Conformable Fractional Differential Equations Research Articles

Solving Higher-Order Fractional Differential Equations by the Fuzzy Generalized Conformable Derivatives

In this paper, the generalized concept of conformable fractional derivatives of order q ∈ n , n + 1 for fuzzy functions is introduced. We presented the definition and proved properties and theorems of these derivatives. The fuzzy conformable fractional differential equations and the properties of the fuzzy solution are investigated, developed, and proved. Some examples are provided for both the new solutions.

Conformable Double Sumudu Transform with Applications

Recently, a new deformation of the Sumudu transforms namely, conformable Sumudu transform has been introduced. In this article, we generalize the concept of one-dimensional conformable Sumudu transform into two-dimensional conformable Sumudu transform namely, conformable double Sumudu transform. Further, we present and prove some main properties and theorems related to the introduced transform. In order to illustrate the validity, efficiency, and applicability of the proposed transform, we apply the conformable double Sumudu transform to solve initial and boundary value problems of conformable fractional partial differential equations.

Monotone Iterative Technique for Nonlinear Impulsive Conformable Fractional Differential Equations With Delay

In this paper, we investigate the existence of solutions for boundary value problems of nonlinear impulsive conformable fractional differential equations with delay. By establishing the associate Green's function and a comparison result for the linear impulsive problem, we obtain that the lower and upper solutions converge to the extremal solutions via the monotone iterative technique. An example is also presented in the last section.

New analytical methods for solving a class of conformable fractional differential equations by fractional Laplace transform

In this paper, new analytical solutions for a class of conformable fractional differential equations (CFDE) and some more results about Laplace transform introduced by Abdeljawad cite{abdeljawad2015conformable} are investigated. The Laplace transform method is developed to get the exact solution of conformable fractional differential equations. The aim of this paper is to convert the conformable fractional differential equations into ordinary differential equations (ODE), this is done by using the fractional Laplace transformation of $(alpha+beta)$ order.

Conformable Fractional Models of the Stellar Helium Burning via Artificial Neural Networks

The helium burning phase represents the second stage that the star used to consume nuclear fuel in its interior. In this stage, the three elements, carbon, oxygen, and neon, are synthesized. The present paper is twofold: firstly, it develops an analytical solution to the system of the conformable fractional differential equations of the helium burning network, where we used, for this purpose, the series expansion method and obtained recurrence relations for the product abundances, that is, helium, carbon, oxygen, and neon. Using four different initial abundances, we calculated 44 gas models covering the range of the fractional parameterα=0.5−1with stepΔα=0.05. We found that the effects of the fractional parameter on the product abundances are small which coincides with the results obtained by a previous study. Secondly, we introduced the mathematical model of the neural network (NN) and developed a neural network algorithm to simulate the helium burning network using a feed-forward process. A comparison between the NN and the analytical models revealed very good agreement for all gas models. We found that NN could be considered as a powerful tool to solve and model nuclear burning networks and could be applied to the other nuclear stellar burning networks.

Semi-Analytic Technique for Solving Fractional Partial Differential Equations with Conformable Derivatives

In this work, we present a semi-analytical technique to find an approximate result of the conformable fractional partial differential equations (CFPDEs). The fractional order derivative will be in the conformable (CFD) sense. This definition is effective and simple in the solution of the fractional differential equations that have intricate solution with classical fractional derivative definition like Riemann-Liouville and Caputo. Furthermore, the result obtained by the proposed technique is like those in previous studies that used other types of approximate methods like (Homotopy analysis method) but it has the advantage of being simpler than the rest of these methods. In addition, results demonstrate obtained the Precision and effectiveness of the suggested technique.

Existence of Positive Solutions for a Class of Conformable Fractional Differential Equations with Parameterized Integral Boundary Conditions

Existence of Positive Solutions for a Class of Conformable Fractional Differential Equations with Parameterized Integral Boundary Conditions

Hille and Nehari Type Oscillation Criteria for Conformable Fractional Differential Equations

In this paper, we develop the Hille and Nehari Type criteria for the oscillation of all solutions to the Fractional Differential Equations involving Conformable fractional derivative. Some new oscillatory criteria are obtained by using the Riccati transformations and comparison technique. We show the validity and effectiveness of our results by providing various examples.

Exact solutions of conformable fractional differential equations

This article is about to formulate exact solutions of the time fractional Dodd-Bullough-Mikhailov (DBM) equation, Sinh-Gordon equation and Liouville equation by utilizing simplest equation method (SEM) in conformable fractional derivative (CFD) sense. Using a proper transformation, the original equations are transformed to nonlinear ordinary differential equations (ODEs). The method is very simple in comparison with the classical techniques and very much effective for solving fractional order partial differential equations (FOPDEs).

Fuzzy Conformable Fractional Differential Equations

In this study, fuzzy conformable fractional differential equations are investigated. We study conformable fractional differentiability, and we define fractional integrability properties of such functions and give an existence and uniqueness theorem for a solution to a fuzzy fractional differential equation by using the concept of conformable differentiability. This concept is based on the enlargement of the class of differentiable fuzzy mappings; for this, we consider the lateral Hukuhara derivatives of order q ∈ 0,1 .

