Approximate Solutions in Linear, Coupled Thermoelasticity

1968 ◽  
Vol 35 (2) ◽  
pp. 255-266 ◽  
Author(s):  
R. E. Nickell ◽  
J. L. Sackman
Keyword(s):  
Half Space ◽  
Rapid Heating ◽  
Ritz Method ◽  
Gaussian Quadrature ◽  
Boundary Surface ◽  
Initial Boundary ◽  
Approximate Solutions ◽  
Coupled Thermoelasticity ◽  
Initial Boundary Value ◽  
Thermally Conducting

A method for obtaining approximate solutions to initial-boundary-value problems in the linear theory of coupled thermoelasticity is developed. This procedure is a direct variational method representing an extension of the Ritz method. As an illustration of the procedure, it is applied to a class of one-dimensional, transient problems involving weak thermal shocks. The problems considered are: (a) Rapid heating of a half space through a thermally conducting boundary layer, and (b) gradual heating of the boundary surface of a half space. The solutions generated by the extended Ritz method are compared, for accuracy, to solutions obtained from a numerical inversion scheme for the Laplace transform based on Gaussian quadrature. These comparisons indicate that the variational procedure developed here can yield accurate results.

Unsteady, Combined Radiation and Conduction in an Absorbing, Scattering, and Emitting Medium

1973 ◽  
Vol 95 (3) ◽  
pp. 357-364 ◽  
Author(s):  
K. C. Weston ◽  
J. L. Hauth
Keyword(s):  
Boundary Value Problem ◽  
Energy Equation ◽  
Numerical Solutions ◽  
Gaussian Quadrature ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Isotropic Scattering ◽  
Radiation Parameter ◽  
Parallel Plates ◽  
Initial Boundary Value

The transient cooldown of a gray, absorbing, isotropic scattering, emitting, and conducting medium bounded by gray, diffusely emitting and reflecting parallel plates is considered. Numerical solutions are obtained for the initial boundary-value problem with a discontinuous decrease in temperature at one boundary. The quasi-steady equation of radiative transfer is solved using Gaussian quadrature and a matrix eigenvector technique together with explicit numerical solution of the unsteady energy equation. Temperature and energy flux distributions are presented for variations of optical thickness, boundary emissivity, albedo, and conduction–radiation parameter.

scholarly journals Exact, approximate solutions and error bounds for coupled implicit systems of partial differential equations

1992 ◽  
Vol 15 (4) ◽  
pp. 663-672
Author(s):  
Lucas Jódar
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Initial Boundary ◽  
Approximate Solutions ◽  
Value Systems ◽  
Finite Domain ◽  
Partial Differential ◽  
Implicit Systems ◽  
Initial Boundary Value ◽  
The Given

In this paper coupled implicit initial-boundary value systems of second order partial differential equations are considered. Given a finite domain and an admissible errorϵan analytic approximate solution whose error is upper bounded byϵin the given domain is constructed in terms of the data.

GLOBAL WEAK SOLUTIONS OF AN INITIAL BOUNDARY VALUE PROBLEM FOR SCREW PINCHES IN PLASMA PHYSICS

2009 ◽  
Vol 19 (06) ◽  
pp. 833-875 ◽  
Author(s):  
JIANWEN ZHANG ◽  
SONG JIANG ◽  
FENG XIE
Keyword(s):  
Magnetic Field ◽  
Boundary Value Problem ◽  
Plasma Physics ◽  
Weak Solutions ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Approximate Solutions ◽  
Field Equations ◽  
Initial Boundary Value

This paper is concerned with an initial-boundary value problem for screw pinches arisen from plasma physics. We prove the global existence of weak solutions to this physically very important problem. The main difficulties in the proof lie in the presence of 1/x-singularity in the equations at the origin and the additional nonlinear terms induced by the magnetic field. Solutions will be obtained as the limit of the approximate solutions in annular regions between two cylinders. Under certain growth assumption on the heat conductivity, we first derive a number of regularities of the approximate physical quantities in the fluid region, as well as a lot of uniform integrability in the entire spacetime domain. By virtue of these estimates we then argue in a similar manner as that in Ref. 20 to take the limit and show that the limiting functions are indeed a weak solution which satisfies the mass, momentum and magnetic field equations in the entire spacetime domain in the sense of distributions, but satisfies the energy equation only in the compact subsets of the fluid region. The analysis in this paper allows the possibility that energy is absorbed into the origin, i.e. the total energy be possibly lost in the limit as the inner radius goes to zero.

scholarly journals Approximate Solutions of Delay Parabolic Equations with the Dirichlet Condition

2012 ◽  
Vol 2012 ◽  
pp. 1-31 ◽  
Author(s):  
Deniz Agirseven
Keyword(s):  
Finite Difference ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Approximate Solutions ◽  
Dirichlet Condition ◽  
Difference Schemes ◽  
Parabolic Partial Differential Equation ◽  
Analysis Methods ◽  
Homotopy Analysis ◽  
Initial Boundary Value

Finite difference and homotopy analysis methods are used for the approximate solution of the initial-boundary value problem for the delay parabolic partial differential equation with the Dirichlet condition. The convergence estimates for the solution of first and second orders of difference schemes in Hölder norms are obtained. A procedure of modified Gauss elimination method is used for the solution of these difference schemes. Homotopy analysis method is applied. Comparison of finite difference and homotopy analysis methods is given on the problem.

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