The differentiability of the discontinuous solution of a system of ordinary differential equations with respect to initial conditions and parameters

Cybernetics ◽  
1974 ◽  
Vol 8 (2) ◽  
pp. 247-252
Author(s):  
I. P. Chentsov
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Initial Conditions ◽  
Discontinuous Solution

Plumes Above Line Fires in a Cross-Wind

1994 ◽  
Vol 4 (4) ◽  
pp. 201 ◽  
Author(s):  
GN Mercer ◽  
RO Weber
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Initial Conditions ◽  
Velocity Profiles ◽  
Ambient Wind ◽  
Leaf Scorch ◽  
Wind Velocities ◽  
Wind Profiles ◽  
Fire Characteristics ◽  
Temperature Time

A model for the plume above a line fire in a cross wind is constructed. This problem is shown to reduce to numerically solving a system of 6 coupled ordinary differential equations for given initial conditions that depend upon the fire characteristics. The model is valid above the flaming zone and takes inputs such as the width, velocity and temperature of the plume at a given height above the flaming zone, Different horizontal ambient wind velocities are allowed for and a comparison is made between some of these representative wind profiles. The plume trajectory, width, velocity and temperature are calculated for these different representative velocity profiles. This model has application to the calculation of temperature-time exposures of vegetation above line fires and hence can be used in models that predict effects such as leaf scorch and canopy stored seed death. On a larger scale it has application to the problem of tracking burning brands which can cause spotting ahead of the fire.

Application of Ambarzumian’s Method to Radiant Interchange in a Rectangular Cavity

1974 ◽  
Vol 96 (2) ◽  
pp. 191-196 ◽  
Author(s):  
A. L. Crosbie ◽  
T. R. Sawheny
Keyword(s):  
Differential Equation ◽  
Heat Flux ◽  
Integral Equation ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Nonlinear Integral Equation ◽  
Initial Conditions ◽  
Rectangular Cavity ◽  
Wide Range ◽  
First Time

Ambarzumian’s method had been used for the first time to solve a radiant interchange problem. A rectangular cavity is defined by two semi-infinite parallel gray surfaces which are subject to an exponentially varying heat flux, i.e., q = q0 exp(−mx). Instead of solving the integral equation for the radiosity for each value of m, solutions for all values of m are obtained simultaneously. Using Ambarzumian’s method, the integral equation for the radiosity is first transformed into an integro-differential equation and then into a system of ordinary differential equations. Initial conditions required to solve the differential equations are the H functions which represent the radiosity at the edge of the cavity for various values of m. This H function is shown to satisfy a nonlinear integral equation which is easily solved by iteration. Numerical results for the H function and radiosity distribution within the cavity are presented for a wide range of m values.

Exact Equations for the Inextensional Deformation of Cantilevered Plates

1979 ◽  
Vol 46 (3) ◽  
pp. 631-636 ◽  
Author(s):  
J. G. Simmonds ◽  
A. Libai
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Initial Conditions ◽  
Independent Set ◽  
Straight Edge ◽  
The Other ◽  
Subsequent Paper ◽  
First Order ◽  
Alternate Forms ◽  
Approximate Equations

A set of first-order ordinary differential equations with initial conditions is derived for the exact, nonlinear, inextensional deformation of a loaded plate bounded by two straight edges and two curved ones. The analysis extends earlier approximate work of Mansfield and Kleeman, Ashwell, and Lin, Lin, and Mazelsky. For a plate clamped along one straight edge and subject to a force and couple along the other, there are 13 differential equations, but an independent set of 9 may be split off. In a subsequent paper, we consider alternate forms of these 9 equations for plates that twist as they deform. Their structure and solutions are compared to Mansfield’s approximate equations and particular attention is given to tip-loaded triangular plates.

scholarly journals The Laplace transform as an alternative general method for solving linear ordinary differential equations

2021 ◽  
Vol 1 (4) ◽  
pp. 309
Author(s):  
William Guo
Keyword(s):  
General Solution ◽  
Differential Equations ◽  
Laplace Transform ◽  
Ordinary Differential Equations ◽  
Case Studies ◽  
Initial Conditions ◽  
Linear Ordinary Differential Equations ◽  
The Laplace Transform ◽  
Linear Odes ◽  
General Method

