable time-fractional parabolic equation with exponential nonlinearity is explored, in this article. First, under the specific transformations, the time-fractional parabolic equation is changed into a nonlinear ODE of integer order, and then, the reduced equation is solved using two lately established techniques called the \({ \exp }\left( { - \varphi \left( \varepsilon \right)} \right)\)-expansion and modified Kudryashov methods. Several exact solutions in various wave forms for the nonlinear conformable time-fractional parabolic equation with exponential nonlinearity are formally constructed.     

Exactly solving some typical Riemann-Liouville fractional models by a general method of separation of variables

Finding exact solutions for Riemann–Liouville(RL) fractional equations is very difficult. We propose a general method of separation of variables to study the problem. We obtain several general results and, as applications, we give nontrivial exact solutions for some typical RL fractional equations such as the fractional Kadomtsev–Petviashvili equation and the fractional Langmuir chain equation. In particular, we obtain non-power functions solutions for a kind of RL time-fractional reaction–diffusion equation. In addition, we find that the separation of variables method is more suited to deal with high-dimensional nonlinear RL fractional equations because we have more freedom to choose undetermined functions.     

Element-free Galerkin (EFG) method for analysis of the time-fractional partial differential equations

The present paper deals with the numerical solution of time-fractional partial differential equations using the element-free Galerkin (EFG) method, which is based on the moving least-square approximation. Compared with numerical methods based on meshes, the EFG method for time-fractional partial differential equations needs only scattered nodes instead of meshing the domain of the problem. It neither requires element connectivity nor suffers much degradation in accuracy when nodal arrangements are very irregular. In this method, the first-order time derivative is replaced by the Caputo fractional derivative of order α (0<α ≤1). The Galerkin weak form is used to obtain the discrete equations, and the essential boundary conditions are enforced by the penalty method. Several numerical examples are presented and the results we obtained are in good agreement with the exact solutions.     

Applications of two reliable methods for solving a nonlinear conformable time-fractional equation

The current work presents analytical solutions of a nonlinear conformable time-fractional equation by using two different techniques. These are the modified simple equation method and the exponential rational function method. Based on the conformable fractional derivative and traveling wave transformation, the fractional partial differential equation is turned into the nonlinear non-fractional ordinary differential equation. Therefore, we implement the algorithms to this nonlinear non-fractional ordinary differential equation. To the best of our knowledge, the exact solutions obtained in this paper might be very useful in various areas of applied mathematics in interpreting some physical phenomena.     

Construction of new exact solutions to time-fractional two-component evolutionary system of order 2 via different methods

This paper is concerned with the applications of five different methods including the sub-equation method, the tanh method, the modified Kudryashov method, the \(\left( \frac{G'}{G}\right)\)-expansion method and the Exp-function method to construct exact solutions of time-fractional two-component evolutionary system of order 2. We first convert this type of fractional equations to the nonlinear ordinary differential equations by means of fractional complex transform. Then, the five methods are adopted to solve the nonlinear ordinary differential equations. As a result, some new exact solutions are obtained. It is also shown that each of the considered methods can be used as an alternative for solving fractional differential equations.     

The $$\phi ^{6}$$-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 Schrödinger 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.     

Optical solitons to the fractional perturbed Radhakrishnan–Kundu–Lakshmanan model

This study reaches the dark, bright, mixed dark-bright, singular, mixed singular optical solitons and singular periodic wave solutions to the time-fractional Radhakrishnan–Kundu–Lakshmanan equation. The parametric conditions that guarantee the existence of valid solitons and other solutions are stated. By choosing some suitable values of parameters, the 2- and 3-dimensional surfaces to some of the reported solutions are plotted. The reported solutions may be useful in expalining the physical meaning of the Radhakrishnan–Kundu–Lakshmanan equation and other related nonlinear models arising in nonlinear sciences.     

Bright and singular soliton solutions of the conformable time-fractional Klein–Gordon equations with different nonlinearities

Finding the exact solutions of nonlinear fractional differential equations has gained considerable attention, during the past two decades. In this paper, the conformable time-fractional Klein–Gordon equations with quadratic and cubic nonlinearities are studied. Several exact soliton solutions, including the bright (non-topological) and singular soliton solutions are formally extracted by making use of the ansatz method. Results demonstrate that the method can efficiently handle the time-fractional Klein–Gordon equations with different nonlinearities.     

New exact solutions of Burgers’ type equations with conformable derivative

In this paper, the new exact solutions for some nonlinear partial differential equations are obtained within the newly established conformable derivative. We use the first integral method to establish the exact solutions for time-fractional Burgers’ equation, modified Burgers’ equation, and Burgers–Korteweg–de Vries equation. We report that this method is efficient and it can be successfully used to obtain new analytical solutions of nonlinear FDEs.     

