Solutions Of Fractional Differential Equations Research Articles

On Shehu Transform with Application of Solutions of Fractional Differential Equations

This review article explains the Shehu transform as a tool used for solving linear differential equations of fractional order, where the definition of the Caputo differential operator of order α>0, is taken into consideration. The transformation is used to convert Initial Value Problems (IVPs) of the fractional order of Caputo sense into simple algebraic equations. Then the inverse of the transform is used to obtain the analytical solution of the problem. We solved some illustrative examples.

Existence and stability analysis of solutions for fractional differential equations with delay

In this manuscript, we establish the existence, uniqueness, and stability of solutions for fractional differential equations with delay. Then we solve the version of the problem with constant delay. We utilize the Bielecki Norm and the Ulam-Hyers stability for our results.

Lacunary interpolation by the Spline Function of Fractional Degree

In almost all fields of numerical analysis, spline functions are the most effective tool for polynomials employed as the fundamental method of approximation theory. Additionally, existence, uniqueness, and error boundaries are required for the spline creation in the g-spline interpolation issue. Mathematics, physics, biology, engineering, signal processing, systems identification, control theory, finance, and fractional dynamics have all shown an interest in fractional differential equations, also work in social sciences including economics, finance, and dietary supplements. It is crucial to find both close and accurate solutions of fractional differential equations. To find solutions of fractional differential equations, several analytical and numerical techniques have been developed. In this paper, we extend the five-degree spline (0,4) lacunary interpolation on uniform meshes. The outcomes, uniqueness and error boundaries for generalize (0,4) Lacunary interpolation using five- degree splines. These generalizes outperform the usage of the (0,4) five splines for interpolation.

On asymptotic periodic solutions of fractional differential equations and applications

In this paper we study the asymptotic behavior of solutions of fractional differential equations of the form D C α u ( t ) = A u ( t ) + f ( t ) , u ( 0 ) = x , 0 > α ≤ 1 , ( ∗ ) D^{\alpha }_Cu(t)=Au(t)+f(t), u(0)=x, 0>\alpha \le 1, ( *) where D C α u ( t ) D^{\alpha }_Cu(t) is the derivative of the function u u in the Caputo’s sense, A A is a linear operator in a Banach space X \mathbb {X} that may be unbounded and f f satisfies the property that lim t → ∞ ( f ( t + 1 ) − f ( t ) ) = 0 \lim _{t\to \infty } (f(t+1)-f(t))=0 which we will call asymptotic 1 1 -periodicity. By using the spectral theory of functions on the half line we derive analogs of Katznelson-Tzafriri and Massera Theorems. Namely, we give sufficient conditions in terms of spectral properties of the operator A A for all asymptotic mild solutions of Eq. (*) to be asymptotic 1 1 -periodic, or there exists an asymptotic mild solution that is asymptotic 1 1 -periodic.

Algorithms for the Numerical Solution of Fractional Differential Equations with Interval Parameters

Algorithms for the Numerical Solution of Fractional Differential Equations with Interval Parameters

Multiple positive solutions of fractional differential equations with improper integral boundary conditions on the half-line

This paper investigates the existence of positive solutions for a class of fractional boundary value problems involving an improper integral and the infinite-point on the half-line by making use of properties of the Green function and Avery–Peterson fixed point theorem. In addition, an example is presented to illustrate the applicability of our main result.

On the solution of fractional differential equations using Atangana’s beta derivative and its applications in chaotic systems

In this research, we examine the solution of ordinary fractional differential equations using Atangana’s beta derivative. Our approach is divided into two parts. First, we establish conditions under which the fractional differential equation has a unique solution. Next, we develop a numerical technique based on the Adams–Bashforth method and demonstrate its convergence. We then apply the numerical method to a test example to evaluate its efficacy. Finally, we analyze three nonlinear chaotic and hyperchaotic fractional dynamical systems, calculating the strange attractors for various fractional orders. We analyzed the fractional chaotic systems and showed that by changing the order of the fractional derivative, some properties of the system change, such as the area of the chaos attracting region.

