Nonlinear Fractional Boundary Value Problem Research Articles

A Spectral Collocation Method for Nonlinear Fractional Boundary Value Problems with a Caputo Derivative

In this paper, we consider the nonlinear boundary value problems involving the Caputo fractional derivatives of order \(\alpha \in (1,2)\) on the interval (0, T). We present a Legendre spectral collocation method for the Caputo fractional boundary value problems. We derive the error bounds of the Legendre collocation method under the \(L^2\)- and \(L^\infty \)-norms. Numerical experiments are included to illustrate the theoretical results.

A New Formulation of the Fractional Optimal Control Problems Involving Mittag–Leffler Nonsingular Kernel

The aim of this paper is to propose a new formulation of the fractional optimal control problems involving Mittag–Leffler nonsingular kernel. By using the Lagrange multiplier within the calculus of variations and by applying the fractional integration by parts, the necessary optimality conditions are derived in terms of a nonlinear two-point fractional boundary value problem. Based on the convolution formula and generalized discrete Gronwall’s inequality, the numerical scheme for solving this problem is developed and its convergence is proved. Numerical simulations and comparative results show that the suggested technique is efficient and provides satisfactory results.

Unique Solutions for Fractional q-Difference Boundary Value Problems Via a Fixed Point Method

In this paper, by applying the cone theory in ordered Banach spaces associated with the characters of increasing $$\varphi -(h,e)$$ -concave operators, we investigate the existence and uniqueness of nontrivial solutions for a nonlinear fractional q-difference equation boundary value problem. The main results show that we can construct an iterative scheme approximating the unique nontrivial solution. Relying on an example, we show the efficiency and applicability of the main result.

Positive Solutions in the Space of Lipschitz Functions for Fractional Boundary Value Problems with Integral Boundary Conditions

In this paper, we study the existence of positive solutions for a class of nonlinear fractional boundary value problems with integral boundary conditions in Holder spaces. Our analysis relies on a sufficient condition for the relative compactness in Holder spaces and the classical Schauder fixed point theorem.

Classifying First Extremal Points for a Fractional Boundary Value Problem with a Fractional Boundary Condition

Let $$n\in \mathbb {N}$$ , $$n\ge 2$$ , $$\beta >0$$ fixed, and $$0<b\le \beta $$ . For $$n-1<\alpha \le n$$ , we look to classify extremal points for the fractional differential equation $$D_{0^+}^{\alpha }u+p(t) u=0$$ , satisfying the boundary conditions $$u^{(i)}(0)=0$$ , $$i=0,\ldots ,n-2$$ , $$D_{0^+}^\gamma u(b)=0$$ , where p(t) is a continuous nonnegative function on $$[0,\beta ]$$ which does not vanish identically on any nondegenerate compact subinterval of $$[0,\beta ]$$ . Using the theory of Krein and Rutman, first extremal points of this boundary value problem are classified. As an application, the results are applied, along with a fixed-point theorem, to show the existence of a solution of a nonlinear fractional boundary value problem.

On the uniqueness of solutions for a class of fractional differential equations

We are concerned with the uniqueness of solutions for the following nonlinear fractional boundary value problem: D p x ( t ) + f ( t , x ( t ) ) = 0 , 2 < p ≤ 3 , t ∈ ( 0 , 1 ) , x ( 0 ) = x ′ ( 0 ) = 0 , x ( 1 ) = 0 where D 0 + p denotes the standard Riemann–Liouville fractional derivative. Our analysis relies on the theory of linear operators and the ‖ ⋅ ‖ e norm.

New Modified Adomian Decomposition Recursion Schemes for Solving Certain Types of Nonlinear Fractional Two-Point Boundary Value Problems

We apply new modified recursion schemes obtained by the Adomian decomposition method (ADM) to analytically solve specific types of two-point boundary value problems for nonlinear fractional order ordinary and partial differential equations. The new modified recursion schemes, which sometimes utilize the technique of Duan’s convergence parameter, are derived using the Duan-Rach modified ADM. The Duan-Rach modified ADM employs all of the given boundary conditions to compute the remaining unknown constants of integration, which are then embedded in the integral solution form before constructing recursion schemes for the solution components. New modified recursion schemes obtained by the method are generated in order to analytically solve nonlinear fractional order boundary value problems with a variety of two-point boundary conditions such as Robin and separated boundary conditions. Some numerical examples of such problems are demonstrated graphically. In addition, the maximal errors (MEn) or the error remainder functions (ERn(x)) of each problem are calculated.

Existence of three solutions for impulsive nonlinear fractional boundary value problems

In this work we present new criteria on the existence of three solutions for a class of impulsive nonlinear fractional boundary-value problems depending on two parameters. We use variational methods for smooth functionals defined on reflexive Banach spaces in order to achieve our results.

