Nonlinear Fractional Differential Equations Research Articles

Some properties and stability of Helmholtz model involved with nonlinear fractional difference equations and its relevance with quadcopter

This study is devoted to developing mathematical models associated with the Helmholtz equation as a second-order oscillator involved with nonlinear Caputo fractional difference equations. This study also focuses on determining the approximate solution of this model via the Ulam stability conception. Some properties of the mathematical model dealt with in this study are also presented. Numerical simulations are presented to justify the existence of stability results.

Fractional View Study of the Brusselator Reaction–Diffusion Model Occurring in Chemical Reactions

In this paper, we study a fractional Brusselator reaction–diffusion model with the help of the residual power series transform method. Specific reaction–diffusion chemical processes are modeled by applying the fractional Brusselator reaction–diffusion model. It should be mentioned that many problems in nonlinear science are characterized by fractional differential equations, where an unknown term occurs when a fractional-order derivative is operating on it. The analytic method of this problem is rarely discussed in the literature, despite numerous scholars having researched its application and usefulness. To validate our proposed method’s accuracy, we compare the numerical results of the residual power series transform method and the exact result with different fractional orders. The solution shows that the introduced approach is a good tool for solving linear and nonlinear fractional system differential equations. Finally, we provide two and three-dimensional graphical plots to support the impact of the fractional derivative on the behavior of the achieved profile results to the proposed equations.

Direct Power Series Approach for Solving Nonlinear Initial Value Problems

In this research, a new approach for solving fractional initial value problems is presented. The main goal of this study focuses on establishing direct formulas to find the coefficients of approximate series solutions of target problems. The new method provides analytical series solutions for both fractional and ordinary differential equations or systems directly, without complicated computations. To show the reliability and efficiency of the presented technique, interesting examples of systems and fractional linear and nonlinear differential equations of ordinary and fractional orders are presented and solved directly by the new approach. This new method is faster and better than other analytical methods in establishing many terms of analytic solutions. The main motivation of this work is to introduce general new formulas that express the series solutions of some types of differential equations in a simple way and with less calculations compared to other numerical power series methods, that is, there is no need for differentiation, discretization, or taking limits while constructing the approximate solution.

Theories of tempered fractional calculus applied to tempered fractional Langevin and Vasicek equations

Our main goal in the current research work is to explore proofs of newly discovered theorems related to tempered fractional calculus. We use a new mechanism, namely, the natural tempered fractional transformation method, which can be used to solve important tempered fractional differential equations that are important in science, such as the linear and nonlinear tempered fractional differential equations. Indeed, we found new exact solutions to both tempered fractional Langevin and Vasicek differential equations and an exact solution for the famous tempered fractional diffusion equation. The new method makes it easier to do the calculations than with the traditional methods, and all you need is a few simple manipulations. Our new research technique is straightforward to use and highly accurate.

Solving a nonlinear fractional SPDE with spatially inhomogeneous white noise

In this paper, we study the following nonlinear fractional stochastic partial differential equation where denotes the Markovian generator of a stable-like Feller process with variable order and is a measurable function. The forcing noise denoted by is a spatially inhomogeneous white noise. Under some assumptions on the catalytic measure of the inhomogeneous Brownian sheet , we study the moment bounds for the solution. As a byproduct, we prove that the solution is weakly full intermittent based on the moment estimates of the solution. We also study the Hölder regularity of the solution with respect to the temporal and spatial variables, respectively.

Boundary-Value Problem for Nonlinear Fractional Differential Equations of Variable Order with Finite Delay via Kuratowski Measure of Noncompactness

This paper is devoted to boundary-value problems for Riemann–Liouville-type fractional differential equations of variable order involving finite delays. The existence of solutions is first studied using a Darbo’s fixed-point theorem and the Kuratowski measure of noncompactness. Secondly, the Ulam–Hyers stability criteria are examined. All of the results in this study are established with the help of generalized intervals and piecewise constant functions. We convert the Riemann–Liouville fractional variable-order problem to equivalent standard Riemann–Liouville problems of fractional-constant orders. Finally, two examples are constructed to illustrate the validity of the observed results.

Investigation of Fractional Nonlinear Regularized Long-Wave Models via Novel Techniques

The main goal of the current work is to develop numerical approaches that use the Yang transform, the homotopy perturbation method (HPM), and the Adomian decomposition method to analyze the fractional model of the regularized long-wave equation. The shallow-water waves and ion-acoustic waves in plasma are both explained by the regularized long-wave equation. The first method combines the Yang transform with the homotopy perturbation method and He’s polynomials. In contrast, the second method combines the Yang transform with the Adomian polynomials and the decomposition method. The Caputo sense is applied to the fractional derivatives. The strategy’s effectiveness is shown by providing a variety of fractional and integer-order graphs and tables. To confirm the validity of each result, the technique was substituted into the equation. The described methods can be used to find the solutions to these kinds of equations as infinite series, and when these series are in closed form, they give the precise solution. The results support the claim that this approach is simple, strong, and efficient for obtaining exact solutions for nonlinear fractional differential equations. The method is a strong contender to contribute to the existing literature.