On the Solitary Waves and Nonlinear Oscillations to the Fractional Schrödinger–KdV Equation in the Framework of the Caputo Operator

The fractional Schrödinger–Korteweg-de Vries (S-KdV) equation is an important mathematical model that incorporates the nonlinear dynamics of the KdV equation with the quantum mechanical effects described by the Schrödinger equation. Motivated by the several applications of the mentioned evolution equation, in this investigation, the Laplace residual power series method (LRPSM) is employed to analyze the fractional S-KdV equation in the framework of the Caputo operator. By incorporating both the Caputo operator and fractional derivatives into the mentioned evolution equation, we can account for memory effects and non-local behavior. The LRPSM is a powerful analytical technique for the solution of fractional differential equations and therefore it is adapted in our current study. In this study, we prove that the combination of the residual power series expansion with the Laplace transform yields precise and efficient solutions. Moreover, we investigate the behavior and properties of the (un)symmetric solutions to the fractional S-KdV equation using extensive numerical simulations and by considering various fractional orders and initial fractional values. The results contribute to the greater comprehension of the interplay between quantum mechanics and nonlinear dynamics in fractional systems and shed light on wave phenomena and symmetry soliton solutions in such equations. In addition, the proposed method successfully solves fractional differential equations with the Caputo operator, providing a valuable computational instrument for the analysis of complex physical systems. Moreover, the obtained results can describe many of the mysteries associated with the mechanism of nonlinear wave propagation in plasma physics.

Modeling and Analysis of a Fractional Visceral Leishmaniosis with Caputo and Caputo–Fabrizio derivatives

Visceral leishmaniosis is one recent example of a global illness that demands our best efforts at understanding. Thus, mathematical modeling may be utilized to learn more about and make better epidemic forecasts. By taking into account the Caputo and Caputo-Fabrizio derivatives, a frictional model of visceral leishmaniosis was mathematically examined based on real data from Gedaref State, Sudan. The stability analysis for Caputo and Caputo-Fabrizio derivatives is analyzed. The suggested ordinary and fractional differential mathematical models are then simulated numerically. Using the Adams-Bashforth method, numerical simulations are conducted. The results demonstrate that the Caputo-Fabrizio derivative yields more precise solutions for fractional differential equations.

New Solutions of the Fractional Differential Equations With Modified Mittag-Leffler Kernel

Abstract This paper is concerned with some features of the modified Caputo-type Mittag-Leffler fractional derivative operator and its associated fractional integral operator. Mainly, new types of solutions for fractional differential equations with Mittag-Leffler kernel are generated based on a numerical algorithm developed in this paper. The suggested algorithm is used to describe the solution behavior of models involving modified Caputo-type Mittag-Leffler fractional derivatives. The results described in this paper are expected to be effectively employed in the area of simulating related fractional models.

Approximate analytical solutions and applications of pantograph-type equations with Caputo derivative and variable orders

This study presents an efficient method that is suitable for differential equations, both with integer-order and fractional derivatives. This study examines the construction of solutions of fractional differential equations that are associated with varying delay proportional to the independent variable using a hybrid of Sumudu transform method. This study considers differential equations with Caputo derivatives of fractional variable orders and their applications in Nuclear Physics. The application indicates that fractional differential equations that have varying delay proportional to the independent variable are useful as tools for the modelling of many anomalous phenomena in nature and in the theory of complex systems.

M-Lauricella hypergeometric functions: integral representations and solutions of fractional differential equations

In this paper, using the modified beta function involving the generalized M-series in its kernel, we described new extensions for the Lauricella hypergeometric functions $F_{A}^{(r)}$, $F_{B}^{(r)}$, $F_{C}^{(r)}$ and $F_{D}^{(r)}$. Furthermore, we obtained various integral representations for the newly defined extended Lauricella hypergeometric functions. Then, we obtained solution of fractional differential equations involving new extensions of Lauricella hypergeometric functions, as examples.

