Higher-order Fractional Differential Equations Research Articles

Relatively exact controllability for higher-order fractional stochastic delay differential equations

In this paper, our main purpose is to study a class of higher-order fractional stochastic delay differential equations (FSDDEs). We first define a more generalized delay Grammian matrix involving delayed matrix functions, and then we obtain the relatively exact controllability of linear higher-order FSDDEs by using such matrix. Subsequently, by constructing a suitable control function, we discuss relatively exact controllability of the nonlinear addressed equations. The results we presented are based on a new approach, Carathéodory approximation, which differs from previous literature. The main results are obtained by using fractional calculus, nonlinear analysis, Jensen inequality, Grönwall-Bellman inequality, Hölder inequality, Bihari inequality and Burkholder-Davis-Gundy inequality. Finally, the theoretical conclusions are supported through examples.

Positive solutions for a class of fractional differential equations with infinite-point boundary conditions on infinite intervals

In this paper, the existence of the multiple positive solutions for a class of higher-order fractional differential equations on infinite intervals with infinite-point boundary value conditions is mainly studied. First, we construct the Green function and analyze its properties, and then by using the Leggett–Williams fixed point theorem, some new results on the existence of positive solutions for the boundary value problem are obtained. Finally, we illustrate the application of our conclusion by an example.

Using Laplace Residual Power Series Method in Solving Coupled Fractional Neutron Diffusion Equations with Delayed Neutrons System

In this paper, a system of coupled fractional neutron diffusion equations with delayed neutrons was solved efficiently by using a combination of residual power series and Laplace transform techniques, and the anomalous diffusion was considered by taking the non-Gaussian case with different values of fractional parameter α. The Laplace residual power series method (LRPSM) does not require differentiation, conversion, or discretization for the assumed conditions, so the approach is simple and suitable for solving higher-order fractional differential equations. To assure the theoretical results, two different neutron flux initial conditions were presented numerically, where the needed Mathematica codes were performed using essential nuclear reactor cross-section data, and the results for different values of times were tabulated and graphically figured out. Finally, it must be noted that the results align with the Adomian decomposition method.

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.

Linear higher-order fractional differential and integral equations

We study the equivalences and the implications between linear (or homogeneous) nth order fractional differential equations (FDEs) and integral equations in the spaces L1(a,b) and C[a,b] when n≥ 2. We establish the equivalence in C[a,b] of the IVP of the nth order FDE subject to the initial condition u(i)(a)=ui for i in {0,1,...,n-1} when n≥2. The difficulty is that the known conditions for such equivalence for the first order FDEs are not sufficient for equivalence in the nth order FDEs with n≥2. In this article we provide additional conditions to ensure the equivalence for the nth order FDEs with n≥2. In particular, we obtain conditions under which solutions of the integral equations are solutions of the linear nth order FDEs. These results are keys for further studying the existence of solutions and nonnegative solutions to initial and boundary value problems of nonlinear nth order FDEs. This is done via the corresponding integral equations by topological methods such as the Banach contraction principle, fixed point index theory, and degree theory.

The existence of positive solutions for high order fractional differential equations with sign changing nonlinearity and parameters

<abstract><p>By constructing an auxiliary boundary value problem, the difficulty caused by sign changing nonlinearity terms is overcome by means of the linear superposition principle. Using the Guo-Krasnosel'skii fixed point theorem, the results of the existence of positive solutions for boundary value problems of high order fractional differential equation are obtained in different parameter intervals under a more relaxed condition compared with the existing literature. As an application, we give two examples to illustrate our results.</p></abstract>

Some generalized numerical methods for solving higher-order of fractional partial differential equations with application

This article recommends a computer method for solving some classes of higher-order fractional partial differential equations (FrPDEs). The updated techniques were made by combining the method of lines (MOL) and the numerical RK-type method. For solving certain types of FrPDEs that aren’t in the order, you might expect. Some problems with second-, third-, and fourth-order FrPDEs are solved with the methods made, and the numerical solutions are compared with the analytical answers. Also, a new type of application in classical physics called a wave FrPDE has been made for mechanical waves like water, earthquake, sound, and electromagnetic waves. Many examples are given to show how flexible and useful the method is. The numerical results are seen as the same as the exact solutions of the implementations to show how accurate and useful the suggested methods are. Also, the numerical implementations compare the suggested methods with exact solutions to show that they work and are accurate. The suggested methods used RKM analytical solutions to come up with numerical solutions to the test problems. Some examples are given to show how flexible and useful the method is. Also, the numerical implementations are compared with exact results to show that the proposed methods work and are accurate. R.K. techniques were used to find numerical answers to the suggested test problems.

