A block-centred finite difference method for the distributed-order differential equation with Neumann boundary condition

2018 ◽  
Vol 96 (3) ◽  
pp. 622-639 ◽  
Author(s):  
Xiaoli Li ◽  
Hongxing Rui
Keyword(s):  
Differential Equation ◽  
Boundary Condition ◽  
Finite Difference ◽  
Finite Difference Method ◽  
Difference Method ◽  
Neumann Boundary Condition ◽  
Order Differential Equation ◽  
Neumann Boundary ◽  
Distributed Order

Finite Difference Method for a Second-order Ordinary Differential Equation with a Boundary Condition of the Third Kind

2010 ◽  
Vol 10 (1) ◽  
pp. 109-116 ◽  
Author(s):  
P.K. Pandey
Keyword(s):  
Differential Equation ◽  
Boundary Condition ◽  
Finite Difference ◽  
Finite Difference Method ◽  
Difference Method ◽  
Second Order ◽  
Point Boundary ◽  
Order Ordinary Differential Equation ◽  
Numerical Examples ◽  
The Third

Abstract We present a second-order finite difference method for obtaining a solution of a second order two-point boundary value problem subject to Sturm's boundary conditions. We use equidistant discretization points, and the discretization of the differential equation at an interior point is based on just two evaluations of the function. Numerical examples are considered and the convergence of the proposed method is proved computationally.

Numerical Dynamics of Nonstandard Finite Difference Method for Nonlinear Delay Differential Equation

2018 ◽  
Vol 28 (11) ◽  
pp. 1850133 ◽  
Author(s):  
Xiaolan Zhuang ◽  
Qi Wang ◽  
Jiechang Wen
Keyword(s):  
Differential Equation ◽  
Finite Difference ◽  
Finite Difference Method ◽  
Delay Differential Equation ◽  
Difference Method ◽  
Discrete System ◽  
Delay Differential ◽  
Nonlinear Delay Differential Equation ◽  
Nonlinear Delay ◽  
Nonstandard Finite Difference

In this paper, we study the dynamics of a nonlinear delay differential equation applied in a nonstandard finite difference method. By analyzing the numerical discrete system, we show that a sequence of Neimark–Sacker bifurcations occur at the equilibrium as the delay increases. Moreover, the existence of local Neimark–Sacker bifurcations is considered, and the direction and stability of periodic solutions bifurcating from the Neimark–Sacker bifurcation of the discrete model are determined by the Neimark–Sacker bifurcation theory of discrete system. Finally, some numerical simulations are adopted to illustrate the corresponding theoretical results.

scholarly journals Wellposedness of Neumann boundary-value problems of space-fractional differential equations

2017 ◽  
Vol 20 (6) ◽  
Author(s):  
Hong Wang ◽  
Danping Yang
Keyword(s):  
Differential Equation ◽  
Boundary Condition ◽  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Neumann Boundary Condition ◽  
Boundary Value ◽  
Neumann Boundary ◽  
Fractional Differential ◽  
Neumann Boundary Value Problems ◽  
Well Posed

AbstractFractional differential equation (FDE) provides an accurate description of transport processes that exhibit anomalous diffusion but introduces new mathematical difficulties that have not been encountered in the context of integer-order differential equation. For example, the wellposedness of the Dirichlet boundary-value problem of one-dimensional variable-coefficient FDE is not fully resolved yet. In addition, Neumann boundary-value problem of FDE poses significant challenges, partly due to the fact that different forms of FDE and different types of Neumann boundary condition have been proposed in the literature depending on different applications.We conduct preliminary mathematical analysis of the wellposedness of different Neumann boundary-value problems of the FDEs. We prove that five out of the nine combinations of three different forms of FDEs that are closed by three types of Neumann boundary conditions are well posed and the remaining four do not admit a solution. In particular, for each form of the FDE there is at least one type of Neumann boundary condition such that the corresponding boundary-value problem is well posed, but there is also at least one type of Neumann boundary condition such that the corresponding boundary-value problem is ill posed. This fully demonstrates the subtlety of the study of FDE, and, in particular, the crucial mathematical modeling question: which combination of FDE and fractional Neumann boundary condition, rather than which form of FDE or fractional Neumann boundary condition, should be used and studied in applications.

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