Conformable Fractional Differential Equations Research Articles

ANALYTICAL SOLUTIONS FOR TIME-FRACTIONAL RADHAKRISHNAN–KUNDU–LAKSHMANAN EQUATION

In this paper, two algebraic methods are applied for solving a class of conformable fractional partial differential equations (FPDEs). We use these methods for the time-fractional Radhakrishnan–Kundu–Lakshmanan equation. With these methods, further solutions can be obtained compared with other approaches and techniques. The exact particular solutions include the exponential solution, trigonometric function solution, rational solution and hyperbolic function solution. These methods are very effective to obtain exact solutions of many fractional differential equations.

Lyapunov type inequalities for nonlinear fractional Hamiltonian systems in the frame of conformable derivatives

Lyapunov type integral inequality is one of the important inequalities in qualitative analysis of differential equations. In this paper, using properties of conformable derivatives and integral, and the solution of initial value problem for conformable fractional differential equation, some new Lyapunov type integral inequalities for higher order nonlinear fractional Hamiltonian systems are obtained, which improve and unify some known results in the literatures.

Exact Solutions and Finite Time Stability of Linear Conformable Fractional Systems with Pure Delay

We study nonhomogeneous systems of linear conformable fractional differential equations with pure delay. By using new conformable delayed matrix functions and the method of variation, we obtain a representation of their solutions. As an application, we derive a finite time stability result using the representation of solutions and a norm estimation of the conformable delayed matrix functions. The obtained results are new, and they extend and improve some existing ones. Finally, an example is presented to illustrate the validity of our theoretical results.

Uniqueness criteria for initial value problem of conformable fractional differential equation

<abstract><p>This paper presents four uniqueness criteria for the initial value problem of a differential equation which depends on conformable fractional derivative. Among them is the generalization of Nagumo-type uniqueness theory and Lipschitz conditional theory, and advances its development in proving fractional differential equations. Finally, we verify the main conclusions of this paper by providing four concrete examples.</p></abstract>

Existence and stability theory of pantograph conformable fractional differential problem

The purpose of this paper is to investigate the existence and uniqueness (EU) of solutions to a class of conformable fractional differential equations (DE) with delay term using Krasnoselskii's fixed point theorem. The proposed problem is devoted to non-local initial value problems. Such problems are increasingly occurred in applications like in the filed of quantum mechanics and electrodynamics. The theoretical analysis is further enriched by establishing stability theory due to Ulam and its different kinds including ?Ulam-Hyers (UH), generalized Ulam-Hyers (GUH), Ulam-Hyers-Rassias (UHR), and generalized Ulam-Hyers-Rassias (GUHR)? stability for the considered class. The obtain analysis is then testified by an example.

Ulam's stability of conformable neutral fractional differential equations

This article is concerned with the existence and uniqueness of solutions of a nonlinear neutral conformable fractional differential system with infinite delay, involving conformable fractional derivative. Additionally, we study the Ulam--Hyres stability, Ulam--Hyres--Mittag--Leffler stability, Ulam--Hyres--Mittag--Leffler--Rassias stability for the solutions of considered system using Picard operator. For application of the theory, we add an example at the end.

Solving Conformable Fractional Differential Equations with ``EJS Software and Visualization of Sub-diffusion Process

In this work, we numerically solve ordinary and conformable fractional differential equations using Easy Java Simulations software. Their solutions, including homogeneous and non-homogeneous parts, are compared in various time intervals. Using software’s visualization andsimulation features, we may better examine, compare, and evaluate solutions of analytical and numerical fractional differential equations. A kind of oscillatory behavior is seen in long enough times. In simulation of diffusion and sub-diffusion processes, two intriguing events have been observed.

MODIFIED FINITE ELEMENT NUMERICAL METHOD FOR SOLVING CONFORMABLE SPACE-TIME FRACTIONAL NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS

This paper shows how to approximate the solution to a nonlinear conformable space-time fractional partial differential equations. The proposed method is based on the Cubic B-spline polynomials and Galerkin method. Two test problems show that the approach we use to approximate the proposed equation is accurate and efficient. We apply the Von Neumann approach to show that stability requires some conditions.

ANALYTICAL AND APPROXIMATE SOLUTIONS FOR CONFORMABLE FRACTIONAL ORDER CORONA-VIRUS (COVID-19) EPIDEMIC MODEL

:In this investigation, we discussed the SARS-CoV-2 virus into a system of equations and we apply the Conformable Fractional Differential Transformation Method (CFDTM) to COVID-19 mathematical model described by the system of non-linear conformable fractional order differential equations. The aspire of this study is to estimate the effectiveness of preventive measures, predicting future outbreaks and potential control strategies using the mathematical model. The impacts of various biological parameters on transmission dynamics of COVID-19 is examined. These results are based on different values of the fractional parameter and serve as a control parameter to identify the significant strategies for the control of the disease. In the end, the obtained results are demonstrated graphically to justify our theoretical findings.

On Implicit Impulsive Conformable Fractional Differential Equations with Infinite Delay in b-Metric Spaces

On Implicit Impulsive Conformable Fractional Differential Equations with Infinite Delay in b-Metric Spaces

Controllability of Mild Solution of Nonlocal Conformable Fractional Differential Equations

In many research works Bouaouid et al. have proved the existence of mild solutions of an abstract class of nonlocal conformable fractional Cauchy problem of the form: d α x t / d t α = A x t + f t , x t , x 0 = x 0 + g x , t ∈ 0 , τ . The present paper is a continuation of these works in order to study the controllability of mild solution of the above Cauchy problem. Precisely, we shall be concerned with the controllability of mild solution of the following Cauchy problem d α x t / d t α = A x t + f t , x t + B u t , x 0 = x 0 + g x , t ∈ 0 , τ , where d α . / d t α is the vectorial conformable fractional derivative of order α ∈ 0 , 1 in a Banach space X and A is the infinitesimal generator of a semigroup T t t ≥ 0 on X . The element x 0 is a fixed vector in X and f , g are given functions. The control function u is an element of L 2 0 , τ , U with U is a Banach space and B is a bounded linear operator from U into X .

