Nonlocal Fractional Differential Equations Research Articles

Positive solutions for a nonlocal fractional differential equation

In this paper, we study the following singular boundary value problem of a nonlocal fractional differential equation { D 0 + α u ( t ) + q ( t ) f ( t , u ( t ) ) = 0 , 0 < t < 1 , n − 1 < α ≤ n , u ( 0 ) = u ′ ( 0 ) = ⋯ = u ( n − 2 ) ( 0 ) = 0 , u ( 1 ) = ∫ 0 1 u ( s ) d A ( s ) , where α ≥ 2 , D 0 + α is the standard Riemann–Liouville derivative, ∫ 0 1 u ( s ) d A ( s ) is given by Riemann–Stieltjes integral with a signed measure, q may be singular at t = 0 and/or t = 1 , f ( t , x ) may also have singularity at x = 0 . The existence and multiplicity of positive solutions are obtained by means of the fixed point index theory in cones.

Positive Solutions of Nonlinear Eigenvalue Problems for a Nonlocal Fractional Differential Equation

By using the fixed point theorem, positive solutions of nonlinear eigenvalue problems for a nonlocal fractional differential equation are considered, where 1 &lt; α ≤ 2 is a real number, λ is a positive parameter, is the standard Riemann‐Liouville differentiation, and ξi ∈ (0,1), αi ∈ [0, ∞) with , a(t) ∈ C([0,1], [0, ∞)), f(t, u) ∈ C([0, ∞), [0, ∞)).

On integral solutions of some nonlocal fractional differential equations with nondense domain

In this paper, we prove the existence and uniqueness of integral solution for some nondensely defined fractional semilinear differential equation with nonlocal conditions. The results are obtained by means of fixed point methods. We also give some examples of such problems.

Vector Additive Decomposition for 2D Fractional Diffusion Equation

Such physical processes as the diffusion in the environments with fractal geometry and the particles’ subdiffusion lead to the initial value problems for the nonlocal fractional order partial differential equations. These equations are the generalization of the classical integer order differential equations. An analytical solution for fractional order differential equation with the constant coefficients is obtained in [1] by using Laplace-Fourier transform. However, nowadays many of the practical problems are described by the models with variable coefficients. In this paper we discuss the numerical vector decomposition model which is based on a shifted version of usual Gr¨unwald finite-difference approximation [2] for the non-local fractional order operators. We prove the unconditional stability of the method for the fractional diffusion equation with Dirichlet boundary conditions. Moreover, a numerical example using a finite difference algorithm for 2D fractional order partial differential equations is also presented and compared with the exact analytical solution.