Abundant wave solutions of conformable space-time fractional order Fokas wave model arising in physical sciences

The higher dimensional Fokas equation is the integrable expansion of the Kadomtsev-Petviashvili (KP) and Davey-Stewartson (DS) equations. The Fokas model has an important role in wave theory, to describe the physical phenomena of waves on the surface and inside the water. This article deals with the (4+1)-dimensional conformable space-time fractional-order Fokas partial differential equation. Two efficient methods, namely the generalized exp(-ϕ(ξ))-expansion and improved F-expansion methods, are formulated for conformable fractional-order partial differential equation and new wave structures of fractional order Fokas model are constructed. The different kinds of new solitons are achieved such as bright soliton, dark soliton, Kink and anti-kink solitons, periodic solitary waves, and traveling waves. These new soliton waves are constructed at some values of fractional order α and using different parametric values of the methods by using the software package Mathematica. Newly obtained soliton solutions are compared with the available soliton solutions with different fractional derivatives in the literature. Some of the achieved results are explained 2D and 3D graphically. The new results interpreting that these obtained solutions can be a part, to complete the family of solutions and considered methods are effective, simple, and easy to use. Furthermore, this paper gives an idea, how can reduce the conformable fractional order higher dimensional partial differential equation into an ODE of one variable to obtain the exact solutions. These results and methods can be help to investigate the other higher-dimensional conformable fractional-order models which appear in nonlinear wave theory such as optics, quantum gases, hydrodynamics, photonics, plasmas, and solid-state physics.

Solutions of Conformable Fractional-Order SIR Epidemic Model

In this paper, the conformable fractional-order SIR epidemic model are solved by means of an analytic technique for nonlinear problems, namely, the conformable fractional differential transformation method (CFDTM) and variational iteration method (VIM). These models are nonlinear system of conformable fractional differential equation (CFDE) that has no analytic solution. The VIM is based on conformable fractional derivative and proved. The result revealed that both methods are in agreement and are accurate and efficient for solving systems of OFDE.

Further study on the conformable fractional Gauss hypergeometric function

<abstract> This article presents an exhaustive study on the conformable fractional Gauss hypergeometric function (CFGHF). We start by solving the conformable fractional Gauss hypergeometric differential equation (CFGHDE) about the fractional regular singular points $ x = 1 $ and $ x = \infty $. Next, various generating functions of the CFGHF are established. We also develop some differential forms for the CFGHF. Subsequently, differential operators and contiguous relations are reported. Furthermore, we introduce the conformable fractional integral representation and the fractional Laplace transform of CFGHF. As an application, and after making a suitable change of the independent variable, we provide general solutions of some known conformable fractional differential equations, which could be written by means of the CFGHF. </abstract>

New generalized conformable fractional impulsive delay differential equations with some illustrative examples

<abstract> This article is concerned to develop a novel generalized class of fractional impulsive delay differential equations. These equations are defined using newly discovered generalized conformable fractional operators which unify various previously-defined operators into a single form. The successive approximation method is employed and a sufficient criterion for the existence and uniqueness of the solution is developed. For the sake of illustration, three examples are provided at the end of the main results. </abstract>

Some results and applications on conformable fractional Kamal transform

In this paper, we derive conformable fractional Kamal transform from the classical Kamal transform and we present their properties. We discuss the conformable fractional kamal transform of some functions and relationship with conformable fractional Laplace transform. Moreover we solve the general analytical solution of a general conformable Bernoulli’s fractional differential equation by this new transform and Adomain polynomial method. Also we use this method to find the solution of linear and nonlinear conformable fractional differential equations.

Forced oscillation of conformable fractional partial delay differential equations with impulses

In this paper, we establish some interval oscillation criteria for impulsive conformable fractional partial delay differential equations with a forced term. The main results will be obtained by employing Riccati technique. Our results extend and improve some results reported in the literature for the classical differential equations without impulses. An example is provided to illustrate the relevance of the new theorems.

Stability of conformable linear differential systems: a behavioural framework with applications in fractional‐order control

The authors study the stability of linear systems whose dynamics are described by systems of conformable fractional differential equations. They provide a framework in terms of behavioural system theory to develop a general modelling specification as well as stability conditions for conformable linear systems with a fractional differential order. They also provide sufficient conditions and tests for stability based on linear matrix inequalities. To do so, they elaborate on the relationship between the conformable derivative and the traditional integer derivative as well as on a conformable Laplace transform and the use of polynomial algebra as the pivotal figure for analysis. They demonstrate the advantages of the proposed approach by designing a stabilising fractional-order controller for a DC–DC converter feeding a constant power load.

Monotone Iterative Technique for Conformable Fractional Differential Equations with Deviating Arguments

This paper is concerned with the existence of extremal solutions for periodic boundary value problems for conformable fractional differential equations with deviating arguments. We first build two comparison principles for the corresponding linear equation with deviating arguments. With the help of new comparison principles, some sufficient conditions for the existence of extremal solutions are established by combining the method of lower and upper solutions and the monotone iterative technique. As an application, an example is presented to enrich the main results of this article.

Monotone Iterative Technique for Nonlinear Periodic Time Fractional Parabolic Problems

In this paper, the existence and uniqueness of the weak solution for a linear parabolic equation with conformable derivative are proved, the existence of weak periodic solutions for conformable fractional parabolic nonlinear differential equation is proved by using a more generalized monotone iterative method combined with the method of upper and lower solutions. We prove the monotone sequence converge to weak periodic minimal and maximal solutions. Moreover, the conformable version of the Lions-Magness and Aubin–Lions lemmas are also proved.

New theoretical results and applications on conformable fractional Natural transform

In this paper, we proceed on to develop the classical Natural transform to fractional order in the sense of conformable derivative and set the basic concepts in this new interesting fractional transform version. The conformable fractional Natural transform of certain functions, properties and relationships are derived and discussed. Furthermore, we present the general analytical solution of a generalized conformable Bernoulli’s fractional differential equation based on the new version and Adomain polynomial method. This method is also applied successfully to find the general solutions of some linear and nonlinear conformable fractional differential equations. Finally, the results show that our new approach is simple, accurate and effective.