<p style='text-indent:20px;'>The Laplace transform is a popular approach in solving ordinary differential equations (ODEs), particularly solving initial value problems (IVPs) of ODEs. Such stereotype may confuse students when they face a task of solving ODEs without explicit initial condition(s). In this paper, four case studies of solving ODEs by the Laplace transform are used to demonstrate that, firstly, how much influence of the stereotype of the Laplace transform was on student's perception of utilizing this method to solve ODEs under different initial conditions; secondly, how the generalization of the Laplace transform for solving linear ODEs with generic initial conditions can not only break down the stereotype but also broaden the applicability of the Laplace transform for solving constant-coefficient linear ODEs. These case studies also show that the Laplace transform is even more robust for obtaining the specific solutions directly from the general solution once the initial values are assigned later. This implies that the generic initial conditions in the general solution obtained by the Laplace transform could be used as a point of control for some dynamic systems.</p>

scholarly journals Modified nonlinearities distribution Homotopy Perturbation method as a tool to find power series solutions to ordinary differential equations

Nova Scientia ◽  
2014 ◽  
Vol 6 (12) ◽  
pp. 13 ◽  
Author(s):  
Umberto Filobello-Nino ◽  
Héctor Vázquez-Leal ◽  
Yasir Khan ◽  
D. Pereyra-Díaz ◽  
A. Pérez-Sesma ◽  
...  
Keyword(s):  
Power Series ◽  
Differential Equations ◽  
Perturbation Method ◽  
Ordinary Differential Equations ◽  
Homotopy Perturbation Method ◽  
Initial Conditions ◽  
Series Solutions ◽  
Homotopy Perturbation ◽  
Polynomial Coefficients ◽  
Power Series Solutions

In this article, modified non-linearities distribution homotopy perturbation method (MNDHPM) is used in order to find power series solutions to ordinary differential equations with initial conditions, both linear and nonlinear. We will see that the method is particularly relevant in some cases of equations with non-polynomial coefficients and inhomogeneous non-polynomial terms

A method for the estimation of errors propagated in the numerical solution of a system of ordinary differential equations

1966 ◽  
Vol 25 ◽  
pp. 281-287 ◽  
Author(s):  
P. E. Zadunaisky
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Differential Equations ◽  
Asymptotic Behaviour ◽  
Numerical Solution ◽  
Ordinary Differential Equations ◽  
Difference Method ◽  
Initial Conditions ◽  
Current Theory ◽  
Numerical Process

Let bex′=f(t,x) a system of ordinary differential equations, with initial conditionsx(a) =s, which is integrated numerically by a finite difference method of orderpand constant steph.To estimate the truncation and round-off errors accumulated during the numerical process it is established a method based on the current theory of the asymptotic behaviour (whenh→0) of errors. This method should avoid the main difficulties that arise when the results of the theory must be applied to practical cases. The method has been successfully tested and applied to estimate the errors accumulated in a numerical computation of planetary perturbations on the orbit of a comet.

Full-size mathematical model of the fuel metering unit

2011 ◽  
Vol 8 (1) ◽  
pp. 249-256
Author(s):  
E.Sh. Nasibullaeva ◽  
E.V. Denisova ◽  
I.Sh. Nasibullayev
Keyword(s):  
Mathematical Model ◽  
Differential Equations ◽  
Pressure Gradient ◽  
Ordinary Differential Equations ◽  
Constant Pressure ◽  
Initial Conditions ◽  
Control Valve ◽  
Nonlinear Mathematical Model ◽  
Full Size

The paper presents a nonlinear mathematical model for the operation of the fuel metering unit, which takes into account the operation of the control valve, which includes two pistons and three fuel circuits. A technique for determining the initial conditions for a system of ordinary differential equations describing the movements of a servo piston, a piston of a constant pressure gradient valve and a piston of a control valve is proposed.

Dynamics of a Basketball Rolling Around the Rim

2005 ◽  
Vol 128 (2) ◽  
pp. 359-364
Author(s):  
C. Q. Liu ◽  
Fang Li ◽  
R. L. Huston
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Initial Conditions ◽  
Equations Of Motion ◽  
Dynamical Equations ◽  
First Order ◽  
Coaching Strategies ◽  
Ball Motion

Governing dynamical equations of motion for a basketball rolling on the rim of a basket are developed and presented. These equations form a system of five first-order, ordinary differential equations. Given suitable initial conditions, these equations are readily integrated numerically. The results of these integrations predict the success (into the basket) or failure (off the outside of the rim) of the basketball shot. A series of examples are presented. The examples show that minor changes in the initial conditions can produce major changes in the subsequent ball motion. Shooting and coaching strategies are recommended.

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