Nonlinear Fractional Differential Equations Research Articles

Existence results for a coupled system of nonlinear fractional functional differential equations with infinite delay and nonlocal integral boundary conditions

<abstract><p>This article is devoted to studying a new class of nonlinear coupled systems of fractional differential equations supplemented with nonlocal integro-coupled boundary conditions and affected by infinite delay. We first transform the boundary value problem into a fixed-point problem, and, with the aid of the theory of infinite delay, we assume an appropriate phase space to deal with the obtained problem. Then, the existence result of solutions to the given system is investigated by employing Schaefer's fixed-point theorem, while the uniqueness result is established in view of the Banach contraction mapping principle. The illustrative examples are constructed to ensure the availability of the main results.</p></abstract>

Utilizing Schaefer's fixed point theorem in nonlinear Caputo sequential fractional differential equation systems

<abstract><p>In the present study, established fixed-point theories are utilized to explore the requisite conditions for the existence and uniqueness of solutions within the realm of sequential fractional differential equations, incorporating both Caputo fractional operators and nonlocal boundary conditions. Subsequently, the stability of these solutions is assessed through the Ulam-Hyers stability method. The research findings are validated with a practical example that corroborate and reinforce the theoretical results.</p></abstract>

Green's functions and existence of solutions of nonlinear fractional implicit difference equations with Dirichlet boundary conditions

This article is devoted to deduce the expression of the Green's function related to a general constant coefficients fractional difference equation coupled to Dirichlet conditions. In this case, due to the points where some of the fractional operators are applied, we are in presence of an implicit fractional difference equation. So, due to such a property, it is more complicated to calculate and manage the expression of the Green's function than in the explicit case studied in a previous work of the authors. Contrary to the explicit case, where it is shown that the Green's function is constructed as finite sums, the Green's function constructed here is an infinite series. This fact makes necessary to impose more restrictive assumptions on the parameters that appear in the equation. The expression of the Green's function will be deduced from the Laplace transform on the time scales of the integers. We point out that, despite the implicit character of the considered equation, we can have an explicit expression of the solution by means of the expression of the Green's function. These two facts are not incompatible. Even more, this method allows us to have an explicit expression of the solution of an implicit problem. Finally, we prove two existence results for nonlinear problems, via suitable fixed point theorems.

Positive solutions for integral boundary value problems of nonlinear fractional differential equations

In this paper, we consider integral boundary value problems of nonlinear fractional differential equations. Existence results of positive solutions for the problem are obtained based on the Guo-Krasnoselskii theorem and the Five functional fixed point theorem. Simple examples follow the main results in successive sections.

On the Initial Value Problem for Fuzzy Nonlinear Fractional Differential Equations

In this paper, we study the existence result of solutions for fuzzy nonlinear fractional differential equations involving Caputo differentiability of an arbitrary order 0 < q < 1. As application, an example is included to show the applicability of our result.

Solving systems of fractional differential equations by modified fractional differential transform method with adomian polynomials

In this article, we use a new algorithm as FDT with Adomian polynomials is carried out for solving systems of non-linear fractional differential equations. We approximate solutions to the system of nonlinear equations using this innovative method and discuss the suggested technique of convergence analysis. The theoretical findings and the performance of the numerical approximation are supported by numerical examples. Moreover, using the Python programming, we do numerical and graphical analysis.

A study for a higher order Riemann-Liouville fractional differential equation with weakly singularity

<abstract><p>In this paper, we study an initial value problem with a weakly singular nonlinear fractional differential equation of higher order. First, we establish the existence of global solutions to the problem within the appropriate function space. We then introduce a generalized Riemann-Liouville mean value theorem. Using this theorem, we prove the Nagumo-type uniqueness theorem for the stated problem. Moreover, we give two examples to illustrate the applicability of the existence and uniqueness theorems.</p></abstract>

Stability Results for Nonlinear Implicit ϑ-Caputo Fractional Differential Equations with Fractional Integral Boundary Conditions

This article examines the necessary conditions for the unique existence of solutions to nonlinear implicit ϑ-Caputo fractional differential equations accompanied by fractional order integral boundary conditions. The analysis draws upon Banach’s contraction principle and Krasnoselskii’s fixed point theorem. Furthermore, the circumstances leading to the attainment of Ulam–Hyers–Rassias forms of stability are established. An illustrative example is provided to demonstrate the derived findings.

