Conformable Fractional Differential Equations Research Articles

Analytical Solutions for Nonlinear Systems of Conformable Space-Time Fractional Partial Differential Equations via Generalized Fractional Differential Transform

The present paper introduces approximate analytical solutions for nonlinear systems of conformable space-time fractional partial differential equations (CSTFPDEs) by using a generalized conformable fractional partial differential transform (GCFPDT). The GCFPDT is a modified version of the conformable fractional partial differential transform introduced in our recent work and it can be used as an efficient alternative transform to find analytical solutions for nonlinear systems of CSTFPDEs available in the literature. The convergence and error estimation of the proposed GCFPDT are also considered. Moreover, approximate analytical solutions to nonlinear system of gas dynamic equations, nonlinear system of KdV equations, and nonlinear system of approximate long water wave equations in the sense of conformable space-time fractional partial derivatives are successfully obtained to confirm the effectiveness and efficiency of the proposed GCFPDT.

Closed-form solution of optimal control problem of a fractional order system

A formulation and an analytical solution technique of fractional optimal control problem (FOCP) is presented in this paper. The performance index (PI) considered is a conformable fractional integral (CFI) function and is a function of both the state and the control variables. Dynamic behaviour of the system is described by conformable fractional differential equation (FDE). The necessary conditions of optimality and the general transversality condition in terms of Hamiltonian are obtained using variational approach. Both the fixed and free final end point conditions have been considered. An analytical solution technique is presented for solving the conformable FOCPs. To validate the formulation and solution scheme numerical examples are presented.

The Extremal Solution To Conformable Fractional Differential Equations Involving Integral Boundary Condition

In this article, by using the monotone iterative technique coupled with the method of upper and lower solution, we obtain the existence of extremal iteration solutions to conformable fractional differential equations involving Riemann-Stieltjes integral boundary conditions. At the same time, the comparison principle of solving such problems is investigated. Finally, an example is given to illustrate our main results. It should be noted that the conformal fractional derivative is essentially a modified version of the first-order derivative. Our results show that such known results can be translated and stated in the setting of the so-called conformal fractional derivative.

New periodic wave solutions of a time fractional integrable shallow water equation

In this paper, author employed Jacobi elliptic function expansion method to build the new wave solutions of time fractional modified Camassa–Holm equation which is completely integrable dispersive shallow-water equation. In ocean engineering, Camassa–Holm equation is generally used as a tool in computer simulation of the water waves in shallow sees, coastal and harbors. The obtained solutions show that the Jacobi elliptic function expansion method (JEFEM) which based on Jacobi elliptic functions is an efficient, reliable, applicable and accurate tool for analytic approximation of a wide variety of nonlinear conformable time fractional partial differential equations.

Generalized solitary wave solutions to the time fractional generalized Hirota-Satsuma coupled KdV via new definition for wave transformation

In this paper, the (G′/G)-expansion method has been applied to explore new solitary wave solutions for the time fractional generalized Hirota-Satsuma coupled KdV (FGHSC KdV) system. The fractional derivative is described with the use of conformable derivative. The results show that this method is a very useful and effective mathematical tool for solving nonlinear conformable fractional equations arising in mathematical physics. As a result, this method can also be applied to other nonlinear conformable fractional differential equations.

Inverse Problems in Conformable Fractional Differential Equation

The main objective of this article is to study the effect of replacing classical derivative with conformable fractional derivative. Direct and inverse problems are studied for projectile motion, heat equation and mixture problems.

Existence of Solutions for Boundary Value Problems of Conformable Fractional Differential Equations

In this paper, we study a class of boundary value problems for conformable fractional differential equations under a new definition. Firstly, by using the monotone iterative technique and the method of coupled upper and lower solution, the sufficient condition for the existence of the boundary value problem is obtained, and the range of the solution is determined. Then the existence and uniqueness of the solution are proved by the proof by contradiction. Finally, a concrete example is given to illustrate the wide applicability of our main results.

New Results on the Conformable Fractional Sumudu Transform: Theories and Applications

In this paper, we generalize the formula of Sumudu transform to the conformable fractional order and some interesting and important rules of this transform and conformable fractional Laplace transform are derived and discussed. Moreover, we present the general analytical solution of a singular and nonlinear conformable fractional Poisson- Boltzmann differential equation based on the conformable fractional Sumudu transform. Also, our proposed method is applied successfully for obtaining the general solutions of some linear and nonhomogeneous conformable fractional differential equations. Finally, the results show that our proposed method is an efficient and can be applied for finding the general solutions of the all cases realted to the conformable fractional differential equations.

Existence results for systems of conformable fractional differential equations

In this article, we study the existence of solutions to systems of conformable fractional differential equations with periodic boundary value or initial value conditions. where the right member of the system is $L^{1}_{\alpha }$-caratheodory function. We employ the method of solution-tube and Schauder’s fixed-point theorem.

The analytical solution of Van der Pol and Lienard differential equations within conformable fractional operator by retarded integral inequalities

Abstract In this study we introduced and tested retarded conformable fractional integral inequalities utilizing non-integer order derivatives and integrals. In line with this purpose, we used the Katugampola type conformable fractional calculus which has several practical properties. These inequalities generalize some famous integral inequalities which provide explicit bounds on unknown functions. The results provided here had been implemented to the global existence of solutions to the conformable fractional differential equations with time delay.

