A Sparse Kernel Approximate Method for Fractional Boundary Value Problems

In this paper,the weak pre-orthogonal adaptive Fourier decomposition(W-POAFD)method is applied to solve fractional boundary value problems(FBVPs)in the reproducing kernel Hilbert spaces(RKHSs)W_(0)^(4)[0,1] and W^(1)[0,1].The process of the W-POAFD is as follows:(i)choose a dictionary and implement the pre-orthogonalization to all the dictionary elements;(ii)select points in[0,1]by the weak maximal selection principle to determine the corresponding orthonormalized dictionary elements iteratively;(iii)express the analytical solution as a linear combination of these determined dictionary elements.Convergence properties of numerical solutions are also discussed.The numerical experiments are carried out to illustrate the accuracy and efficiency of W-POAFD for solving FBVPs.

Iterative Technology in a Singular Fractional Boundary Value Problem with q -Difference

In this paper, we apply the iterative technology to establish the existence of solutions for a fractional boundary value problem with q-difference. Explicit iterative sequences are given to approxinate the solutions and the error estimations are also given.

Existence of Positive Solutions to Boundary Value Problem for a Nonlinear Fractional Differential Equation

In this paper,we study a Dirichlet-type boundary value problem(BVP) of nonlinear fractional differential equation with an order α∈(3,4],where the fractional derivative D~α_(o^+)is the standard Riemann-Liouville fractional derivative.By constructing the Green function and investigating its properties,we obtain some criteria for the existence of one positive solution and two positive solutions for the above BVP.The Krasnosel'skii fixedpoint theorem in cones is used here.We also give an example to illustrate the applicability of our results.

EXISTENCE OF SOLUTION FOR BOUNDARY VALUE PROBLEM OF NONLINEAR FRACTIONAL DIFFERENTIAL EQUATION

This paper is concerned with the boundary value problem of a nonlinear fractional differential equation. By means of Schauder fixed-point theorem, an existence result of solution is obtained.

FRACTIONAL ORDER BOUNDARY VALUE PROBLEM WITH INTEGRAL BOUNDARY CONDITIONS INVOLVING PETTIS INTEGRAL

In this article, we investigate the existence of Pseudo solutions for some fractional order boundary value problem with integral boundary conditions in the Banach space of continuous function equipped with its weak topology. The class of such problems constitute a very interesting and important class of problems. They include two, three, multi-point and nonlocal boundary-value problems as special cases. In our investigation, the right hand side of the above problem is assumed to be Pettis integrable function. To encompass the full scope of this article, we give an example illustrating the main result.

EXISTENCE AND UNIQUENESS RESULTS FOR BOUNDARY VALUE PROBLEMS OF HIGHER ORDER FRACTIONAL INTEGRO-DIFFERENTIAL EQUATIONS INVOLVING GRONWALL'S INEQUALITY IN BANACH SPACES

We study boundary value problems for fractional integro-differential equations involving Caputo derivative of order α∈ （n-1, n） in Banach spaces. Existence and uniqueness results of solutions are established by virtue of the Holder＇s inequality, a suitable singular Cronwall＇s inequality and fixed point theorem via a priori estimate method. At last, examples are given to illustrate the results.

Existence of Positive Solutions of Three-point Boundary Value Problem for Higher Order Nonlinear Fractional Differential Equations

In this paper, we consider the positive solutions of fractional three-point boundary value problem of the form Dο^α＋u（t）＋f（t,u（t）,u＇（t）,…,u^（n-3）（5）,u^（n-2）（t））=0,u^（i）（0）=0,0≤i≤n-2,u^（n-2）（1）-βu^（n-2）（ξ）=0,where 0〈t〈1,n-1〈α≤n,n≥2,ξ Е（0,1）,βξ^a-n〈1. We first transform it into another equivalent boundary value problem. Then, we derive the Green＇s function for the equivalent boundary value problem and show that it satisfies certain properties. At last, by using some fixed-point theorems, we obtain the existence of positive solution for this problem. Example is given to illustrate the effectiveness of our result.

ON SOME BOUNDARY VALUE PROBLEMS FOR NONHOMOGENOUS POLYHARMONIC EQUATION WITH BOUNDARY OPERATORS OF FRACTIONAL ORDER

In the paper we study questions about solvability of some boundary value prob- lems for a non-homogenous poly-harmonic equation. As a boundary operator we consider differentiation operator of fractional order in Miller-Ross sense. The considered problem is a generalization of well-known Dirichlet and Neumann problems.

Solutions to Boundary Value Problem of Nonlinear Impulsive Differential Equation of Fractional Order

This paper is devoted to study the existence and uniqueness of solutions to a boundary value problem of nonlinear fractional differential equation with impulsive effects.The arguments are based upon Schauder and Banach fixed-point theorems. We improve and generalize the results presented in[B.Ahmad,S.Sivasundaram, Existence results for nonlinear impulsive hybrid boundary value problems involving fractional differential equations,Nonlinear Analysis：Hybrid Systems,3（2009）,251- 258].