Genocchi collocation method for accurate solution of nonlinear fractional differential equations with error analysis

In this study, we introduce an innovative fractional Genocchi collocation method for solving nonlinear fractional differential equations, which have significant applications in science and engineering. The fractional derivative is defined in the Caputo sense and by leveraging fractional-order Genocchi polynomials, we transform the nonlinear problem into a system of nonlinear algebraic equations. A novel technique is employed to solve this system, enabling the determination of unknown coefficients and ultimately the solution. We derive the error bound for our proposed method and validate its efficacy through several test problems. Our results demonstrate superior accuracy compared to existing techniques in the literature, suggesting the potential for extending this approach to tackle more complex problems of critical physical significance.

A Novel Approach for Solving Nonlinear Time Fractional Fisher Partial Differential Equations

This study focuses on solving non-linear time fractional Fisher partial differential equations using analytical series solutions. The authors consider the Caputo fractional derivative in their formulas, which adds accuracy to the results. They introduce a novel method called LRPS which proves to be a powerful tool for obtaining precise analytical and numerical solutions for these equations. The LRPS method emphasizes precision, effectiveness, and practical application, making it suitable for various fields such as physics, engineering, and finance. Due to the importance of accuracy, effectiveness and method of application in this approach, it is highlighted that accurate solutions can be obtained when there is a pattern in the parts of the series, while approximate estimates are provided otherwise. The LRPS method is presented as a powerful technique specifically designed for solving nonlinear fractional Fisher partial differential equations.

Bifurcation from interval and multiple positive solutions for a nonlinear fractional differential equations with integral boundary conditions

This paper studies the bifurcation from interval and Multiple positive solutions for a boundary-value problem of nonlinear fractional differential equations with integral boundary conditions. Using the bifurcation technique and the topological degree theory, the existence of Multiple positive solutions is investigated and some sufficient conditions are obtained.

Solving Nonlinear Fractional PDEs with Applications to Physics and Engineering Using the Laplace Residual Power Series Method

The Laplace residual power series (LRPS) method uses the Caputo fractional derivative definition to solve nonlinear fractional partial differential equations. This technique has been applied successfully to solve equations such as the fractional Kuramoto–Sivashinsky equation (FKSE) and the fractional generalized regularized long wave equation (GRLWE). By transforming the equation into the Laplace domain and replacing fractional derivatives with integer derivatives, the LRPS method can solve the resulting equation using a power series expansion. The resulting solution is accurate and convergent, as demonstrated in this paper by comparing it with other analytical methods. The LRPS approach offers both computational efficiency and solution accuracy, making it an effective technique for solving nonlinear fractional partial differential equations (NFPDEs). The results are presented in the form of graphs for various values of the order of the fractional derivative and time, and the essential objective is to reduce computation effort.

Caputo-Fabrizio type fractional differential equations with non-instantaneous impulses: Existence and stability results

In the current article, we establish the existence and uniqueness of solutions to the initial value problem for nonlinear implicit fractional differential equations with non-instantaneous impulses, including the Caputo-Fabrizio fractional derivative. This derivative encapsulates non-local and memory-dependent effects in the system's behavior. The problem's complexity arises from its implicit nature, where the relationships between variables are not directly specified. This inherent intricacy is further compounded by the introduction of non-instantaneous impulses, signifying abrupt, intermittent changes within the system at specific junctures, which in turn trigger sudden transitions. At its core, the initial value problem is concerned with deciphering a system's behavior based on its initial conditions, necessitating a solution that adheres to the given equation while considering the presence of non-instantaneous impulses. To derive the existence results, the study leverages established mathematical tools, specifically, the standard Banach's fixed point theorem and Darbo's fixed point theorem connected to Kuratowski's measure of noncompactness (KMN). Furthermore, the Hyers-Ulam stability (HU-stability) of the given problem is discussed, which, in turn, enhances the utility and reliability of these solutions in practical applications. In order to demonstrate the applicability of the discovered results for various values of δ, we construct some examples towards the end. The practical implications of these results extend to a wide range of fields, offering improved modeling, control, and understanding of complex systems.