Eigenvalue problems for nonlinear conformable fractional differential equations with multi-point boundary conditions

Eigenvalue problems for nonlinear conformable fractional differential equations with multi-point boundary conditions

Lie symmetry analysis of some conformable fractional partial differential equations

In this article, Lie symmetry analysis is used to investigate invariance properties of some nonlinear fractional partial differential equations with conformable fractional time and space derivatives. The analysis is applied to Korteweg–de Vries, modified Korteweg–de Vries, Burgers, and modified Burgers equations with conformable fractional time and space derivatives. For each equation, all the vector fields and the Lie symmetries are obtained. Moreover, exact solutions are given to these equations in terms of solutions of ordinary differential equations. In particular, it is shown that the fractional Korteweg–de Vries can be reduced to the first Painlevé equation and to the fractional second Painlevé equation. In addition, a solution of the fractional modified Korteweg–de Vries is given in terms of solutions of the fractional second Painlevé equation.

Discretization of conformable fractional differential equations by a piecewise constant approximation

ABSTRACTIn this paper, a conformable fractional-order logistic differential equation including both discrete and continuous time is taken into account. By using a piecewise constant approximation, a discretization method which transforms a fractional-order differential equation into a difference equation is introduced. Necessary and sufficient conditions for both local and global stability of the discretized system are obtained. The control space diagrams and with the fractional-order parameter α, a discretization parameter and the growth parameter are obtained and these diagrams illustrate the regions where the solutions of the system approach to the positive equilibrium point with monotonic and damped oscillations. Finally, the existence of flip bifurcation is proved using the centre manifold theory and these theoretical results are supported by numerical calculations.

Convergence analysis for iterative learning control of conformable fractional differential equations

This paper mainly deals with iterative learning control for the conformable fractional differential equations. The standard P‐type, Dα‐type, and conformable PIαDα‐type learning updating laws are proposed to derive the convergence results for nonlinear and linear problems varying with the initial state is (not) coincident with the desired initial state. Finally, numerical examples are given to illustrate the results.

A new approach for seeking coefficient function solutions of conformable fractional partial differential equations based on the Jacobi elliptic equation

In this paper, we present an approach for seeking exact solutions with coefficient function forms of conformable fractional partial differential equations. By a combination of an under-determined fractional transformation and the Jacobi elliptic equation, exact solutions with coefficient function forms can be obtained for fractional partial differential equations. The innovation point of the present approach lies in two aspects. One is the fractional transformation, which involve the traveling wave transformations used by many articles as special cases. The other is that more general exact solutions with coefficient function forms can be found, and traveling wave solutions with constants coefficients are only special cases of our results. As of applications, we apply this method to the space-time fractional (2+1)-dimensional dispersive long wave equations and the time fractional Bogoyavlenskii equations. As a result, some exact solutions with coefficient function forms for the two equations are successfully found.

Positive solutions of conformable fractional differential equations with integral boundary conditions

In this paper, we discuss the existence of positive solutions of the conformable fractional differential equation T_{alpha }x(t)+f(t,x(t))=0, tin [0,1], subject to the boundary conditions x(0)=0 and x(1)= lambda int_{0}^{1}x(t),mathrm{d}t, where the order α belongs to (1,2], T_{alpha }x(t) denotes the conformable fractional derivative of a function x(t) of order α, and f:[0,1]times [0,infty)mapsto [0,infty) is continuous. By use of the fixed point theorem in a cone, some criteria for the existence of at least one positive solution are established. The obtained conditions are generally weaker than those derived by using the classical norm-type expansion and compression theorem. A concrete example is given to illustrate the possible application of the obtained results.

Basic theory of initial value problems of conformable fractional differential equations

In this paper, we discuss the basic theory of the conformable fractional differential equation T^{a}_{alpha}x(t)=f(t,x(t)), tin[a,infty), subject to the local initial condition x(a)=x_{a} or the nonlocal initial condition x(a)+g(x)=x_{a}, where 0<alpha<1, T^{a}_{alpha}x(t) denotes the conformable fractional derivative of a function x(t) of order α, f:[a,infty)timesmathbb{R}mapstomathbb{R} is continuous and g is a given functional defined on an appropriate space of functions. The theory of global existence, extension, boundedness, and stability of solutions is considered; by virtue of the theory of the conformable fractional calculus and by the use of fixed point theorems, some criteria are established. Several concrete examples are given to illustrate the possible application of our analytical results.

Oscillation of solutions to a generalized forced nonlinear conformable fractional differential equation

By using averaging functions, we present some new oscillation criteria for the solution of a generalized forced nonlinear conformable fractional differential equation. The results obtained here extend and improve on some existing results. Examples are also given to show the validity of our results.

Exact Solutions with Variable Coefficient Function Forms for Conformable Fractional Partial Differential Equations by an Auxiliary Equation Method

In this paper, an auxiliary equation method is introduced for seeking exact solutions expressed in variable coefficient function forms for fractional partial differential equations, where the concerned fractional derivative is defined by the conformable fractional derivative. By the use of certain fractional transformation, the fractional derivative in the equations can be converted into integer order case with respect to a new variable. As for applications, we apply this method to the time fractional two-dimensional Boussinesq equation and the space-time fractional (2+1)-dimensional breaking soliton equation. As a result, some exact solutions including variable coefficient function solutions as well as solitary wave solutions for the two equations are found.

Hybrid Method for Solution of Fractional Order Linear Differential Equation with Variable Coefficients

Abstract In this paper, we proposed a new analytical hybrid methods for the solution of conformable fractional differential equations (CFDE), which are based on the recently proposed conformable fractional derivative (CFD) in R. Khalil, M. Al Horani, A. Yusuf and M. Sababhed, A New definition of fractional derivative, J. Comput. Appl. 264 (2014). Moreover, we use the method of variation of parameters and reduction of order based on CFD, for the CFDE. Furthermore, to show the efficiency of the proposed analytical hybrid method, some examples are also presented.