Conformable Fractional Differential Equations Research Articles

Conformable Double Laplace Transform Method (CDLTM) and Homotopy Perturbation Method (HPM) for Solving Conformable Fractional Partial Differential Equations

In the present article, the method which was obtained from a combination of the conformable fractional double Laplace transform method (CFDLTM) and the homotopy perturbation method (HPM) was successfully applied to solve linear and nonlinear conformable fractional partial differential equations (CFPDEs). We included three examples to help our presented technique. Moreover, the results show that the proposed method is efficient, dependable, and easy to use for certain problems in PDEs compared with existing methods. The solution graphs show close contact between the exact and CFDLTM solutions. The outcome obtained by the conformable fractional double Laplace transform method is symmetrical to the gain using the double Laplace transform.

Some important results for the conformable fractional stochastic pantograph differential equations in the L p space

Some important results for the conformable fractional stochastic pantograph differential equations in the L p space

Coupled systems of conformable fractional differential equations

This paper deals with some existence of solutions for some classes of coupled systems of conformable fractional differential equations with initial and boundary conditions in Banach and Fréchet spaces. Our results are based on some fixed point theorems. Some illustrative examples are presented in the last section.

Initial Value Problem for Hybrid Improved Conformable Fractional Differential Equations with Retardation and Anticipation

This manuscript is devoted to proving some results concerning the existence of solutions for a class of initial value problem for nonlinear implicit fractional Hybrid differential equations and improved conformable fractional derivative. The result is based on a fixed point theorem due to Dhage. Further, an example is provided for the justification of our main result.

Thenovel numerical solutions for conformable fractional Kuramoto-Sivashinsky equations by using Cq-HATM and CHPETM

The present study used the novel conformable fractional methods to examine the nonlinear Kuramoto-Sivashinsky equations of the conformable fractional derivative. The Kuramoto-Sivashinsky equation is a mathematical model employed to elucidate a diverse range of chemical and physical phenomena, encompassing chemical reaction-diffusion processes, plasma instabilities, viscous flow issues, flame front propagation, and magnetized plasmas. The present inquiry provides an examination of convergence and error for the future scheme. The conformable q-homotopy analysis transform method (Cq-HATM) provides h-curves that illustrate the convergence range of the series solution achieved. In order to validate the effectiveness and suitability of the Cq-HATM, we examine separate instances. In this study, we introduce an application that demonstrates the potential benefits and effectiveness of the proposed approach. Furthermore, an error analysis is performed in order to verify the accuracy of the scheme. Numerical simulations are conducted to validate the precision of the forthcoming approach. The findings obtained from the numerical and graphical analyses are presented in this study. The method provided in this study showcases a notable degree of computational precision and ease in examining and resolving intricate phenomena related to conformable fractional nonlinear partial differential equations within the domains of science and technology.

Existence results and stabilization of homogeneous conformable fractional order systems

In this paper, we give results on the stability of homogeneous conformable fractional order systems using assumptions on a family of compact sets and provide the stabilization of an affine control system and given an explicit homogeneous feedback control with the requirement that a control Lyapunov function exists and satisfying a homogeneous condition and we present existence and uniqueness theorems for sequential linear conformable fractional differential equations.

Conformable Triple Sumudu Transform with Applications

One of the important topics in applied mathematics is the topic of integral transformations, due to their importance in electrical engineering applications, including communications in particular, and other sciences. In this work, one of the most important transformations in its three dimensions was presented, which is the triple Sumudu transform, including solving some real-life applications of physics, some of which have not been solved using such an integral transform before. In this work, we extend the Sumudu transform formula to the conformable fractional order, as well as other interesting and significant rules. The general analytical solution of a singular and nonlinear conformable fractional differential equation based on the conformable fractional Sumudu transform is also presented in this paper. The general solutions of several linear and nonhomogeneous conformable fractional differential equations can be obtained using the method we’ve proposed. As a result, our results reveal that our proposed method is an efficient one that can be used for solving all conformable fractional differential equations. The relationship between the Sumudu integral transform and other important and recently proposed integral transforms are also discussed. Finally, the triple Sumudu transform is used to solve boundary value problems, such as the heat equation with boundary values. The triple Sumudu integral transform is also used to solve linear partial integro-differential equations. The transform capability to handle such equations has been proven via its utilization in three applications.