A System of High-Order Fractional Differential Equations with Integral Boundary Conditions

The existence of a solution for a system of two nonlinear high-order fractional differential equations including the Atangana-Baleanu-Caputo derivative with integral boundary conditions, is proved. Simultaneously, we discuss the existence of a solution by applying the Schauder fixed point theorem and a generalized Darbo fixed point theorem, which involves the concept of measure of noncompactness. The paper also contains some examples that illustrate the application of the main result.

A New Perspective on the Exact Solutions of the Local Fractional Modified Benjamin–Bona–Mahony Equation on Cantor Sets

A new local fractional modified Benjamin–Bona–Mahony equation is proposed within the local fractional derivative in this study for the first time. By defining some elementary functions via the Mittag–Leffler function (MLF) on the Cantor sets (CSs), a set of nonlinear local fractional ordinary differential equations (NLFODEs) is constructed. Then, a fast algorithm namely Yang’s special function method is employed to find the non-differentiable (ND) exact solutions. By this method, we can extract abundant exact solutions in just one step. Finally, the obtained solutions on the CS are outlined in the form of the 3-D plot. The whole calculation process clearly shows that Yang’s special function method is simple and effective, and can be applied to investigate the exact ND solutions of the other local fractional PDEs.

On the Existence and Stability of Solutions for a Class of Fractional Riemann–Liouville Initial Value Problems

This article deals with a class of nonlinear fractional differential equations, with initial conditions, involving the Riemann–Liouville fractional derivative of order α∈(1,2). The main objectives are to obtain conditions for the existence and uniqueness of solutions (within appropriate spaces), and to analyze the stabilities of Ulam–Hyers and Ulam–Hyers–Rassias types. In fact, different conditions for the existence and uniqueness of solutions are obtained based on the analysis of an associated class of fractional integral equations and distinct fixed-point arguments. Additionally, using a Bielecki-type metric and some additional contractive arguments, conditions are also obtained to guarantee Ulam–Hyers and Ulam–Hyers–Rassias stabilities for the problems under analysis. Examples are also included to illustrate the theory.

Chaos control and numerical solution of time-varying fractional Newton-Leipnik system using fractional Atangana-Baleanu derivatives

<abstract><p>Nonlinear fractional differential equations and chaotic systems can be modeled with variable-order differential operators. We propose a generalized numerical scheme to simulate variable-order fractional differential operators. Fractional calculus' fundamental theorem and Lagrange polynomial interpolation are used. Two methods, Atangana-Baleanu-Caputo and Atangana-Seda derivatives, were used to solve a chaotic Newton-Leipnik system problem with fractional operators. Our scheme examined the existence and uniqueness of the solution. We analyze the model qualitatively using its equivalent integral through an iterative convergence sequence. This novel method is illustrated with numerical examples. Simulated and analytical results agree. We contribute to real-world mathematical applications. Finally, we applied a numerical successive approximation method to solve the fractional model.</p></abstract>

Solving Fractional Differential Equations via Fixed Points of Chatterjea Maps

In this paper, we present the existence and uniqueness of fixed points and common fixed points for Reich and Chatterjea pairs of self-maps in complete metric spaces. Furthermore, we study fixed point theorems for Reich and Chatterjea nonexpansive mappings in a Banach space using the Krasnoselskii-Ishikawa iteration method associated with and consider some applications of our results to prove the existence of solutions for nonlinear integral and nonlinear fractional differential equations. We also establish certain interesting examples to illustrate the usability of our results.

EXISTENCE AND STABILITY RESULTS FOR COUPLED SYSTEM OF FRACTIONAL DIFFERENTIAL EQUATIONS INVOLVING AB-CAPUTO DERIVATIVE

In this paper, we use Krasnoselskii’s fixed point theorem to find existence results for the solution of the following nonlinear fractional differential equations (FDEs) for a coupled system involving AB-Caputo fractional derivative [Formula: see text] with boundary conditions [Formula: see text] We discuss uniqueness with the help of the Banach contraction principle. The criteria for Hyers–Ulam stability of given AB-Caputo fractional-coupled boundary value problem (BVP) is also discussed. Some examples are provided to validate our results. In Example 1, we find a unique and stable solution of AB-Caputo fractional-coupled BVP. In Example 2, the analysis of approximate and exact solutions with errors of nonlinear integral equations is elaborated with graphs.

