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Re: matching conditions in PDE

  • To: mathgroup at smc.vnet.net
  • Subject: [mg42179] Re: [mg42155] matching conditions in PDE
  • From: CAP F <Ferdinand.Cap at eunet.at>
  • Date: Sat, 21 Jun 2003 20:56:58 -0400 (EDT)
  • References: <200306210649.CAA13186@smc.vnet.net>
  • Sender: owner-wri-mathgroup at wolfram.com

Vladimir Petrov wrote:

> Hi,
>
> I would like to know how to solve numerically a partial
> differential equation (the heat equation) in a medium
> consisting of two subregions characterized by different
> physical parameters. How to set in NDSolve the matching
> conditions at the boundary between the two subregions ?
> Could you give an example.
> Thank you.
> Regards
>
> Vladimir

Mathematical Methods of Physics and Engineering with Mathematica
Ferdinand F . Cap, CRC-Press/Chapman and Hall, 2003,
ISBN 1584884029,  56 Codes  to be downloaded from
www.crcpress.com

1 Introduction
1.1 What is a boundary problem?
1.2 Classification of partial differential equations
1.3 Types of boundary conditions and the collocation method
1.4 Differential equations as models for nature

2 Boundary problems of ordinary differential equations
2.1 Linear differential equations
2.2 Solving linear differential equations
2.3 Differential equations of physics and engineering
2.4 Boundary value problems and eigenvalues
2.5 Boundary value problem as initial value problem
2.6 Nonlinear ordinary differential equations
2.7 Solutions of nonlinear differential equations

3 Partial differential equations
3.1 Coordinate systems and separability
3.2 Other methods to reduce partial to ordinary differential
equations
3.3 The method of characteristics
3.4 Nonlinear partial differential equations

4 Boundary problems with one closed boundary defined by
coordinate lines

4.1 Laplace and Poisson equation
4.2 Conformal mapping in two and three dimensions
4.3 D'Alembert wave equation and string vibrations
4.4 Helmholtz equation and membrane vibrations
4.5 Rods and the plate equation
4.6 Approximation methods
4.7 Variational calculus
4.8 Collocation methods

5 Boundary problems with two closed boundaries
5.1 Inseparable problems
5.2 Holes in the domain.Two boundaries belonging to different
coordinate systems
5.3 Corners in the boundary

6 Nonlinear boundary problems
6.1 Some definitions and examples
6.2 Moving and free boundaries
6.3 Waves of large amplitudes. Solitons
6.4 Rupture of an embankment-type dam
6.5 Gas flow in a combustion engine

List of codes (1 - 56)
see:  www.crcpress.com/downloads
(F.Cap,  Mathematical Methods in Physics and Engineering with
Mathematica, ccrcpress and Chapman and Hall,  2003, ISBN
1584884029
c1 Equation of motion of a parachutist
c2 Differentiate and integrate
c3 The  differential  equation describing the spread of an
epidemic disease
c4 The Wronskian of exp(x), exp(2x)
c5 The most general linear differential equation of second order
c6 Inhomogeneous equation  of oscillations
c7 An initial value problem
c8 Homogeneous boundary value  problem
c9 Inhomogeneous boundary value problem
c10 Pick out values of a numerical solution of a problem with
varying boundary values
c11 Preparing the shooting method for inhomogeneous boundary
value problems
c12 Series expansion of a Lie-series solution
c13 Learn a loop for the shooting method
c14 Limit cycle  of the Van der Pol equation
c15 High  voltage electrostatic parametric generator  US patent
c16 Phase portrait of the Duffing equation
c17 Phase portrait Mathieu equation
c18 Phase portrait of the pendulum equation
c19 Jacobian  matrix of spherical coordinates
c20 Vector  analysis: curl  in spherical coordinates
c21 Separation setup for  the  Laplacian
c22 Laplace transformation
c23 Characteristics of one-dimensional flow (3.3.27), see page
119 of the book
c24 Solution of the heat conduction equation using a similarity
transformation
c25 Harmonic polynomials as solution of the  Laplacian
c26 Biharmonic polynomials solve the static homogeneous plate
equation
c27 Generalized Bessel and Kummer equations
c28 Whittaker, Gegenbauer and Weber equation
c29 Legendre equation with Rodriguez formula
c30 Laguerre equation with Rodriguez formula
c31 Hermite equation with Rodriguez formula
c32 Vector field - cover of the book
c33 Conformal  mapping
c34 2D flow around a cylinder
c35 Solution of the damped Helmholtz equation
c36 Free bending vibrations of a rod
c37 Two boundaries for the  Laplace equation
c38 Inhomogeneous boundary problem for the Laplace equation
c39 Nontrivial homogeneous boundary problem of the Laplace
equation
c40 COLLOC, a FORTRAN PROGRAM to calculate the eigenvalues of a
 circular membrane in CARTESIAN COORDINATES
c41 Eigenvalue problem for a circular membrane.
 Calculation in cartesian coordinates
c42 Clamped Cassini membrane in cartesian coordinates
c43 Circular membrane with varying surface mass density.
 Includes four-fold loop
c44 Circular plate with two  homogeneous boundary conditions
c45 Laplace equation in polar coordinates with a singularity
c46 Laplace equation with two different boundaries, inner
inhomogeneous on a square and outer homogeneous on circle
c47 Membrane with an homogeneous boundary  condition  on an inner
circle and an inhomogeneous outer boundary condition
c48 Corner in a boundary curve
c49 Steepening up of a large-amplitude wave
c50 Envelope soliton
c51 Laplace equation with outer inhomogeneous boundary  values on
square and inner homogeneous boundary values on circle
c52 Gaussian elimination
c53 The shooting method
c54 Navier solutions of the plate equation
c55 From Maxwell's to vector Helmholtz equations
c56 Toroidal wave guides


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