Oscillation of higher order fractional differential equations

The oscillation for nonlinear two-term fractional differential equations of real orders is investigated. We explore the impact of varying derivative orders on oscillatory and asymptotic properties of studied equations, with an emphasis put on dissimilarities between integer and non-integer order cases. This enables us to reach a better understanding of how even and odd order equations affect analogous oscillation results.

New Efficient Computations with Symmetrical and Dynamic Analysis for Solving Higher-Order Fractional Partial Differential Equations

Due to the rapid development of theoretical and computational techniques in the recent years, the role of nonlinearity in dynamical systems has attracted increasing interest and has been intensely investigated. A study of nonlinear waves in shallow water is presented in this paper. The classic form of the Korteweg–de Vries (KdV) equation is based on oceanography theory, shallow water waves in the sea, and internal ion-acoustic waves in plasma. A shallow fluid assumption is shown in the framework by a sequence of nonlinear fractional partial differential equations. Indeed, the primary purpose of this study is to use a semi-analytical technique based on Fractional Taylor Series to achieve numerical results for nonlinear fifth-order KdV models of non-integer order. Caputo is the operator used for dealing with fractional derivatives. The generated solutions of nonlinear fifth-order KdV models of non-integer order for modeling turbulence processes in the field of ocean engineering are compared analytically and numerically, to demonstrate the behaviors of several parameters of the current model. We verified the method’s convergence analysis and provided an error estimate by showing 2D and 3D graphs to further confirm its efficacy.

Superstability of higher-order fractional differential equations

Using generalized Taylor's formula, this work investigate the superstability for a class of fractional differential equations with Caputo derivative. In this way, some interesting results are generalized.

On the eventual periodicity of fractional order dispersive wave equations using RBFS and transform

In this research work, let’s focus on the construction of numerical scheme based on radial basis functions finite difference (RBF-FD) method combined with the Laplace transform for the solution of fractional order dispersive wave equations. The numerical scheme is then applied to examine the eventual periodicity of the proposed model subject to the periodic boundary conditions. The implementation of proposed technique for high order fractional and integer type nonlinear partial differential equations (PDEs) is beneficial because this method is local in nature, therefore it yields and resulted in sparse differentiation matrices instead of full and dense matrices. Only small dimensions of linear systems of equations are to be solved for every center in the domain and hence this procedure is more reliable and efficient to solve large scale physical and engineering problems in complex domain. Laplace transform is utilized for obtaining the equivalent time-independent equation in Laplace space and also valuable to handle time-fractional derivatives in the Caputo sense. Application of Laplace transform avoids the time steeping procedure which commonly encounters the time instability issues. The solution to the transformed model is then obtained by computing the inversion of Laplace transform with an appropriate contour in a complex space, which is approximated by trapezoidal rule with high accuracy. Also since the Laplace transform operator is linear, it cannot be used to transform non-linear terms therefore let’s use a linearization approach and an appropriate iterative scheme. The proposed approach is tasted for some nonlinear fractional order KdV and Burgers equations. The capacity, high order accuracy and efficiency of our approach are demonstrated using examples and resultsRBFs Methods

Nonoscillatory Solutions of Higher-Order Fractional Differential Equations

Nonoscillatory Solutions of Higher-Order Fractional Differential Equations

Exact and numerical solutions of higher-order fractional partial differential equations: A new analytical method and some applications

