In this paper, a delayed fractional order financial system is proposed and the complex dynamical behaviors of such a system are discussed by numerical simulations. A great variety of interesting dynamical behaviors of such a system including single-periodic, multiple-periodic, and chaotic motions are displayed. In particular, the effect of time delay on the chaotic behavior is investigated, it is found that an approximate time delay can enhance or suppress the emergence of chaos. Meanwhile, corresponding to different values of delay, the lowest orders for chaos to exist in the delayed fractional order financial systems are determined, respectively.

Analysis of nonlinear dynamics and chaos in a fractional order financial system with time delay

In this paper, we develop the theory of fractional hybrid differential equations involving Riemann-Liouville differential operators of order 0

Theory of fractional hybrid differential equations

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 for a nonlocal fractional differential equation

We investigate the existence of nontrivial solutions for a multi-point boundary value problem for fractional differential equations. Under certain growth conditions on the nonlinearity, several sufficient conditions for the existence of nontrivial solution are obtained by using Leray-Schauder nonlinear alternative. As an application, some examples to illustrate our results are given.

Nontrivial solutions for a nonlinear multi-point boundary value problem of fractional order

This paper investigates the existence and multiplicity of positive solutions for a class of higher-order nonlinear fractional differential equations with integral boundary conditions. The results are established by converting the problem into an equivalent integral equation and applying Krasnoselskii's fixed-point theorem in cones. The nonexistence of positive solutions is also studied.

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New Existence Results for Higher-Order Nonlinear Fractional Differential Equation with Integral Boundary Conditions

In this paper, we are concerned with the nonlinear differential equation of fractional order D 0 + α u ( t ) + f ( t , u ( t ) ) = 0 , 0 t 1 , 1 α ≤ 2 , where D 0 + α is the standard Riemann–Liouville fractional order derivative, subject to the boundary conditions u ( 0 ) = 0 , D 0 + β u ( 1 ) = a D 0 + β u ( ξ ) . We obtain the existence and multiplicity results of positive solutions by using some fixed point theorems.

Existence of positive solutions of the boundary value problem for nonlinear fractional differential equations

Direct numerical simulation (DNS) is performed to solve the governing equations for fluid flow and heat transfer around a sphere which is placed on the bottom wall of a flume. Two way coupling is used to account for the effect of the sphere on the structure of the near-wall turbulence and on the main stream. The calculation of the thermal field is done with the same grid system used for the velocity field. Water and transformer oil were used as test fluids, with the Prandtl numbers Pr &#61; 5.4 and Pr &#61; 55, respectively. The heat transfer calculations were carried out at the constant mean heat flux along the bottom. For both Prandtl numbers the DNS results indicate enhancement of heat transfer coefficient associated with a flow motion toward the wall. The thermal pattern around the sphere is obtained and the effect of Prandtl number is discussed. 

C. F. Li

Papers

Existence of positive solutions of the boundary value problem for nonlinear fractional differential equations

The influence of prandtl number on heat transfer effects around a sphere placed in a turbulent boundary layer