Nonlinear fractional boundary value problem with not instantaneous impulse

In this article, the main focus is to propose the solution for the nonlinear fractional boundary system with non-instantaneous impulse under some weak conditions. By applying well known classical fixed point theorems, we obtained the existence and uniqueness outcomes of the solution for the proposed problem. Moreover, an example is also discussed to explain the present work.

Multiple Solutions for a Nonlinear Fractional Boundary Value Problem via Critical Point Theory

This paper is concerned with the existence of multiple solutions for the following nonlinear fractional boundary value problem: DT-αaxD0+αux=fx,ux, x∈0,T, u0=uT=0, where α∈1/2,1, ax∈L∞0,T with a0=ess infx∈0,Tax&gt;0, DT-α and D0+α stand for the left and right Riemann-Liouville fractional derivatives of order α, respectively, and f:0,T×R→R is continuous. The existence of infinitely many nontrivial high or small energy solutions is obtained by using variant fountain theorems.

Solvability for a system of nonlinear fractional higher-order three-point boundary value problem

Solvability for a system of nonlinear fractional higher-order three-point boundary value problem

Fixed points of mixed monotone operators for existence and uniqueness of nonlinear fractional differential equations

In this paper we study the existence and uniqueness of positive solutions for nonlinear fractional differential equation boundary value problems by using new fixed point results of mixed monotone operators on cones.

Solvability for a system of nonlinear fractional higher-order three-point boundary value problem

Solvability for a system of nonlinear fractional higher-order three-point boundary value problem

Solvability for nonlinear fractional q-difference equations with nonlocal conditions

In this paper, we study a class of nonlinear fractional q-difference equations boundary value problem. We obtain the existence and uniqueness of positive solutions for this problem by Banach's contraction mapping principle and Krasnoselskii's fixed point theorem on cone. Lastly, we give two examples to illuminate the use of the main results.

New Fixed Point Theorems for Mixed Monotone Operators with Perturbation and Applications

By using the properties of cone and the fixed point theorem for mixed monotone operators in ordered Banach spaces, we investigate the mixed monotone operators of a new type with perturbation. We establish some sufficient conditions for such operators to have a new existence and uniqueness fixed point and provide monotone iterative techniques which give sequences convergent to the fixed point. Finally, as applications, we apple the results obtained in this paper to study the existence and uniqueness of positive solutions for nonlinear fractional differential equation boundary value problems.

Smoothing transformation and spline collocation for nonlinear fractional initial and boundary value problems

We construct and justify a class of high order methods for the numerical solution of initial and boundary value problems for nonlinear fractional differential equations of the form (D∗αy)(t)=f(t,y(t)) with Caputo type fractional derivatives D∗αy of order α>0. Using an integral equation reformulation of the underlying problem we first regularize the solution by a suitable smoothing transformation. After that we solve the transformed equation by a piecewise polynomial collocation method on a mildly graded or uniform grid. Optimal global convergence estimates are derived and a superconvergence result for a special choice of collocation parameters is established. To illustrate the reliability of the proposed algorithms two numerical examples are given.

Solvability of a boundary value problem at resonance.

This paper concerns the solvability of a nonlinear fractional boundary value problem at resonance. By using fixed point theorems we prove that the perturbed problem has a solution, then by some ideas from analysis we show that the original problem is solvable. An example is given to illustrate the obatined results.

Infinitely many solutions for nonlinear fractional boundary value problems via variational methods

In this paper, the author considers the following nonlinear fractional boundary value problem: $$\left \{ \textstyle\begin{array}{l} \frac{d}{dt} ( {\frac{1}{2}{ }_{0}D_{t}^{-\beta} ({u}'(t))+\frac{1}{2}{ }_{t}D_{T}^{-\beta} ({u}'(t))} )+\nabla F(t,u(t))=0, \quad\mbox{a.e. } t\in[0,T], u(0)=u(T)=0, \end{array}\displaystyle \right . $$ where ${ }_{0}D_{t}^{-\beta} $ and ${ }_{t}D_{T}^{-\beta} $ are the left and right Riemann-Liouville fractional integrals of order $0\le\beta<1$ , respectively, $\nabla F(t,x)$ is the gradient of F at x. By applying the variant fountain theorems, the author obtains the existence of infinitely many small or high energy solutions to the above boundary value problem.

Infinitely many solutions for impulsive nonlinear fractional boundary value problems

Based on variational methods and critical point theory, the existence of infinitely many classical solutions for impulsive nonlinear fractional boundary value problems is ensured.

A generalized Lyapunov’s inequality for a fractional boundary value problem

We prove existence of positive solutions to a nonlinear fractional boundary value problem. Then, under some mild assumptions on the nonlinear term, we obtain a smart generalization of Lyapunov’s inequality. The new results are illustrated through examples.