Higher-order Fractional Differential Equations Research Articles

Positive Solutions for a Higher-Order Semipositone Nonlocal Fractional Differential Equation with Singularities on Both Time and Space Variable

In this paper, we consider the following higher-order semipositone nonlocal Riemann-Liouville fractional differential equation D0+αx(t)+f(t,x(t),D0+βx(t))+e(t)=0, 0<t<1,D0+βx(0)=D0+β+1x(0)=⋯=D0+n+β-2x(0)=0, and D0+βx(1)=∑i=1m-2ηiD0+βx(ξi), where D0+α and D0+β are the standard Riemann-Liouville fractional derivatives. The existence results of positive solution are given by Guo-krasnosel’skii fixed point theorem and Schauder’s fixed point theorem.

Positive solutions of higher order fractional integral boundary value problem with a parameter

In this paper, we study a higher-order fractional differential equation with integral boundary conditions and a parameter. Under different conditions of nonlinearity, existence and nonexistence results for positive solutions are derived in terms of different intervals of parameter. Our approach relies on the Guo–Krasnoselskii fixed point theorem on cones.

Some Existence Results For Higher Order Fractional Differential Equations with Multi-Strip Riemann-Liouville Type Fractional Integral Boundary Conditions

Some Existence Results For Higher Order Fractional Differential Equations with Multi-Strip Riemann-Liouville Type Fractional Integral Boundary Conditions

Shehu Transform and Applications to Caputo-Fractional Differential Equations

In this manuscript we establish the expressions of the Shehu transform for fractional Riemann-Liouville and Caputo operators. With the help of this new integral transform we solve higher order fractional differential equations in the Caputo sense. Three illustrative examples are discussed to show our approach.

New unique existence criteria for higher-order nonlinear singular fractional differential equations

In this paper, a nonlinear three-point boundary value problem of higher-order singular fractional differential equations is discussed. By applying the properties of Green function and some fixed point theorems for sum-type operator on cone, some new criteria on the existence and uniqueness of solutions are obtained. Moreover, two iterative sequences are given for uniformly approximating the positive solution, which are important for practical application. At last, we give two examples to illustrate the main results.

Existence of solutions for several higher-order Hadamard-type fractional differential equations with integral boundary conditions on infinite interval

In this paper, we investigate the existence of solutions for several higher-order integral boundary value problems of Hadamard-type fractional differential equations on an infinite interval by using the monotone iterative technique and Mawhin’s continuation theorem. The results enrich and extend some known conclusions of Hadamard-type fractional boundary value problems. Moreover, we give two concrete examples to illustrate the theoretical results.

Uniqueness of Successive Positive Solution for Nonlocal Singular Higher-Order Fractional Differential Equations Involving Arbitrary Derivatives

In this article, by means of fixed point theorem on mixed monotone operator, we establish the uniqueness of positive solution for some nonlocal singular higher-order fractional differential equations involving arbitrary derivatives. We also give iterative schemes for approximating this unique positive solution.

Hermite collocation method for fractional order differential equations

This paper focuses on the approximate solutions of the higher order fractional differential equations with multi terms by the help of Hermite Collocation method (HCM). This new method is an adaptation of Taylor's collocation method in terms of truncated Hermite Series. With this method, the differential equation is transformed into an algebraic equation and the unknowns of the equation are the coefficients of the Hermite series solution of the problem. This method appears as a useful tool for solving fractional differential equations with variable coefficients. To show the pertinent feature of the proposed method, we test the accuracy of the method with some illustrative examples and check the error bounds for numerical calculations.

Iterative unique positive solutions for singular p-Laplacian fractional differential equation system with several parameters

By using the method of reducing the order of a derivative, the higher-order fractional differential equation is transformed into the lower-order fractional differential equation and combined with the mixed monotone operator, a unique positive solution is obtained in this paper for a singular p-Laplacian boundary value system with the Riemann–Stieltjes integral boundary conditions. This equation system is very wide because there are many parameters, which can be changeable in the equation system in this paper, and the nonlinearity is allowed to be singular in regard to not only the time variable but also the space variable. Moreover, the unique positive solution that we obtained in this paper is dependent on λ, and an iterative sequence and convergence rate are given, which are important for practical application. An example is given to demonstrate the application of our main results.