A Study on Some Conformable Fractional Implicit Hybrid Differential Equations with Delay

This paper deals with some existence results for a class of conformable implicit fractional differential Hybrid equations with delay. The results are based on some suitable fixed point theorems. In the last section, we provide different examples to illustrate our obtained results.

On a conformable fractional differential equations with maxima

This work is concerned with the existence and uniqueness of solutions for a class of first order conformable fractional differential equations with maxima. We also give some examples illustrating the application of ourresults.

Lie symmetry analysis and exact solutions for conformable time fractional partial differential equations with variable coefficients

Lie symmetry analysis and exact solutions for conformable time fractional partial differential equations with variable coefficients

Traveling Wave Solutions of the Conformable Fractional Klein–Gordon Equation With Power Law Nonlinearity

This article investigates the construction of new traveling wave solutions for the conformable fractional Klein–Gordon equation, which is a well‐known mathematical and physical model that can be used to explain spinless pion and de Broglie waves. In order to accomplish this task, a classic and effective analysis method, namely, the extended tanh–coth method, was utilized. By introducing appropriate transformations, the conformable fractional Klein–Gordon equations are reduced to ordinary differential equations, and then the solutions with hyperbolic function form are obtained. In addition, the effect of fractional parameters on waveform is analyzed by drawing two‐ and three‐dimensional graphics. These results contribute to a deeper understanding of the dynamics of the conformable fractional Klein–Gordon equation. The research in this article also indicates that the extended tanh–coth method is a straightforward and concise technique that has the potential to be applicable to many other conformable fractional partial differential equations that appear in mathematical physics.

The Efficient Method to Solve the Conformable Time Fractional Benney Equation

The current study employed the innovative conformable fractional method to analyze the nonlinear Benney equations involving the conformable fractional derivative. Conformable fractional Benney equations have been examined by the conformable q‐Shehu analysis transform method. By including nonlinear factors, it offers a more precise depiction of wave propagation compared to linear models. Various natural phenomena, including ocean waves, plasma waves, and some forms of solitons, display nonlinear behavior that cannot be precisely explained by linear equations. The fractional Benney equation is important because it extends the classical Benney equation, which describes the evolution of weakly nonlinear and weakly dispersive long waves in shallow water. By incorporating fractional calculus operators, the fractional Benney equation provides a more accurate description of wave propagation phenomena in certain physical systems characterized by nonlocal or memory‐dependent behavior. The utilization of the Benney equation enables researchers to simulate these occurrences with greater realism. This study investigates the convergence and inaccuracy of the future scheme. The conformable q‐Shehu homotopy analysis transform method (Cq‐SHATM) generates h‐curves that demonstrate the convergence interval of the series solution obtained. In order to determine the effectiveness and suitability of the Cq‐SHATM, uniqueness and convergence theorems have been proven. This study presents an application that showcases the potential advantages and efficacy of the suggested method. Moreover, an error analysis is conducted to validate the precision of the scheme. Computational simulations are performed to verify the accuracy of the upcoming method. This study presents the results gained from the numerical and graphical analysis. The method presented in this work demonstrates a high level of computational accuracy and simplicity in analyzing and solving complex phenomena associated with conformable fractional nonlinear partial differential equations in the fields of science and technology.

Analysis of stochastic delay differential equations in the framework of conformable fractional derivatives

<abstract><p>In numerous domains, fractional stochastic delay differential equations are used to model various physical phenomena, and the study of well-posedness ensures that the mathematical models accurately represent physical systems, allowing for meaningful predictions and analysis. A fractional stochastic differential equation is considered well-posed if its solution satisfies the existence, uniqueness, and continuous dependency properties. We established the well-posedness and regularity of solutions of conformable fractional stochastic delay differential equations (CFrSDDEs) of order $ \gamma\in(\frac{1}{2}, 1) $ in $ \mathbb{L}^{\mathrm{p}} $ spaces with $ \mathrm{p}\geq2 $, whose coefficients satisfied a standard Lipschitz condition. More specifically, we first demonstrated the existence and uniqueness of solutions; after that, we demonstrated the continuous dependency of solutions on both the initial values and fractional exponent $ \gamma $. The second section was devoted to examining the regularity of time. As a result, we found that, for each $ \Phi\in(0, \gamma-\frac{1}{2}) $, the solution to the considered problem has a $ \Phi- $H$ \ddot o $lder continuous version. Lastly, two examples that highlighted our findings were provided. The two main elements of the proof were the Burkholder-Davis-Gundy inequality and the weighted norm.</p></abstract>

