Higher-order Fractional Differential Equations Research Articles

Using the iterative reproducing kernel method for solving a class of nonlinear fractional differential equations

ABSTRACTIn previous works, we have devoted to using the reproducing kernel methods solving integer order differential equations, based on the review of previous works, in this paper, we mainly present a method for solving a class of higher order fractional differential equations with general boundary value problems by using Taylor formula into reproducing kernel space. Its analytical solution is represented in the form of series. The analytical solution and approximate solution obtained by this method is given and it is uniformly converge to the exact solution and its corresponding derivatives. The numerical examples are studied to demonstrate the accuracy of the present method.

Asymptotic Behavior of Solutions of Nonlinear Fractional Differential Equations with Caputo-Type Hadamard Derivatives

AbstractIn this paper, the authors study the asymptotic behavior of solutions of higher order fractional differential equations with Caputo-type Hadamard derivatives of the formwhere

A survey and new investigation on $(n,n-k)$-type boundary value problems for higher order impulsive fractional differential equations

A survey for studies on boundary value problems of higher order ordinary differential equations is given firstly. Secondly a simple review for studies on solvability of boundary value problems for impulsive fractional differential equations is presented. Thirdly by using a general method for converting an impulsive fractional differential equation with the Riemann-Liouville fractional derivatives to an equivalent integral equation and employing fixed point theorems in Banach space, we establish existence results of solutions for three classes of boundary value problems ($(n,n-k)$ type BVPs) of impulsive higher order fractional differential equations. Some examples are presented to illustrate the efficiency of the results obtained and some mistakes are also corrected at the end of the paper finally. A conclusion section is given at the end of the paper.

Lyapunov-type inequalities for a higher order fractional differential equation with fractional integral boundary conditions

The research of J. J. Nieto was partially supported by the Ministerio de Economia y Competitividad of Spain under grant MTM2013-43014-P, co-financed by the European Community fund FEDER, and XUNTA de Galicia under grant GRC2015-004. The third author extends his appreciation to Distinguished Scientist Fellowship Program (DSFP) at King Saud University (Saudi Arabia)

Existence and non-existence of positive solutions for integral boundary value problems of high-order fractional differential equations with generalised p-Laplacian operator

In this paper, we investigate the non-linear integral boundary value problem of fractional differential equations with generalised p-Laplacian operator. By means of the monotone iteration method, we obtain the existence of positive solutions and establish the iterative sequence for approximating the solutions. Moreover, the non-existence of positive solution is also considered. An example is given to show the applicability of our main results.

Existence and non-existence of positive solutions for integral boundary value problems of high-order fractional differential equations with generalised p-Laplacian operator

In this paper, we investigate the non-linear integral boundary value problem of fractional differential equations with generalised p-Laplacian operator. By means of the monotone iteration method, we obtain the existence of positive solutions and establish the iterative sequence for approximating the solutions. Moreover, the non-existence of positive solution is also considered. An example is given to show the applicability of our main results.

Existence of solutions for a class of nonlinear higher-order fractional differential equation with fractional nonlocal boundary condition

In this paper, we study the existence of solutions for a class of nonlinear higher-order fractional differential equation with fractional nonlocal boundary condition by using the monotone iterative technique based on the method of upper and lower solutions and give a specific iterative equation about its solutions.

Solvability of impulsive periodic boundary value problems for higher order fractional differential equations

A class of periodic boundary value problems for higher order fractional differential equations with impulse effects is considered. We first convert the problem to an equivalent integral equation. Then, using a fixed-point theorem in Banach space, we establish existence results of solutions for this kind of boundary value problem for impulsive singular higher order fractional differential equations. Two examples are presented to illustrate the efficiency of the results obtained.

Positive solutions of a system for nonlinear singular higher-order fractional differential equations with fractional multi-point boundary conditions

This paper deals with the existence and multiplicity of positive solutions for a system of nonlinear singular higher-order fractional differential equations with fractional multi-point boundary conditions. The main tool used in the proof is fixed point index theory. Some limit type conditions for ensuring the existence of positive solutions are given, and our conditions are suitable for more general functions.

Existence of positive solutions for singular higher-order fractional differential equations with infinite-point boundary conditions

In this paper, we consider a class of singular fractional differential equations with infinite-point boundary conditions. The fractional orders are involved in the nonlinearity of the boundary value problem, and the nonlinearity is allowed to be singular with respect to not only the time variable but also to the space variable. Firstly, we give Green’s function and establish its properties. Then, we utilize the sequential technique and regularization to investigate the existence of positive solutions. Finally, we give an example of application of our result.

Solvability of impulsive $$(n,n-p)$$ ( n , n - p ) boundary value problems for higher order fractional differential equations

We present a new general method for converting an impulsive fractional differential equation to an equivalent integral equation. Using this method and employing a fixed point theorem in Banach space, we establish existence results of solutions for a boundary value problem of impulsive singular higher order fractional differential equation. An example is presented to illustrate the efficiency of the results obtained. A conclusion section is given at the end of the paper.

