Numerical solutions of the parabolic wave equation: An ordinary‐differential‐equation approach

1980 ◽  
Vol 68 (5) ◽  
pp. 1482-1488 ◽  
Author(s):  
Ding Lee ◽  
John S. Papadakis
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Wave Equation ◽  
Numerical Solutions ◽  
Equation Approach

scholarly journals From time series analysis to a modified ordinary differential equation

2018 ◽  
Vol 12 (2) ◽  
pp. 85-90
Author(s):  
Meiyu Xue ◽  
Choi-Hong Lai
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Time Series ◽  
Time Series Analysis ◽  
Stock Prices ◽  
Time Frame ◽  
Equation Approach ◽  
Advantages And Disadvantages ◽  
Series Analysis ◽  
Bank Stock

In understanding Big Data, people are interested to obtain the trend and dynamics of a given set of temporal data, which in turn can be used to predict possible futures. This paper examines a time series analysis method and an ordinary differential equation approach in modeling the price movements of petroleum price and of three different bank stock prices over a time frame of three years. Computational tests consist of a range of data fitting models in order to understand the advantages and disadvantages of these two approaches. A modified ordinary differential equation model, with different forms of polynomials and periodic functions, is proposed. Numerical tests demonstrated the advantage of the modified ordinary differential equation approach. Computational properties of the modified ordinary differential equation are studied.

scholarly journals Large diffusivity finite-dimensional asymptotic behaviour of a semilinear wave equation

2003 ◽  
Vol 2003 (8) ◽  
pp. 409-427 ◽  
Author(s):  
Robert Willie
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Wave Equation ◽  
Asymptotic Behaviour ◽  
Nonlinear Term ◽  
Damped Wave Equation ◽  
Order Ordinary Differential Equation ◽  
Finite Dimensional ◽  
Type Boundary ◽  
Large Diffusion

We study the effects of large diffusivity in all parts of the domain in a linearly damped wave equation subject to standard zero Robin-type boundary conditions. In the linear case, we show in a given sense that the asymptotic behaviour of solutions verifies a second-order ordinary differential equation. In the semilinear case, under suitable dissipative assumptions on the nonlinear term, we prove the existence of a global attractor for fixed diffusion and that the limiting attractor for large diffusion is finite dimensional.

scholarly journals Differential Transform Algorithm for Functional Differential Equations with Time-Dependent Delays

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-12 ◽  
Author(s):  
Josef Rebenda ◽  
Zuzana Pátíková
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Differential Equations ◽  
Recurrence Relation ◽  
Numerical Solutions ◽  
Functional Differential Equations ◽  
Neutral System ◽  
Initial Value ◽  
Differential Transformation ◽  
Functional Differential

An algorithm using the differential transformation which is convenient for finding numerical solutions to initial value problems for functional differential equations is proposed in this paper. We focus on retarded equations with delays which in general are functions of the independent variable. The delayed differential equation is turned into an ordinary differential equation using the method of steps. The ordinary differential equation is transformed into a recurrence relation in one variable using the differential transformation. Approximate solution has the form of a Taylor polynomial whose coefficients are determined by solving the recurrence relation. Practical implementation of the presented algorithm is demonstrated in an example of the initial value problem for a differential equation with nonlinear nonconstant delay. A two-dimensional neutral system of higher complexity with constant, nonconstant, and proportional delays has been chosen to show numerical performance of the algorithm. Results are compared against Matlab function DDENSD.

scholarly journals Nonlinear electron solutions and their characteristics at infinity

2002 ◽  
Vol 44 (1) ◽  
pp. 51-59 ◽  
Author(s):  
Hilary Booth
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Numerical Solutions ◽  
Coulomb Field ◽  
Stationary Case ◽  
Order Ordinary Differential Equation ◽  
Dirac Equations ◽  
Electron Field ◽  
Axis Of Symmetry ◽  
Dirac Current