A new analytical algorithm for uncertain fractional differential equations in the fuzzy conformable sense

<abstract><p>This paper aims to explore and examine a fractional differential equation in the fuzzy conformable derivative sense. To achieve this goal, a novel analytical algorithm is formulated based on the Laplace-residual power series method to solve the fuzzy conformable fractional differential equations. The methodology being used to discover the fuzzy solutions depends on converting the desired equations into two fractional crisp systems expressed in $ \wp $-cut form. The main objective of our algorithm is to transform the systems into fuzzy conformable Laplace space. The transformation simplifies the system by reducing its order and turning it into an easy-to-solve algorithmic equation. The solutions of three important applications are provided in a fuzzy convergent conformable fractional series. Both the theoretical and numerical implications of the fuzzy conformable concept are explored about the consequential outcomes. The convergence analysis and theorems of the developed algorithm are also studied and analyzed in this regard. Additionally, this article showcases a selection of results through the use of both two-dimensional and three-dimensional graphs. Ultimately, the findings of this study underscore the efficacy, speed, and ease of the Laplace-residual power series algorithm in finding solutions for uncertain models that arise in various physical phenomena.</p></abstract>

Random solutions for improved conformable fractional differential equations with retardation and anticipation

Random solutions for improved conformable fractional differential equations with retardation and anticipation

Fundamental solutions and conservation laws for conformable time fractional partial differential equation

In this paper, the connections between fundamental solutions and Lie symmetry groups for a class of conformable time fractional partial differential equations (PDEs) with variable coefficient are investigated for the first time. The group-invariant solutions to the considered equations are constructed by means of symmetry group method. Then, the corresponding fundamental solutions for these PDEs are established by taking the inverse Laplace transform of the group invariant solutions. In addition, some examples are introduced to illustrate the effectiveness of this approach. Furthermore, the conservation laws of these fractional PDEs are obtained making use of new Noether theorem.

Mild Solutions of a Class of Conformable Fractional Differential Equations with Nonlocal Conditions

Mild Solutions of a Class of Conformable Fractional Differential Equations with Nonlocal Conditions

Nonlinear two conformable fractional differential equation with integral boundary condition

"This paper deals with a boundary value problem for a nonlinear differential equation with two conformable fractional derivatives and integral boundary conditions. The results of existence, uniqueness and stability of positive solutions are proved by using the Banach contraction principle, Guo-Krasnoselskii's fixed point theorem and Hyers-Ulam type stability. Two concrete examples are given to illustrate the main results."

Conformable finite element method for conformable fractional partial differential equations

<abstract><p>The finite element (FE) method is a widely used numerical technique for approximating solutions to various problems in different fields such as thermal diffusion, mechanics of continuous media, electromagnetism and multi-physics problems. Recently, there has been growing interest among researchers in the application of fractional derivatives. In this paper, we present a generalization of the FE method known as the conformable finite element method, which is specifically designed to solve conformable fractional partial differential equations (CF-PDE). We introduce the basis functions that are used to approximate the solution of CF-PDE and provide error estimation techniques. Furthermore, we provide an illustrative example to demonstrate the effectiveness of the proposed method. This work serves as a starting point for tackling more complex problems involving fractional derivatives.</p></abstract>

Existence and controllability for conformable fractional stochastic differential equations with infinite delay via measures of noncompactness.

In this article, we consider conformable fractional stochastic differential equations (CFSDEs) driven by fBm with infinite delay via measures of noncompactness (MNC). As far as we know, there are few papers considering this issue. First, by virtue of a Mönch fixed point theorem and MNC, we explore the existence of solutions for CFSDEs. Subsequently, with the aid of Jensen inequality, Hölder inequality, stochastic analysis techniques, and semigroup theory, the controllability for this considered CFSDEs is investigated by employing a Mönch fixed point theorem. Thereafter, the controllability of CFSDEs with nonlocal conditions is discussed. Finally, the theoretical result is supported through an example.