Problems For Fractional Differential Equations Research Articles

Solutions to fractional differential equations involving integral boundary conditions

ABSTRACT In this article, we discuss the existence and uniqueness of solutions to a class of integral boundary value problem of fractional differential equations. Different fractional order derivations are included in the nonlinear term and boundary conditions. The unique solution is derived by a new fixed point theorem of increasing -concave operators defined on ordered sets. Further, we construct a monotone iterative scheme to approximate the unique solution. An example is given to illustrate the validity of our main results.

On One Problems of Spectral Theory for Ordinary Differential Equations of Fractional Order

The present paper is devoted to the spectral analysis of operators induced by fractional differential equations and boundary conditions of Sturm-Liouville type. It should be noted that these operators are non-self-adjoint. The spectral structure of such operators has been insufficiently explored. In particular, a study of the completeness of systems of eigenfunctions and associated functions has begun relatively recently. In this paper, the completeness of the system of eigenfunctions and associated functions of one class of non-self-adjoint integral operators corresponding boundary value problems for fractional differential equations is established. The proof is based on the well-known Theorem of M.S. Livshits on the spectral decomposition of linear non-self-adjoint operators, as well as on the sectoriality of the fractional differentiation operator. The results of Dzhrbashian-Nersesian on the asymptotics of the zeros of the Mittag-Leffler function are used.

Existence and iteration of positive solution for fractional integral boundary value problems with p-Laplacian operator

This paper is concerned with an integral boundary value problem of fractional differential equations with p-Laplacian operator. Sufficient conditions ensuring the existence of extremal solutions for the given problem are obtained. Our results are based on the method of upper and lower solutions and monotone iterative technique.

Positive solutions for a boundary value problem of fractional differential equation with p-Laplacian operator

In this paper, the existence of a positive solution to a boundary value problem of fractional differential equations with the p-Laplacian operator is studied. By applying a monotone iterative method, some existence results of positive solutions are obtained. In addition, an example is included to illustrate the main results.

Unbounded solutions to abstract boundary value problems of fractional differential equations on a half line

In this paper, we investigate the existence of solutions to abstract boundary value problems of fractional differential equations on . The choice of the Banach space allows the solutions to be unbounded. Our approach mainly depends on the technique of Hausdorff measure of noncompactness in conjunction with the criterion of compactness in . Finally, an example in the Banach sequence space ℓ1 is given to illustrate our results.

Variational approach to fractional Dirichlet problem with instantaneous and non-instantaneous impulses

In this paper, we study the existence of solutions to the Dirichlet boundary value problem for fractional differential equations with instantaneous and non-instantaneous impulses, establish the existence result based on the variational method, and provide an example that illustrates the main result.

A class of nonlocal problems of fractional differential equations with composition of derivative and parameters

In this paper, we study existence and nonexistence of positive solutions for a class of Riemann–Stieltjes integral boundary value problems of fractional differential equations with parameters. By using the fixed point index theory, some new sufficient conditions for the existence of at least one, two and the nonexistence of positive solutions are obtained. The results we obtain show the influence of parameter λ and parameter a on the existence of positive solutions. Finally, some examples are given to illustrate our main results.

Impulsive Problems for Fractional Differential Equations with Functional Boundary Value Conditions at Resonance

We establish novel results on the existence of impulsive problems for fractional differential equations with functional boundary value conditions at resonance with dim⁡Ker L=1. Our results are based on the degree theory due to Mawhin, which requires appropriate Banach spaces and suitable projection schemes.

Investigating a Class of Nonlinear Fractional Differential Equations and Its Hyers-Ulam Stability by Means of Topological Degree Theory

The aim of this article is to seek some adequate conditions via a prior estimate method (topological degree method) to derive the existence of solution to a nonlinear boundary value problem of fractional differential equations (FDEs). With the help of topological degree method which has been applied in many articles, we establish the required results for existence and uniqueness of solution to a class of FDEs. Moreover, we also formulate sufficient conditions for Hyers-Ulam stability to the solution of the considered problem. Finally, an appropriate example is provided to justify the relevant results.

