Boundary tracing and boundary value problems: I. Theory

2007 ◽  
Vol 463 (2084) ◽  
pp. 1909-1924 ◽  
Author(s):  
Michael L Anderson ◽  
Andrew P Bassom ◽  
Neville Fowkes
Keyword(s):  
Differential Equation ◽  
Nonlinear Problems ◽  
Boundary Curve ◽  
Two Dimensions ◽  
Two Dimensional ◽  
Partial Differential ◽  
Boundary Constraints ◽  
Linear And Nonlinear ◽  
Boundary Tracing ◽  
Analytical Tools

Given an exact solution of a partial differential equation in two dimensions, which satisfies suitable conditions on the boundary of the domain of interest, it is possible to deform the boundary curve so that the conditions remain fulfilled. The curves obtained in this manner can be patched together in various ways to generate a remarkably broad range of domains for which the boundary constraints remain satisfied by the initial solution. This process is referred to as boundary tracing and works for both linear and nonlinear problems. This article presents a general theoretical framework for implementing the technique for two-dimensional, second-order, partial differential equations with a general flux condition imposed around the boundary. A couple of simple examples are presented that serve to demonstrate the analytical tools in action. Applications of more intrinsic interest are discussed in the following paper.

GLOBAL C1 MAPS ON GENERAL DOMAINS

2009 ◽  
Vol 19 (05) ◽  
pp. 803-832 ◽  
Author(s):  
ALF EMIL LØVGREN ◽  
YVON MADAY ◽  
EINAR M. RØNQUIST
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Vector Fields ◽  
Two Dimensions ◽  
Arbitrary Lagrangian Eulerian ◽  
Reduced Basis ◽  
Partial Differential ◽  
Harmonic Extension ◽  
Transfinite Interpolation ◽  
Grid Points

In many contexts, there is a need to construct C1 maps from a given reference domain to a family of deformed domains. In our case, the motivation comes from the application of the Arbitrary Lagrangian Eulerian (ALE) method and also the reduced basis element method. In these methods, the maps are used to construct the grid-points needed on the deformed domains, and the corresponding Jacobian of the map is used to map vector fields from one domain to another. In order to keep the continuity of the mapped vector fields, the Jacobian must be continuous, and thus the maps need to be C1. In addition, the constructed grids on the deformed domains should be quality grids in the sense that, for a given partial differential equation defined on any of the deformed domains, the solution should be accurate. Since we are interested in a family of deformed domains, we consider the solutions of the partial differential equation to be a family of solutions governed by the geometry of the domains. Different mapping strategies are discussed and compared: the transfinite interpolation proposed by Gordon and Hall,12 the pseudo-harmonic extension proposed by Gordon and Wixom,13 a new generalization of the Gordon–Hall method (e.g., to general polygons in two dimensions), the harmonic extension, and the mean-valued extension proposed by Floater.8

Heat Conduction Equation Finite Volume Method to Achieve on MATLAB

2012 ◽  
Vol 195-196 ◽  
pp. 712-717
Author(s):  
Qiong Xue ◽  
Xiao Feng Xiao ◽  
Niang Zhi Fan
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Heat Conduction ◽  
Finite Volume Method ◽  
Finite Volume ◽  
Heat Conduction Equation ◽  
Conduction Equation ◽  
Two Dimensional ◽  
Partial Differential ◽  
Volume Method

Diffusion only, two dimensional heat conduction has been described on partial differential equation. Based on Finite Volume Method, Discretized algebraic Equation of partial differential equation have been deduced. different coefficients and source terms have been discussed under different boundary conditions, which include prescribed heat flux, prescribed temperature, convection and insulated. Transient heat conduction analysises of infinite plate with uniform thickness and two dimensional rectangle region have been realized by programming using MATLAB. It is useful to make the heat conduction equation more understandable by its solution with graphical expression, feasibility and stability of numerical method have been demonstrated by running result.

Identification of Systems Described by Partial Differential Equations

1966 ◽  
Vol 88 (2) ◽  
pp. 463-468 ◽  
Author(s):  
F. J. Perdreauville ◽  
R. E. Goodson
Keyword(s):  
Differential Equation ◽  
Experimental Data ◽  
Equation Model ◽  
Distributed Parameter ◽  
Differential Equation Model ◽  
Partial Differential ◽  
Varying Coefficients ◽  
Partial Differential Equation Model ◽  
Linear And Nonlinear

A method is given for the identification of distributed parameter systems. Normal operating records or experimental data may be used. The method involves the determination of arbitrary parameters in an assumed partial differential-equation model of the system. The method applies equally well to linear and nonlinear equations, and equations with varying coefficients. The accuracy of the results depends upon the exactness of the model, the amount of data used, the error in numerical integration, and the amount of noise which is present in the data. Examples are given which illustrate the application of the method. Results using the method for the identification of a physical system are given.

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