The Hamilton-Jacobi Equation Applied to Continuum

1997 ◽  
Vol 64 (3) ◽  
pp. 658-663 ◽  
Author(s):  
C. M. Leech
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Jacobi Equation ◽  
Principal Function ◽  
Partial Differential ◽  
One Dimensional ◽  
Linear Elastic ◽  
Elastic Bar ◽  
Continuum Systems ◽  
Hamilton Jacobi Equation

The Hamilton-Jacobi partial differential equation is established for continuum systems; to do this a new concept in material distributions is introduced. The Lagrangian and Hamiltonian are developed, so that the Hamilton-Jacobi equation can be formulated and the principal function defined. Finally the principal function is constructed for the dynamics of a one-dimensional linear elastic bar; the solution for its’ vibrations is then established following the differentiation of the principal function.

scholarly journals A Numerical Well-Balanced Scheme for One-Dimensional Heat Transfer in Longitudinal Triangular Fins

2013 ◽  
Vol 2013 ◽  
pp. 1-9 ◽  
Author(s):  
I. Rusagara ◽  
C. Harley
Keyword(s):  
Heat Transfer ◽  
Differential Equation ◽  
Partial Differential Equation ◽  
Nonlinear Partial Differential Equation ◽  
Transfer Coefficients ◽  
Partial Differential ◽  
One Dimensional ◽  
Heat Transfer Coefficients ◽  
Highly Nonlinear ◽  
Triangular Fin

The temperature profile for fins with temperature-dependent thermal conductivity and heat transfer coefficients will be considered. Assuming such forms for these coefficients leads to a highly nonlinear partial differential equation (PDE) which cannot easily be solved analytically. We establish a numerical balance rule which can assist in getting a well-balanced numerical scheme. When coupled with the zero-flux condition, this scheme can be used to solve this nonlinear partial differential equation (PDE) modelling the temperature distribution in a one-dimensional longitudinal triangular fin without requiring any additional assumptions or simplifications of the fin profile.

The Hamilton-Jacobi Equations for a Relativistic Charged Particle

1963 ◽  
Vol 6 (3) ◽  
pp. 341-350 ◽  
Author(s):  
J. R. Vanstone
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Jacobi Equation ◽  
Classical Particle ◽  
Quantum Mechanical ◽  
Order Partial Differential Equation ◽  
First Order ◽  
Relativistic Charged Particle ◽  
Jacobi Equations ◽  
Hamilton Jacobi Equation

In the problem of finding the motion of a classical particle one has the choice of dealing with a system of second order ordinary differential equations (Lagrange's equations) or a single first order partial differential equation (the Hamilton-Jacobi equation, henceforth referred to as the H-J equation). In practice the latter method is less frequently used because of the difficulty in finding complete integrals. When these are obtainable, however, the method leads rapidly to the equations of the trajectories. Furthermore it is of fundamental theoretical importance and it provides a basis for quantum mechanical analogues.

Unsteady Flow of Gas Through a Semi-Infinite Porous Medium

1957 ◽  
Vol 24 (3) ◽  
pp. 329-332
Author(s):  
R. E. Kidder
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Porous Medium ◽  
Gas Flow ◽  
Transient Flow ◽  
Order Term ◽  
Nonlinear Partial Differential Equation ◽  
Flow In Porous Media ◽  
Partial Differential ◽  
One Dimensional

Abstract This paper presents an analytic solution to a problem of the transient flow of gas within a one-dimensional semi-infinite porous medium. A perturbation method, carried out to include terms of the second order, is employed to obtain a solution of the nonlinear partial differential equation describing the flow of gas. The zero-order term of the solution represents the solution of the linearized partial differential equation of gas flow in porous media given by Green and Wilts (1).