Monotone Iterative Technique and Ulam-Hyers Stability Analysis for Nonlinear Fractional Order Differential Equations with Integral Boundary Value Conditions

In this manuscript, the monotone iterative scheme has been extended to the nature of solution to boundary value problem of fractional differential equation that consist integral boundary conditions. In this concern, some sufficient conditions are developed in this manuscript. On the base of sufficient conditions, the monotone iterative scheme combined with lower and upper solution method for the existence, uniqueness, error estimates and various view plots of the extremal solutions to boundary value problem of nonlinear fractional differential equations have been studied. The obtain results have clarified the nature of the extremal solutions. Further, the Ulam--Hyers and Ulam--Hyers--Rassias stability have been investigated for the considered problem. Two illustrative examples of the BVP of the nonlinear fractional differential equations have been provided to justify our contribution.

Monotone Iterative Technique for Periodic Boundary Value Problem of Fractional Differential Equation in Banach Spaces

Abstract In this paper, we are concerned with the periodic boundary value problem of fractional differential equations on ordered Banach spaces. By introducing a concept of upper and lower solutions, we construct a new monotone iterative technique for the periodic boundary value problems of fractional differential equation, and obtain the existence of solutions between lower and upper solutions.

Anti-periodic boundary value problems with Riesz\u2013Caputo derivative

This paper is concerned with a class of anti-periodic boundary value problems for fractional differential equations with the Riesz–Caputo derivative, which can reflected both the past and the future nonlocal memory effects. By means of new fractional Gronwall inequalities and some fixed point theorems, we obtain some existence results of solutions under the Lipschitz condition, the sublinear growth condition, the nonlinear growth condition and the comparison condition. Three examples are given to illustrate the results.

Unique Existence Result of Approximate Solution to Initial Value Problem for Fractional Differential Equation of Variable Order Involving the Derivative Arguments on the Half-Axis

The semigroup properties of the Riemann–Liouville fractional integral have played a key role in dealing with the existence of solutions to differential equations of fractional order. Based on some results of some experts’, we know that the Riemann–Liouville variable order fractional integral does not have semigroup property, thus the transform between the variable order fractional integral and derivative is not clear. These judgments bring us extreme difficulties in considering the existence of solutions of variable order fractional differential equations. In this work, we will introduce the concept of approximate solution to an initial value problem for differential equations of variable order involving the derivative argument on half-axis. Then, by our discussion and analysis, we investigate the unique existence of approximate solution to this initial value problem for differential equation of variable order involving the derivative argument on half-axis. Finally, we give examples to illustrate our results.

Stability theory and existence of solution to a multi-point boundary value problem of fractional differential equations

The aims and objectives of this manuscript are concerned with the investigation of some appropriate conditions to establish existence theory of solutions to a class of nonlinear four-point boundary value problem (BVP) corresponding to fractional order differential equations (FODEs) provided as cDωy(t)=Ft,y(t),cDω-1y(t),1<ω≤2,t∈J=[0,1],y(0)=ζy(α),y(1)=ξy(β),ξ,ζ,α,β∈(0,1),\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} \\left\\{ \\begin{aligned} ^{c}&{\\mathscr {D}}^{\\omega }y(t)={\\mathcal {F}}\\left(t, y(t), {^{c}{\\mathscr {D}}^{\\omega -1}y(t)}\\right),1<\\omega \\le 2,\\,\\, t\\in {\\mathbf{J }}=[0, 1],\\\\&y(0)=\\zeta y(\\alpha ), \\,\\, y(1)=\\xi y(\\beta ),\\,\\xi ,\\ \\zeta ,\\ \\alpha ,\\,\\beta \\in (0,1), \\end{aligned}\\right. \\end{aligned}$$\\end{document}where {^{c}{mathscr {D}}^{omega }} is Caputo’s fractional derivative of order q and {mathcal {F}}in ({mathbf {J}}times {mathbf {R}} times {mathbf {R}} ,{mathbf {R}}) may be nonlinear. The required conditions are obtained by using classical results of functional analysis and fixed point theory. Further, we establish some adequate conditions for the Ulam–Hyers stability and generalized Ulam–Hyers stability for the solutions to the considered BVP of nonlinear FODEs. We include a proper problem to illustrate our established results.