Existence of positive solutions for singular higher-order fractional differential equation via spectral analysis

In this paper, the existence of positive solutions for a class of fractional differential equations with Riemann–Stieltjes integral boundary conditions are investigated under some weak conditions concerning the spectral analysis of the relevant linear operator and Gelfand’s formula by means of the fixed point index theorem in cones. The nonlinearity permits singularities not only at \(t=0,1,\) but also at \(x_{i}=0\ (i=1,2,\ldots ,n-1)\). The results obtained herein generalize and improve some known results including singular and non-singular cases.

An efficient technique for higher order fractional differential equation.

In this study, we establish exact solutions of fractional Kawahara equation by using the idea of exp ,left( { - varphi left( eta right)} right)-expansion method. The results of different studies show that the method is very effective and can be used as an alternative for finding exact solutions of nonlinear evolution equations (NLEEs) in mathematical physics. The solitary wave solutions are expressed by the hyperbolic, trigonometric, exponential and rational functions. Graphical representations along with the numerical data reinforce the efficacy of the used procedure. The specified idea is very effective, expedient for fractional PDEs, and could be extended to other physical problems.

Multiple monotone positive solutions of integral BVPs for a higher-order fractional differential equation with monotone homomorphism

This paper is concerned with the integral boundary value problems of higher-order fractional differential equation with. In the sense of a monotone homomorphism, some sufficient criteria are established to guarantee the existence of at least two monotone positive solutions by employing the fixed point theorem of cone expansion and compression of functional type proposed by Avery, Henderson and O’Regan. As applications, some examples are provided to illustrate the validity of our main results.

Positive solution for higher-order singular infinite-point fractional differential equation with p-Laplacian

By means of the method of upper and lower solutions together with the Schauder fixed point theorem, the conditions for the existence of at least one positive solution are established for some higher-order singular infinite-point fractional differential equation with p-Laplacian. The nonlinear term may be singular with respect to both the time and the space variables.

Successively iterative method for a class of high-order fractional differential equations with multi-point boundary value conditions on half-line

In this paper, we study the existence of the positive solutions for a class of high-order differential equations with multi-point boundary value conditions involving the Caputo fractional derivative on infinite interval. Moreover, we develop two computable explicit monotone iterative sequences for approximating the two minimal and maximal positive solutions. An example is given to show the applicability of our main results.

The hight order Lane-Emden fractional differential system: Existence, uniqueness and Ulam type stabilities

In this paper, by considering a more general Lane-Emden system of high order fractional differential equations with two arbitrary orders in each equation, we obtain some results on the existence and uniqueness of solutions using some fixed point theorems. Furthermore, we define and study some types of Ulam stability. Some examples are presented to illustrate the main results.

Impulsive integral boundary value problems of the higher-order fractional differential equation with eigenvalue arguments

We consider a class of higher-order nonlinear Riemann-Liouville fractional differential equation with Riemann-Stieltjes integral boundary value conditions and impulses as follows: $$ \left \{\textstyle\begin{array}{@{}l} -D_{0^{+}}^{\alpha}u(t)=\lambda a(t)f(t,u(t)),\quad t\in(0,1)\setminus\{ t_{k}\}_{k=1}^{m}, \Delta u(t_{k})=I_{k}(u(t_{k})), \quad t=t_{k}, u(0)=u'(0)=\cdots=u^{(n-2)}(0),\qquad u'(1)=\int_{0}^{1}u(s)\,dH(s). \end{array}\displaystyle \right . $$ By converting the boundary value problem into an equivalent integral equation and applying the Schauder fixed-point theorem, fixed-point index theorem, we have established sufficient conditions for the existence and multiplicity of positive solutions. The eigenvalue intervals are also given. Some examples are presented to illustrate the validity of our main results.

Positive solutions of fractional differential equation nonlocal boundary value problems

In this paper, we study the existence and uniqueness of positive solutions for a class of higher-order nonlocal fractional differential equations with Riemann-Stieltjes integral boundary conditions. We firstly convert the problem to an equivalent integral equation, and then by applying a fixed point theorem of a sum operator, the existence and uniqueness of positive solutions is established. Furthermore, an iterative scheme to approximate the solution is constructed and an example is given to illuminate the application of the main results.

A Numerical Method to Solve Higher-Order Fractional Differential Equations

In this paper, we present a new numerical method to solve fractional differential equations. Given a fractional derivative of arbitrary real order, we present an approximation formula for the fractional operator that involves integer-order derivatives only. With this, we can rewrite FDEs in terms of a classical one and then apply any known technique. With some examples, we show the accuracy of the method.