AbstractThe Maxwell-Dirac equations model an electron in an electromagnetic field. The two equations are coupled via the Dirac current which acts as a source in the Maxwell equation, resulting in a nonlinear system of partial differential equations (PDE's). Well-behaved solutions, within reasonable Sobolev spaces, have been shown to exist globally as recently as 1997 [12]. Exact solutions have not been found—except in some simple cases.We have shown analytically in [6, 18] that any spherical solution surrounds a Coulomb field and any cylindrical solution surrounds a central charged wire; and in [3] and [19] that in any stationary case, the surrounding electron field must be equal and opposite to the central (external) field. Here we extend the numerical solutions in [6] to a family of orbits all of which are well-behaved numerical solutions satisfying the analytic results in [6] and [11]. These solutions die off exponentially with increasing distance from the central axis of symmetry. The results in [18] can be extended in the same way. A third case is included, with dependence on z only yielding a related fourth-order ordinary differential equation (ODE) [3].

Spatial Homogenization of Stochastic Wave Equation with Large Interaction

2016 ◽  
Vol 59 (3) ◽  
pp. 542-552 ◽  
Author(s):  
Yongxin Jiang ◽  
Wei Wang ◽  
Zhaosheng Feng
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Wave Equation ◽  
Invariant Manifold ◽  
Linear Transformation ◽  
Second Order ◽  
Stochastic Wave Equation ◽  
Random Invariant Manifold ◽  
Large Interaction

AbstractA dynamical approximation of a stochastic wave equation with large interaction is derived. A random invariant manifold is discussed. By a key linear transformation, the random invariant manifold is shown to be close to the random invariant manifold of a second-order stochastic ordinary differential equation.

scholarly journals Classroom Note: Numerical solutions of Nagumo's equation

1999 ◽  
Vol 3 (2) ◽  
pp. 189-193 ◽  
Author(s):  
M. Iqbal
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Numerical Solutions ◽  
Linear Ordinary Differential Equation ◽  
Bounded Solutions ◽  
Third Order ◽  
Non Linear

Nagumo's equation is a third order non-linear ordinary differential equation d3udx3−cd2udx2+f′(u)dudx−(b/c)u=0 where f(u)=u(1−u)(u−α) , 0<α<1 . In this paper we have developed a technique to determine those values of the parameters a,b and c which permit non-constant bounded solutions.

A Numerical Comparison of Finite Difference and Finite Element Methods for a Stochastic Differential Equation with Polynomial Chaos

2015 ◽  
Vol 5 (2) ◽  
pp. 192-208 ◽  
Author(s):  
Ning Li ◽  
Bo Meng ◽  
Xinlong Feng ◽  
Dongwei Gui
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Finite Element ◽  
Finite Difference ◽  
Stochastic Differential Equation ◽  
Numerical Experiments ◽  
Polynomial Chaos ◽  
Numerical Solutions ◽  
Numerical Comparison ◽  
Fe Method

AbstractA numerical comparison of finite difference (FD) and finite element (FE) methods for a stochastic ordinary differential equation is made. The stochastic ordinary differential equation is turned into a set of ordinary differential equations by applying polynomial chaos, and the FD and FE methods are then implemented. The resulting numerical solutions are all non-negative. When orthogonal polynomials are used for either continuous or discrete processes, numerical experiments also show that the FE method is more accurate and efficient than the FD method.

Numerical Analysis of the Exact Solution of the Wave Equation of the Longitudinal Seismic Oscillations of Soil and the Construction as a Point Insertion

2018 ◽  
Vol 931 ◽  
pp. 152-157 ◽  
Author(s):  
Kamil D. Yaxubayev ◽  
Dinara D. Kochergina
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Partial Differential Equation ◽  
Exact Solution ◽  
Wave Equation ◽  
Numerical Analysis ◽  
Differential Equations ◽  
Partial Differential ◽  
Seismic Oscillations

The numerical analysis of the exact solution of the system of the differential equations which includes the partial differential equation of the longitudinal seismic oscillations of the soil and the ordinary differential equation of oscillations of the construction in the form of a point rigid insertion.

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