Hybrid approximations for fractional calculus

In this paper, a numerical method for solving the fractional differential equations is presented. The method is based upon hybrid functions approximation. The properties of hybrid functions consisting ofblock-pulse functions and Taylor polynomials are presented. The Riemann-Liouville fractional integral operator for hybrid functions is given. This operator is then utilized to reduce the solution of the initial value problems for fractional differential equations to a system of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the technique.

Initial and Boundary Value Problems for Fractional Differential Equations Involving Atangana-Baleanu Derivative

In the present work, an initial value problem involving the Atangana-Baleanu derivative is considered. An explicit solution of the given problem in integral form is obtained by using the Laplace transform. The use of the given initial value problem is illustrated by considering a boundary value problem in which the solution is expressed in the form of a series expansion using an orthogonal basis obtained by separation of variables. Some examples are also given to illustrate the obtained results.

Eigenvalue problems for fractional differential equations with mixed derivatives and generalized p-Laplacian

This paper reports the investigation of eigenvalue problems for two classes of nonlinear fractional differential equations with generalized p-Laplacian operator involving both Riemann–Liouville fractional derivatives and Caputo fractional derivatives. By means of fixed point theorem on cones, some sufficient conditions are derived for the existence, multiplicity and nonexistence of positive solutions to the boundary value problems. Finally, an example is presented to further verify the correctness of the main theoretical results and illustrate the wide range of their potential applications.

Infinitely many solutions to boundary value problem for fractional differential equations

Abstract Variational methods and critical point theorems are used to discuss existence of infinitely many solutions to boundary value problem for fractional order differential equations where Riemann-Liouville fractional derivatives and Caputo fractional derivatives are used. An example is given to illustrate our result.

Mathematical Modeling of Solutes Migration under the Conditions of Groundwater Filtration by the Model with the k-Caputo Fractional Derivative

Within the framework of a new mathematical model of convective diffusion with the k-Caputo derivative, we simulate the dynamics of anomalous soluble substances migration under the conditions of two-dimensional steady-state plane-vertical filtration with a free surface. As a corresponding filtration scheme, we consider the scheme for the spread of pollution from rivers, canals, or storages of industrial wastes. On the base of a locally one-dimensional finite-difference scheme, we develop a numerical method for obtaining solutions of boundary value problem for fractional differential equation with k-Caputo derivative with respect to the time variable that describes the convective diffusion of salt solution. The results of numerical experiments on modeling the dynamics of the considered process are presented. The results that show an existence of a time lag in the process of diffusion field formation are presented.

Bayesian approach to a nonlinear inverse problem for a time-space fractional diffusion equation

Inverse problems for fractional differential equations have become a promising research area because of their wide applications in many scientific and engineering fields. In particular, the correct orders of fractional derivatives are hard to know as they are usually determined by experimental data and contain non-negligible uncertainty. Therefore, research on inverse problems involving the orders is necessary. Furthermore, problems involving the inversion of fractional orders are essentially nonlinear. Since classical methods may find it hard to provide satisfactory approximations and fail to capture the relevant uncertainty, a natural way to solve such inverse problems is through a Bayesian approach. In this paper, we consider an inverse problem of simultaneously recovering the source function and the orders of both time and space fractional derivatives for a time-space fractional diffusion equation. The problem will be formulated in the Bayesian framework, where the solution is the posterior distribution incorporating the prior information about the unknown and the noisy data. Under the considered infinite-dimensional function space setting, we prove that the corresponding Bayesian inverse problem is well-defined based on a proof of the continuity of the forward mapping. In addition, we also prove that the posterior distribution depends continuously on the data with respect to the Hellinger distance. Moreover, we adopt the iterative regularizing ensemble Kalman method to provide a numerical implementation of the considered inverse problem for the one-dimensional case. The numerical results shed light on the viability and efficiency of the method.