SOLUTION OF VARIABLE-ORDER NONLINEAR FRACTIONAL DIFFERENTIAL EQUATIONS USING HAAR WAVELET COLLOCATION TECHNIQUE

A numerical method for the solution of nonlinear variable-order (VO) fractional differential equations (FDEs) is proposed in this paper. To determine the numerical solution of nonlinear VO FDEs, we used the Haar wavelet collocation method (HWCM) with a combination of Caputo fractional derivatives. For checking the efficiency of the HWCM, some examples are given. The maximum absolute error and mean square root errors of each test problem are computed for a different number of collocation points (CPs) to check the validity and applicability of the presented technique. The comparison of the exact and approximate solution is shown in figures for various numbers of CPs.

Nonlinear fractional differential equations and their existence via fixed point theory concerning to Hilfer generalized proportional fractional derivative

<abstract><p>This article adopts a class of nonlinear fractional differential equation associating Hilfer generalized proportional fractional ($ GPF $) derivative with having boundary conditions, which amalgamates the Riemann-Liouville $ (RL) $ and Caputo-$ GPF $ derivative. Taking into consideration the weighted space continuous mappings, we first derive a corresponding integral for the specified boundary value problem. Also, we investigate the existence consequences for a certain problem with a new unified formulation considering the minimal suppositions on nonlinear mapping. Detailed developments hold in the analysis and are dependent on diverse tools involving Schauder's, Schaefer's and Kransnoselskii's fixed point theorems. Finally, we deliver two examples to check the efficiency of the proposed scheme.</p></abstract>

Fixed point results with applications to nonlinear fractional differential equations

<abstract><p>The aim of this paper is to define a Berinde type ($ \rho $, $ \mu $)-$ \vartheta $ contraction and establish some fixed point results for self mappings in the setting of complete metric spaces. We derive new fixed point results, which extend and improve some results in the literature. We also supply a non trivial example to support the obtained results. Finally, we investigate the existence of solutions for the nonlinear fractional differential equation.</p></abstract>

Fixed point theorems via auxiliary functions with applications to two-term fractional differential equations with nonlocal boundary conditions

<abstract><p>In this study, the $ (h $-$ \varphi)_R $ and $ (h $-$ \varphi)_M $-contractions with two metrics endowed with a directed graph are examined using auxiliary functions. We propose a set of criteria that guarantees the existence of common fixed points for our contractions. This leads to a generalization of previous results in the literature. Towards our accomplishments, we establish affirmative results that demonstrate solutions to a class of nonlinear two-term fractional differential equations with nonlocal boundary conditions. To further corroborate our major findings, we also provide instances.</p></abstract>

Solving Nonlinear Multi-Order Fractional Differential Equations Using Bernstein Polynomials

Solving Nonlinear Multi-Order Fractional Differential Equations Using Bernstein Polynomials

NOVEL TRAVELING-WAVE SOLUTIONS OF SPATIAL-TEMPORAL FRACTIONAL MODEL OF DYNAMICAL BENJAMIN–BONA–MAHONY SYSTEM

This paper investigates the dynamics of exact traveling-wave solutions for nonlinear spatial and temporal fractional partial differential equations with conformable order derivatives arising in nonlinear propagation waves of small amplitude including nonlinear fractional modified Benjamin–Bona–Mahony equation, fractional Zakharov–Kuznetsov–Benjamin–Bona–Mahony equation and fractional (2+1)-dimensional Kadomtsev–Petviashvili–Benjamin–Bona–Mahony equation as well. By utilizing the Sine-Gordon expansion method (SGEM), new real- and complex-valued exact traveling-wave solutions are reported by preferring suitable values of physical free parameters. The nonlinear governing equations are reduced into auxiliary nonlinear ordinary differential equations with aid of fractional traveling-wave transformation, in which the fractional derivative is evaluated in a conformable sense. The productivity process of the proposed method for predicting the desirable solutions is also provided. Some of the obtained solutions are simulated graphically in 3D and contour plots. Meanwhile, the effects of the fractional parameter [Formula: see text] in the space and the time direction are illustrated in 2D plots to ensure the novelty, applicability and credibility of the SGEM. These results reveal that the suggested method is general and adequate for dealing with nonlinear models featuring fractional derivatives and can be employed to analyze wide classes of complex phenomena of partial differential equations occurring in engineering and nonlinear dynamics.

Существование и единственность решений стохастических дробных дифференциальных уравнений в нескольких временных шкалах

A novel class of nonlinear stochastic fractional differential equations with delay and the Jumarie and Ito differentials is introduced in the paper. The aim of the study is to prove existence and uniqueness of solutions to these equations. The main results of the paper generalise some previous findings made for the non-delay and three-scale equations under additional restrictions on the fractional order of the Jumarie differentials, which are removed in our analysis. The techniques used in the paper are based on the properties of the singular integral operators in specially designed spaces of stochastic processes, the representation of delay equations as functional differential equations as well as Picard’s iterative method.