Lie Group Classifications and Non-differentiable Solutions for Time-Fractional Burgers Equation

Lie group method provides an efficient tool to solve nonlinear partial differential equations. This paper suggests Lie group method for fractional partial differential equations. A time-fractional Burgers equation is used as an example to illustrate the effectiveness of the Lie group method and some classes of exact solutions are obtained.     

Compacton and solitary pattern solutions for nonlinear dispersive KdV-type equations involving Jumarie?s fractional derivative

In this Letter, the fractional variational iteration method using He?s polynomials is implemented to construct compacton solutions and solitary pattern solutions of nonlinear time-fractional dispersive KdV-type equations involving Jumarie?s modified Riemann-Liouville derivative. The method yields solutions in the forms of convergent series with easily calculable terms. The obtained results show that the considered method is quite effective, promising and convenient for solving fractional nonlinear dispersive equations. It is found that the time-fractional parameter significantly changes the soliton amplitude of the solitary waves.     

Exact solutions to fractional Drinfel’d–Sokolov–Wilson equations

In this paper, the complete discrimination system for polynomial method is applied to retrieve the exact traveling wave solutions of time-fractional coupled Drinfel’d–Sokolov–Wilson equations. All of the possible exact traveling wave solutions which consist of the rational function type solutions, solitary wave solutions, triangle function type periodic solutions and Jacobian elliptic functions solutions are obtained, and some of them are new solutions. Moreover, the concrete examples are presented to ensure the existences of obtained solutions. In addition, four types of representative solutions are depicted to show the nature of solutions.     

Traveling wave solutions of conformable time-fractional Zakharov–Kuznetsov and Zoomeron equations

To model physical phenomena more accurately, fractional order differential equations have been widely used. Investigating exact solutions of the fractional differential equations have become more important because of the applications in applied mathematics, mathematical physics, and other areas. In this work, by means of the trial solution method and complete discrimination system, exact traveling wave solutions of the conformable time-fractional Zakharov–Kuznetsov equation and conformable time-fractional Zoomeron equation have been obtained and also solutions have been illustrated. Finding exact solutions of these equations that are encountered in plasma physics, nonlinear optics, fluid mechanics, and laser physics can help to understand nature of the complex phenomena.     

Lie symmetry analysis,conservation laws and analytical solutions of a time-fractional generalized KdV-type equation

Under investigation in this work is the time-fractional generalized KdV-type equation, which occurs in different contexts in mathematical physics. Lie group analysis method is presented to explicitly study its vector fields and symmetry reductions. Furthermore, two straightforward methods are employed to consider its travelling wave solutions and power series solutions, respectively. Finally, based on the new conservation theorem, conservation laws of the equation are well constructed with a detailed derivation.     

(G'/G)-Expansion Method for Solving Fractional Partial Differential Equations in the Theory of Mathematical Physics

In this paper, the (G'/G)-expansion method is extended to solve fractional partial differential equations in the sense of modified Riemann-Liouville derivative. Based on a nonlinear fractional complex transformation, a certain fractional partial differential equation can be turned into another ordinary differential equation of integer order. For illustrating the validity of this method, we apply it to the space-time fractional generalized Hirota-Satsuma coupled KdV equations and the time-fractional fifth-order Sawada-Kotera equation. As a result, some new exact solutions for them are successfully established.     

Analytical approach for the fractional differential equations by using the extended tanh method

In this study, we consider analytical solutions of space–time fractional derivative foam drainage equation, the nonlinear Korteweg–de Vries equation with time and space-fractional derivatives and time-fractional reaction–diffusion equation by using the extended tanh method. The fractional derivatives are defined in the modified Riemann–Liouville context. As a result, various exact analytical solutions consisting of trigonometric function solutions, kink-shaped soliton solutions and new exact solitary wave solutions are obtained.     

Time-fractional extensions of the Liouville and Zwanzig equations

This paper presents extensions to the classical stochastic Liouville equation of motion that contain the Riemann-Liouville and Caputo time-fractional derivatives. At first, the dynamic equations with the time-fractional derivatives are formally obtained from the classical Liouville equation. A feature of these new equations is that they have the same common formal solution as the classical Liouville equation and therefore may be used for study of the Hamiltonian system dynamics. Two cases of the time-dependent and time-independent Hamiltonian are considered separately. Then, the time-fractional Liouville equations are deduced from the short- and long-time asymptotic expansions of the obtained dynamic equations. The physical meaning of the resulting equations is discussed. The statements of the Cauchy-type problems for the derived time-fractional Liouville equations are given, and the formal solutions of these problems are presented. At last, the projection operator formalism is employed to derive the time-fractional extensions of the Zwanzig kinetic equations and the corresponding formal statistical operators from the time-fractional Liouville equations.