Fractional Differential Problem Research Articles

Existence of Positive Solutions for a Class of Higher-Order Caputo Fractional Differential Equation

In this paper, the following nonlinear fractional ordinary differential boundary value problem $$\begin{aligned} \left\{ \begin{array}{ll} D_{0+}^{\alpha } u(t) +\lambda f(t,u(t))+ q(t)=0, &{} 0<t<1, \\ u(0) = u'(1) =u''(0)= \dots = u^{(n-1)}(0)=0, &{} \\ \end{array} \right. \end{aligned}$$ is considered, where $$\alpha (n-1 <\alpha \le n)$$ is a real number. $$\lambda > 0$$ is a parameter. $$D_{0+}^{\alpha }$$ is the standard Caputo differentiation. Some sufficient conditions for the existence of positive solutions to this boundary value problem of nonlinear fractional differential equation are established by nonlinear alternative of Leray–Schauder type and Guo–Krasnoselskii fixed point theorem on cones. As applications, some examples are provided to illustrate our main results.

Fractional problems with advanced arguments

This paper concerns boundary fractional differential problems with advanced arguments. We investigate the existence of initial value problems when the initial point is given at the end point of an interval. Nonhomogeneous linear fractional differential equations are also studied. The existence of solutions for fractional differential equations with advanced arguments and with boundary value problems has been investigated by using a monotone iterative technique. We also discuss corresponding fractional inequalities. Several examples illustrate the results.

Existence of positive solutions for nonlocal boundary value problem of fractional differential equation

Abstract In this paper, we study a type of nonlinear fractional differential equations multi-point boundary value problem with fractional derivative in the boundary conditions. By using the upper and lower solutions method and fixed point theorems, some results for the existence of positive solutions for the boundary value problem are established. Some examples are also given to illustrate our results.

Uniqueness and existence of positive solutions for a multi-point boundary value problem of singular fractional differential equations

In this paper, we study the uniqueness of a positive solution for the singular nonlinear fractional differential equation boundary value problem , , , , where is a real number, is the standard Riemann-Liouville differentiation and , with . Our analysis relies on a fixed-point theorem in partially ordered set. As an application, an example is presented to illustrate the main result. MSC:26A33, 34B15, 34K37.

Numerical approximations of fractional derivatives with applications

AbstractTwo approximations, derived from continuous expansions of Riemann–Liouville fractional derivatives into series involving integer order derivatives, are studied. Using those series, one can formally transform any problem that contains fractional derivatives into a classical problem in which only derivatives of integer order are present. Corresponding approximations provide useful numerical tools to compute fractional derivatives of functions. Application of such approximations to fractional differential equations and fractional problems of the calculus of variations are discussed. Illustrative examples show the advantages and disadvantages of each approximation.

Bernstein polynomials for solving fractional heat- and wave-like equations

Abstract In this paper, a novel numerical analysis is introduced and performed to obtain the numerical solution of the fractional heat- and wave-like equations. A general formulation for the Bernstein fractional derivatives operational matrix is given. In this approach, a truncated Bernstein series together with the Bernstein operational matrix of fractional derivatives are used to reduce the solution of fractional differential problems to the solution of a system of algebraic equations. Numerical examples are considered aiming to demonstrate the validity and applicability of the proposed techniques and to compare with the existing results.

A sum operator method for the existence and uniqueness of positive solutions to Riemann–Liouville fractional differential equation boundary value problems

In this paper, we are concerned with the existence and uniqueness of positive solutions for the following fractional boundary value problems given by-D0+αu(t)=f(t,u(t))+g(t,u(t)),0<t<1,3<α⩽4,where D0+α is the standard Riemann–Liouville fractional derivative, subject either to the boundary conditions u(0)=u′(0)=u″(0)=u″(1)=0 or u(0)=u′(0)=u″(0)=0,u″(1)=βu″(η) for η,βηα-3∈(0,1). Our analysis relies on a fixed point theorem of a sum operator. Our results can not only guarantee the existence of a unique positive solution, but also be applied to construct an iterative scheme for approximating it. Two examples are given to illustrate the main results.

Positive properties of Green's function for three-point boundary value problems of nonlinear fractional differential equations and its applications

In this article, we consider the properties of Green's function for the nonlinear fractional differential equation boundary value problem where 1 < α < 2, 0 < β, η < 1, is the standard Riemann–Liouville derivative. Here our nonlinearity f may be singular at u = 0. As an application of Green's function, we establish some multiple positive solutions for singular positone and semipositone boundary value problems by means of the Leray–Schauder nonlinear alternative, a fixed-point theorem on cones, and also we give uniqueness of a solution for a singular problem by a mixed monotone method.

Fixed point theorems for mixed monotone operators with perturbation and applications to fractional differential equation boundary value problems

The purpose of this paper is to present some new fixed point theorems for mixed monotone operators with perturbation by using the properties of cones and a fixed point theorem for mixed monotone operators. As applications, we utilize the results obtained in this paper to study the existence and uniqueness of positive solutions for nonlinear fractional differential equation boundary value problems.

Singular integrodifferential inequalities and application to fractional differential problems

In this paper we are concerned with an integrodifferential problem which arises for instance when we study a Cauchy-type fractional differential equation. This problem involves a convolution of a kernel with a nonlinear function of the solution together with its derivatives up to order two. For ordinary (third order) differential equations the kernel is regular while in our case it is singular and nonintegrable. Combining a desingularization technique due to the second author with some other estimations, we find bounds for solutions of the problem with different nonlinearities. Our results are illustrated by an application to fractional differential equations. Mathematics subject classification (2010): 26D15, 26A51, 32F99, 41A17.

Unique positive solutions for fractional differential equation boundary value problems

In this work, we consider the uniqueness of positive solutions for fractional differential equation boundary value problems. Our results can not only guarantee the existence of a unique positive solution, but also be applied to construct an iterative scheme for approximating it.

The critical exponent for an ordinary fractional differential problem

We consider the Cauchy problem for an ordinary fractional differential inequality with a polynomial nonlinearity with variable coefficient. A nonexistence result is proved and the critical exponent separating existence from nonexistence is found. This is proved in the absence of any regularity assumptions.

Variational iteration method: New development and applications

Variational iteration method has been favourably applied to various kinds of nonlinear problems. The main property of the method is in its flexibility and ability to solve nonlinear equations accurately and conveniently. In this paper recent trends and developments in the use of the method are reviewed. Major applications to nonlinear wave equation, nonlinear fractional differential equations, nonlinear oscillations and nonlinear problems arising in various engineering applications are surveyed. The confluence of modern mathematics and symbol computation has posed a challenge to developing technologies capable of handling strongly nonlinear equations which cannot be successfully dealt with by classical methods. Variational iteration method is uniquely qualified to address this challenge. The flexibility and adaptation provided by the method have made the method a strong candidate for approximate analytical solutions. This paper outlines the basic conceptual framework of variational iteration technique with application to nonlinear problems. Both achievements and limitations are discussed with direct reference to approximate solutions for nonlinear equations. A new iteration formulation is suggested to overcome the shortcoming. A very useful formulation for determining approximately the period of a nonlinear oscillator is suggested. Examples are given to illustrate the solution procedure.