Numerical solutions of fractional differential equation with multiple delays via block boundary value method

Numerical solutions of fractional differential equation with multiple delays via block boundary value method

On the numerical solution of fractional differential equations under white noise processes

The aim of this paper is to elucidate some relevant aspects concerning the numerical solution of stochastic differential equations in structural and mechanical applications. Specifically, the attention is focused on those differential problems involving fractional operators to model the viscoelastic behavior of the structural/mechanical components and involving a white noise process as stochastic input. Starting from the consideration that the Grünwald–Letnikov based integration scheme, that is a step-by-step procedure often invoked in literature to discretize and integrate the aforementioned differential equations, is not properly employed due to the discontinuous nature of the input, an alternative numerical integration scheme is proposed. The latter is based on the Riemann–Liouville fractional integral and relies on the parabolic piecewise approximation of the response function to be integrated, leading to a more effective and more advantageous solution than that provided by the Grünwald–Letnikov based integration scheme. This is demonstrated analyzing the case study of a fractional Euler–Bernoulli beam and comparing the numerical results with those obtained by an analytical solution available in literature.

On the zeros of solutions of ordinary and fractional differential equations

This paper is devoted to studying on the locations of zeros of related integral operators and the solutions of some ordinary and fractional differential equations. We generalize Sturm and Picone's theorems and Leighton and Levin's criteria. Moreover, we share some oscillation and disconjugacy criteria for the solutions of ordinary second‐order Sturm–Liouville and fractional differential equations. Finally, we introduce some properties of the solutions of fractional differential equations.

An Efficient and Robust Numerical Solver for Impulsive Control of Fractional Chaotic Systems

This paper derives a computationally efficient and fast-running solver for the approximate solution of fractional differential equations with impulsive effects. In this connection, for approximating the fractional-order integral operator, a B-spline version of interpolation by corresponding equal mesh points is adopted. An illustrative example illustrates the accuracy of the new solver results as compared with those of the previous study. The proposed solver’s performance is evaluated by the fractional Rössler and susceptible-exposed-infectious impulsive systems. Moreover, the effect of impulsive behaviors is shown for various values of impulsive.

Existence of Monotone Positive Solutions for Caputo–Hadamard Nonlinear Fractional Differential Equation with Infinite-Point Boundary Value Conditions

Numerical solutions and approximate solutions of fractional differential equations have been studied by mathematicians recently and approximate solutions and exact solutions of fractional differential equations are obtained in many kinds of ways, such as Lie symmetry, variational method, the optimal ADM method, and so on. In this paper, we obtain the positive solutions by iterative methods for sum operators. Green’s function and the properties of Green’s function are deduced, then based on the properties of Green’s function, the existence of iterative positive solutions for a nonlinear Caputo–Hadamard infinite-point fractional differential equation are obtained by iterative methods for sum operators; an example is proved to illustrate the main result.

Review of solution Fractal Differential Equations via the Riemann-Liouville

In this paper, the definitions, concepts, and basics of linear and nonlinear fractal differential equations of both homogeneous and heterogeneous types were studied, and some methods for solving fractal differential equations were presented with an explanation of each method and how to solve the equation in it An example was taken for each method that was presented to show how to solve in this way. Among the methods that were presented are (the special method, the Laplace transform Method, the Inverse Fractional Shehu Transform Method, The Power Series Method, The generalized Mittag-Leffler method, Spline Interpolation Techniques for Approximating the Solution of Fractional Differential Equations.

Group classification and exact solutions of fractional differential equation with quintic non-Kerr nonlinearity term

Group classification and exact solutions of fractional differential equation with quintic non-Kerr nonlinearity term

Solutions of Fractional differential equations with some modifications of Adomian Decomposition method

Solutions of Fractional differential equations with some modifications of Adomian Decomposition method