Existence and Uniqueness of Solutions for a Coupled System of Higher Order Fractional Differential Equations with Integral Boundary Conditions

Existence and Uniqueness of Solutions for a Coupled System of Higher Order Fractional Differential Equations with Integral Boundary Conditions

Solving Higher-Order Fractional Differential Equations by Reproducing Kernel Hilbert Space Method

Solving Higher-Order Fractional Differential Equations by Reproducing Kernel Hilbert Space Method

Existence of positive solutions for a high order fractional differential equation integral boundary value problem with changing sign nonlinearity

In this paper, we investigate the existence of positive solutions for a class of high order fractional differential equation integral boundary value problems with changing sign nonlinearity. By applying cone expansion and cone compression fixed point theorem, we have obtained and proved theorems related to the existence of positive solutions, which highlight the influences of the parameters in different ranges on the existence of positive solutions. Finally, we also give some examples to illustrate our main results.

A Lyapunov‐type inequality with the Katugampola fractional derivative

In this work we consider the higher order fractional differential equation with derivative defined in the sense of Katugampola. We present some equivalent integral form of the considered boundary value problem and using properties of an appropriate Green function and prove fractional counterpart of the Lyapunov inequality.

Triple positive solutions for nonlocal fractional differential equations with singularities both on time and space variables

In this paper, different height functions of the nonlinear term on special bounded sets together with Leggett–Williams and Krasnosel’skii fixed point theorems are employed to establish the existence of triple positive solutions for a class of higher-order fractional differential equations with integral conditions. The singularities are with respect not only to the time but also to the space variables.

Uniqueness of solution for higher-order fractional differential equations with conjugate type integral conditions

Abstract In this article, we prove the existence and uniqueness of solution for some higher-order fractional differential equations with conjugate type integral conditions. The interesting point lies in that the Lipschitz constant is closely associated with the first eigenvalues corresponding to the relevant linear operator. The discussion is based on the Banach contraction map principle and the theory of u 0-positive linear operator.

Lyapunov-type inequalities for certain higher order fractional differential equations

Lyapunov-type inequalities for certain higher order fractional differential equations

On approximate solutions for two higher-order Caputo-Fabrizio fractional integro-differential equations

We investigate the existence of solutions for two high-order fractional differential equations including the Caputo-Fabrizio derivative. In this way, we introduce some new tools for obtaining solutions for the high-order equations. Also, we discuss two illustrative examples to confirm the reported results. In this way one gets the possibility of utilizing some continuous or discontinuous mappings as coefficients in the fractional differential equations of higher order.

Lyapunov-type Inequalities for Certain Higher Order Fractional Differential Equations

Lyapunov-type Inequalities for Certain Higher Order Fractional Differential Equations

Nonlocal semi-linear fractional-order boundary value problems with strip conditions

This paper is concerned with the question of existence of solutions for one-dimensional higher-order semi-linear fractional differential equations supplemented with nonlocal strip type boundary conditions. The nonlocal strip condition addresses a situation where the linear combination of the values of unknown function at two nonlocal points, located to the left and right hand sides of the strip, respectively, is proportional to its strip value. The case of Stieltjes type strip condition is also discussed. Our results, relying on some standard fixed point theorems are supported with illustrative examples.

Existence of Solutions of Impulsive Lidstone BVPs of Singular Higher Order Fractional Differential Equations

We present a new general method for converting an impulsive fractional differential equation to an equivalent integral equation. By using this method and employing a fixed point theorem in Banach space, we establish existence results of solutions for two boundary value problems of impulsive singular higher order fractional differential equation. An example is presented to illustrate the efficiency of the results obtained.