Existence of Solutions for Infinity-Point Nonlinear Fractional Boundary Value Problem at Resonance

A class of the boundary value problem for fractional order nonlinear differential equation with Riemann-Liouville fractional derivative on the half line was studied. By using the coincidence degree theory due to Mawhin and constructing the suitable operators,the existence theorem of at least one solution has been established. An example is given to illustrate our result.

ON SOLVABILITY OF A BOUNDARY VALUE PROBLEM FOR A NONHOMOGENEOUS BIHARMONIC EQUATION WITH A BOUNDARY OPERATOR OF A FRACTIONAL ORDER

This paper is concerned with the solvability of a boundary value problem for a nonhomogeneous biharmonic equation. The boundary data is determined by a differential operator of fractional order in the Riemann-Liouville sense. The considered problem is a generalization of the known Dirichlet and Neumann problems.

Positive Solutions for Fractional Differential Equations with Multi-Point Boundary Value Problems

In this paper, a fractional multi-point boundary value problem is considered. By using the fixed point index theory and Krein-Rutman theorem, some results on existence are obtained.

Existence and Uniqueness for the Boundary Value Problems of Nonlinear Fractional Differential Equation

This paper studies the existence and uniqueness of solutions for a class of boundary value problems of nonlinear fractional order differential equations involving the Caputo fractional derivative by employing the Banach’s contraction principle and the Schauder’s fixed point theorem. In addition, an example is given to demonstrate the application of our main results.

Existence of Solutions for Boundary Value Problems of Conformable Fractional Differential Equations

In this paper, we study a class of boundary value problems for conformable fractional differential equations under a new definition. Firstly, by using the monotone iterative technique and the method of coupled upper and lower solution, the sufficient condition for the existence of the boundary value problem is obtained, and the range of the solution is determined. Then the existence and uniqueness of the solution are proved by the proof by contradiction. Finally, a concrete example is given to illustrate the wide applicability of our main results.

The Solution to Impulse Boundary Value Problem for a Class of Nonlinear Fractional Functional Differential Equations

In this paper, we investigate the existence of solution for a class of impulse boundary value problem of nonlinear fractional functional differential equation of mixed type. We obtain the existence results of solution by applying some well-known fixed point theorems. An example is given to illustrate the effectiveness of our result.

Existence of Boundary Value Problems for Impulsive Fractional Differential Equations with a Parameter

We investigate a class of boundary value problems for nonlinear impulsive fractional differential equations with a parameter.By the deduction of Altman’s theorem and Krasno-selskii’s fixed point theorem,the existence of this problem is proved.Examples are given to illustrate the effectiveness of our results.

ON SOLVABILITY OF A BOUNDARY VALUE PROBLEM FOR THE POISSON EQUATION WITH A NONLOCAL BOUNDARY OPERATOR

In this work, we investigate the solvability of the boundary value problem for the Poisson equation, involving a generalized Riemann-Liouville and the Caputo derivative of fractional order in the class of smooth functions. The considered problems are generalization of the known Dirichlet and Neumann oroblems with operators of a fractional order.

On a Boundary Value Problem for a Polynomial Pencil of the Sturm-Liouville Equation with Spectral Parameter in Boundary Conditions

The boundary value problem with a spectral parameter in the boundary conditions for a polynomial pencil of the Sturm-Liouville operator is investigated. Using the properties of the transformation operators for such operators, the asymptotic formulas for eigenvalues of the boundary value problem are obtained.

Nontrivial solutions for a class of fractional difference boundary value problems and fixed-point problems

In this work,we use the variant fountain theorem to study the existence of nontrivial solutions for the superquadratic fractional difference boundary value problem:{T△^(v)_(t-1)(t△^(v)_(v-1)x(t))=f(x(t+v-1)),t∈[0,T]N_(0),x(v-2)=[tΔ^(v)_(v-1)x(t)]_(t=T=0.The existence of nontrivial solutions is obtained in the case of super quadratic growth of the nonlinear term f by change of fountain theorem.

On the existence and uniqueness of a nonlinear q-difference boundary value problem of fractional order

In this research collection,we estimate the existence of the unique solution for the boundary value problem of nonlinear fractional q-difference equation having the given form c D^(ζ)_(q) v(t)−h(t,v(t))=0,0≤t≤1,α_(1)v(0)+β_(1)D_(q)v(0)=v(η1),α_(2)v(1)−β_(2)D_(q)v(1)=v(η2),where 1<ζ≤2,(η1,η2)∈(0,1)^(2),α_(i),β_(i)∈R(i=1,2),h∈C([0,1]×R,R)and c Dζq represents the Caputo-type nonclassical q-derivative of orderζ.We use well-known principal of Banach contraction,and Leray–Schauder nonlinear alternative to vindicate the unique solution existence of the given problem.Regarding the applications,some examples are solved to justify our outcomes.