In this paper, the solution methodology of higher-order linear fractional partial deferential equations (FPDEs) as mentioned in eqs (1) and (2) below in Caputo definition relies on a new analytical method which is called the Laplace-residual power series method (L-RPSM). The main idea of our proposed technique is to convert the original FPDE in Laplace space, and then apply the residual power series method (RPSM) by using the concept of limit to obtain the solution. Some interesting and important numerical test applications are given and discussed to illustrate the procedure of our method, and also to confirm that this method is simple, understandable and very fast for obtaining the exact and approximate solutions (ASs) of FPDEs compared with other methods such as RPSM, variational iteration method (VIM), homotopy perturbation method (HPM) and Adomian decomposition method (ADM). The main advantage of the proposed method is its simplicity in computing the coefficients of terms of series solution by using only the concept of limit at infinity and not as the other well-known analytical method such as, RPSM that need to obtain the fractional derivative (FD) each time to determine the unknown coefficients in series solutions, and VIM, ADM, or HPM that need the integration operators which is difficult in fractional case.

On the Solutions of the Higher Order Fractional Differential Equations of Riesz Space Derivative with Anti-Periodic Boundary Conditions

We present existence and uniqueness results for a class of higher order anti-periodic fractional boundary value problems with Riesz space derivative which is two-sided fractional operator. The obtained results are established by applying some fixed point theorems. Various numerical examples are given to illustrate the obtained results.

Existence results for higher order fractional differential equations with integral boundary conditions via Kuratowski measure of noncompactnes

Existence results for higher order fractional differential equations with integral boundary conditions via Kuratowski measure of noncompactnes

Existence of Positive Solutions for a Higher-Order Fractional Differential Equation with Multi-Term Lower-Order Derivatives

This paper deals with the study of the existence of positive solutions for a class of nonlinear higher-order fractional differential equations in which the nonlinear term contains multi-term lower-order derivatives. By reducing the order of the highest derivative, the higher-order fractional differential equation is transformed into a lower-order fractional differential equation. Then, combining with the properties of left-sided Riemann–Liouville integral operators, we obtain the existence of the positive solutions of fractional differential equations utilizing some weaker conditions. Furthermore, some examples are given to demonstrate the validity of our main results.

Informationisation of teaching model for track and field education based on finite element higher-order fractional differential equation

Abstract In order to study the informationised teaching model of track and field education, this paper uses the finite element higher-order fractional differential equation to simulate, to provide high-quality, strong resistance to pressure, physical and mental health talents for the development of the country, through the study of the higher-order fraction of the finite element and track and field course intended to strengthen the application in the innovation of track and field course. The results show that based on the finite element theory of higher-order fractional differential equation, a new teaching model is constructed to solve the current difficulties faced by track and field. Starting from the teaching end, pay attention to the quality of output so as to achieve the purpose of training qualified personnel. Conclusion: In the course of track and field classroom practice based on finite element high-order fractional differential equation, students’ enthusiasm is mobilised, and they take the initiative to learn knowledge and master skills.

Information Teaching Model of Preschool Art Education in Colleges and Universities Based on Finite Element Higher-Order Fractional Differential Equation

Abstract In order to meet the social demand for preschool education professionals, based on the finite element higher-order fractional differential equation, this paper studies the information teaching model of preschool education of fine arts in colleges and universities. Through the scientific and effective fine arts education in the preschool education major in colleges and universities, the students’ fine arts ability can be effectively improved, and the application of information means can make this goal can be better realised so that the teaching needs can be better satisfied and they promote the better development of fine arts education in the preschool education major.

Numerical solution of Bagley–Torvik, nonlinear and higher order fractional differential equations using Haar wavelet

Numerical solution of Bagley–Torvik, nonlinear and higher order fractional differential equations using Haar wavelet

Nonexistence Results for Higher Order Fractional Differential Inequalities with Nonlinearities Involving Caputo Fractional Derivative

Higher order fractional differential equations are important tools to deal with precise models of materials with hereditary and memory effects. Moreover, fractional differential inequalities are useful to establish the properties of solutions of different problems in biomathematics and flow phenomena. In the present work, we are concerned with the nonexistence of global solutions to a higher order fractional differential inequality with a nonlinearity involving Caputo fractional derivative. Namely, using nonlinear capacity estimates, we obtain sufficient conditions for which we have no global solutions. The a priori estimates of the structure of solutions are obtained by a precise analysis of the integral form of the inequality with appropriate choice of test function.