Solution of nonlinear fractional partial differential equations by Shehu transform and Adomian decomposition method (STADM)

Several well-known nonlinear partial differential equations (NPDEs) of integer order, including the Heat Equation and the Korteweg–de Vries (KDV) Equation, as well as NPDEs of fractional order, including the fractional advection equation (FAE), fractional gas dynamic equation (FGDE), diffusion equation, and fractional Burger equation (FBE), are approximated analytically in this study using the Shehu Transform Adomian Decomposition Method, which is a combination of two methods, namely, Shehu Transform (ST) and Adomian Decomposition Method (ADM). The Caputo sense is utilized to the fractional derivatives. This semianalytical method is used to acquire the series solutions to several example problems. The main advantage of the approach is that, in contrast to other similar semianalytical methods, it is straightforward to apply. This method has been widely used to answer any class of equations in the sciences and engineering because it can solve a huge class of linear and nonlinear equations effectively, more quickly, and precisely. The obtained solution matches exactly with the solution obtained by other methods.

Ulam-Type Stability Results for Variable Order Ψ-Tempered Caputo Fractional Differential Equations

An initial value problem for nonlinear fractional differential equations with a tempered Caputo fractional derivative of variable order with respect to another function is studied. The absence of semigroup properties of the considered variable order fractional derivative leads to difficulties in the study of the existence of corresponding differential equations. In this paper, we introduce approximate piecewise constant approximation of the variable order of the considered fractional derivative and approximate solutions of the given initial value problem. Then, we investigate the existence and the Ulam-type stability of the approximate solution of the variable order Ψ-tempered Caputo fractional differential equation. As a partial case of our results, we obtain results for Ulam-type stability for differential equations with a piecewise constant order of the Ψ-tempered Caputo fractional derivative.

On deformable fractional impulsive implicit boundary value problems with delay

This paper deals with some existence and uniqueness results for a class of deformable fractional differential equations. These problems encompassed nonlinear implicit fractional differential equations involving boundary conditions and various types of delays, including finite, infinite, and state-dependent delays. Our approach to proving the existence and uniqueness of solutions relied on the application of the Banach contraction principle and Schauder’s fixed-point theorem. In the last section, we provide different examples to illustrate our obtained results.

NON- INTEGER MATHEMATICAL MODEL OF RESPIRATORY SYNCYTIAL VIRUS

Respiratory syncytial virus (RSV), also known as a human respiratory syncytial virus (hRSV) and human orthopneumovirus, is a common, transmittable virus that roots respiratory tract diseases. It is a negative-sense, single-stranded Ribonucleic acid (RNA) virus. It gets its name from syncytia, which are huge cells that form when infected cells merge. In this paper fractional order model of the respiratory syncytial virus (RSV) virus will be developed. The Caputo fractional derivative operator of the order will be used to generate the model scheme of non-integer differential equations. To calculate an estimated solution of the system of nonlinear fractional differential equations, the Laplace-Adomian Decomposition Method was used. Infinite series was produced as solutions to fractional differential equations. The model's proposed series solution converges quickly to its precise value. The obtained results are compared to the standard case.

Solutions of fractional differential equations by using a blend of variational iteration method with Sumudu transform and application to price adjustment equations

The presence of delays in a mathematical model can improve its vitality and suitability in describing several phenomena. However, in the presence of delays, nonlinear fractional differential equations are more difficult to study. This paper presents the use of a blend of variational iteration method with Sumudu transform for solving delay differential equations with Caputo derivatives of fractional variable order. Moreover, the paper introduces delays into the price adjustment equations to propose new price adjustment models with more potential for vitality and suitability. The paper assigns suitable real values to the parameters for the graphical display and comparison of the obtained solutions. The paper presents the interactions that exist among the price, demand, supply and dependence of supply and demand on the price, which can be applied to estimate the equilibrium price.

A new decomposition transform method for solving nonlinear fractional logistic differential equation

A new decomposition transform method for solving nonlinear fractional logistic differential equation

Solving a Nonlinear Fractional Differential Equation Using Fixed Point Results in Orthogonal Metric Spaces

This research article aims to solve a nonlinear fractional differential equation by fixed point theorems in orthogonal metric spaces. To achieve our goal, we define an orthogonal Θ-contraction and orthogonal (α,Θ)-contraction in the setting of complete orthogonal metric spaces and prove fixed point theorems for such contractions. In this way, we consolidate and amend innumerable celebrated results in fixed point theory. We provide a non-trivial example to show the legitimacy of the established results.