On Nodal Properties for Some Nonlinear Conformable Fractional Differential Equations

A class of conformable fractional differential equations \[ D_x^{\alpha} D_x^{\alpha} u(x) + \omega(x) f(u(x)) = 0 \quad \textrm{on $(0,1)$} \] is considered. We first give a sufficient condition for the existence of sign-changing solutions with the prescribed number of zeros to this problem. On the basis of this result, we turn to a specific case of the above problem and give a uniqueness theorem related to the function $\omega$. Essentially, the main methods using in this work are properties of conformable fractional calculus, the scaling argument and Prüfer-type substitutions.

Construction controllability for conformable fractional stochastic evolution system with noninstantaneous impulse and nonlocal condition

Conformable fractional stochastic differential equation with noninstantaneous impulse and Poisson jump via nonlocal condition is studied. Controllability for the considered problem is constructed. The required results are established based on fractional calculus, stochastic analysis, and Sadovskii’s fixed point theorem. Moreover, an example is provided to illustrate the applicability of the results.

Investigation to analytic solutions of modified conformable time–space fractional mixed partial differential equations

In this paper, the invariant subspace method (ISM) is developed to obtain the exact solution of linear and nonlinear time and space fractional mixed partial differential equations involving modified conformable fractional derivative (MCFD). Moreover, the method is successfully extended to the coupled system of modified conformable fractional differential equations. Variety of time-space fractional mixed PDEs are considered and solved to illustrate the established result of ISM. Finally, a comparison between exact solution of fractional PDE in the sense of MCFD and Caputo fractional derivative (Ca-FD) is presented through some examples of time–space fractional mixed partial differential equations.

The modified simplest equation procedure for conformable time-fractional Boussinesq equations

In this paper, we used the traveling wave transformation to find out the exact solutions of the conformable time fractional differential equations with the help of the Modified simplest equation procedure. We applied the given procedure to two different equations which are conformable time-fractional Boussinesq and coupled Boussinesq equations. First, we reduced the given equations to ordinary differential equations using wave transformations. Then, we applied the procedure to the given equations. We obtained hyperbolic, trigonometric and rational solutions. We gave the 3D plots, 2D plots and contour plots for some obtained solutions. According to our results, we can say the given method is effective, powerful and gives simple, reliable solutions compared to the other methods. All obtained solutions were checked with the help of Mathematica.

Study of Fuzzy Conformable Fractional Differential Equations

It is essential to use fuzzy differential equations in order to simulate a wide range of physical, applied, and engineering phenomena and uncertainties. For some fuzzy differential equations, obtaining precise solutions is quite challenging, hence reliable and effective analytical methods are essential. These fuzzy fractional differential equations will be studied in this research. Conformable fractional differentiability is examined in this paper, with the goal of developing an existence and uniqueness thesis for a fuzzy fractional differential equation using the idea of conformable differentiability, which is based on expanding a fuzzy mapping's class of differentiable fuzzy mappings. For this, we consider lateral Hukuhara derivatives of order q ∈ (0, 1).

Study of Fuzzy Conformable Fractional Differential Equations

Study of Fuzzy Conformable Fractional Differential Equations

Effective Approach to Construct Series Solutions for Uncertain Fractional Differential Equations

Purpose: We construct the analytical approximate resiual power fuzzy series solutions of fuzzy conformable fractional differential equations in an -level depiction in the sense of strongly generalized -fuzzy conformable derivative in which of the all initial conditions are taken to be fuzzy numbers. Methodology: The certain fuzzy conformable fractional differential equation under strongly generalized -fuzzy derivative is converted to a crisp one as a family of differential inclusions and solved via resiual power method. The main drawback concerning the use of differential inclusions is that it does not contain a fuzzification of the differential operator; instead, the solution is not essentially a fuzzy valued function. Findings: (i) To show the efficiency of our proposed method: Several important and attractive test examples, which included the fractional conformable fuzzy integro-differential equation are discussed and solved in detail. (ii) To show the stability of approximate solutions to specific problems: some graphical results, numerical comparisons and tabulate data are created and discussed at different values of Value: Using the residual power series analysis methos is a powerful and easy-to-use analytic tool to solve initial problems on fuzzy conformable fractional differential equations and it successfully applied to solve real life problems such as the inductance–resistance–capacitance, RLC-series circuit.

Çarpımsal Uyumlu Kesirli Diferansiyel Denklemler

In this study, multiplicative conformable fractional differential equations are presented. Wronskian concept, linear dependence-independence concepts are defined on multiplicative conformable fractional calculus and some theorems and results are given among them. Finally, some examples are solved by giving some methods for finding general solutions of multiplicative conformable fractional differential equations.

A novel conformable Laplace transform for conformable fractional Lane–Emden type equations

In the current work, we introduce a relatively new integral operator, a modification of conformable Laplace transform that can handle some conformable fractional differential equations with singularities of a given kind. Some essential properties and theorems are discussed to show the efficiency and applicability of the new transform. We use this transform in conjunction with the Adomian decomposition method to solve various classes of conformable fractional Lane–Emden type equations.