On Nonlinear Modes of Continuous Systems

1994 ◽  
Vol 116 (1) ◽  
pp. 129-136 ◽  
Author(s):  
A. H. Nayfeh ◽  
S. A. Nayfeh
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Boundary Conditions ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Continuous Systems ◽  
Partial Differential ◽  
Nonlinear Beam ◽  
Nonlinear Modes ◽  
One Dimensional

We use several methods to study the nonlinear modes of one-dimensional continuous systems with cubic inertia and geometric nonlinearities. Invariant manifold and perturbation methods applied to the discretized system and the method of multiple scales applied to the partial-differential equation and boundary conditions are discussed and their equivalence is demonstrated. The method of multiple scales is then applied directly to the partial-differential equation and boundary conditions governing several nonlinear beam problems.

scholarly journals Finite Analytic Method for One-Dimensional Nonlinear Consolidation under Time-Dependent Loading

2017 ◽  
Vol 2017 ◽  
pp. 1-12 ◽  
Author(s):  
Dawei Cheng ◽  
Wenke Wang ◽  
Xi Chen ◽  
Zaiyong Zhang
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Numerical Solutions ◽  
Initial Conditions ◽  
Time Dependent ◽  
Analytic Method ◽  
Partial Differential ◽  
One Dimensional ◽  
Finite Analytic Method ◽  
Nonlinear Consolidation

For one-dimensional (1D) nonlinear consolidation, the governing partial differential equation is nonlinear. This paper develops the finite analytic method (FAM) to simulate 1D nonlinear consolidation under different time-dependent loading and initial conditions. To achieve this, the assumption of constant initial effective stress is not considered and the governing partial differential equation is transformed into the diffusion equation. Then, the finite analytic implicit scheme is established. The convergence and stability of finite analytic numerical scheme are proven by a rigorous mathematical analysis. In addition, the paper obtains three corrected semianalytical solutions undergoing suddenly imposed constant loading, single ramp loading, and trapezoidal cyclic loading, respectively. Comparisons of the results of FAM with the three semianalytical solutions and the result of FDM, respectively, show that the FAM can obtain stable and accurate numerical solutions and ensure the convergence of spatial discretization for 1D nonlinear consolidation.

scholarly journals On the numerical solution of the one-dimensional convection-diffusion equation

2005 ◽  
Vol 2005 (1) ◽  
pp. 61-74 ◽  
Author(s):  
Mehdi Dehghan
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Finite Difference ◽  
Diffusion Equation ◽  
Numerical Solution ◽  
Partial Differential ◽  
One Dimensional ◽  
Convection Diffusion ◽  
Convection Diffusion Equation ◽  
The One

The numerical solution of convection-diffusion transport problems arises in many important applications in science and engineering. These problems occur in many applications such as in the transport of air and ground water pollutants, oil reservoir flow, in the modeling of semiconductors, and so forth. This paper describes several finite difference schemes for solving the one-dimensional convection-diffusion equation with constant coefficients. In this research the use of modified equivalent partial differential equation (MEPDE) as a means of estimating the order of accuracy of a given finite difference technique is emphasized. This approach can unify the deduction of arbitrary techniques for the numerical solution of convection-diffusion equation. It is also used to develop new methods of high accuracy. This approach allows simple comparison of the errors associated with the partial differential equation. Various difference approximations are derived for the one-dimensional constant coefficient convection-diffusion equation. The results of a numerical experiment are provided, to verify the efficiency of the designed new algorithms. The paper ends with a concluding remark.

On the integrals of a partial differential equation

1947 ◽  
Vol 43 (3) ◽  
pp. 348-359 ◽  
Author(s):  
F. G. Friedlander
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Wave Equation ◽  
Partial Differential ◽  
One Dimensional ◽  
First Order ◽  
Progressive Waves ◽  
Image Position ◽  
Straight Lines ◽  
Dimensional Wave

The ordinary one-dimensional wave equationhas special integrals of the formwhich satisfy the first-order equationsrespectively, and are often called progressive waves, or progressive integrals, of (1·1). The straight linesin an xt-plane are the characteristics of (1·1). It follows from (1·2) that progressive integrals of (1·1) are constant on some particular characteristic, and are characterized by this property.

Sign in / Sign up

Export Citation Format

Share Document