% Sideways Heat Equation Tools.
%
% Matlab V6 (should also work with Matlab V5)
% Last modified: 14/12-2001.
% Fredrik Berntsson, frber@math.liu.se. 
%
%
% Demonstration:
%
%  shedemo  - A simple script which creates a model problem and
%             solves it.
%  shedemo2 - Same as shedemo.
%  shedemo3 - A short demonstration of other methods for solving 
%             the sideways heat equation. (TSVD and Tikhonov regu.)
%  shedemo4 - The tools developed for solving the sideways heat equation
%             can be used for continuously monitoring the surface 
%             temperature. In this demo we demonstrate a very useful
%             (and simple) technique. 
%  shedemo5 - The surface temperature on a particle board is computed
%             using interior measurements. This test uses actual measured
%             data.
%             
% Test functions: 
% 
%  bell0  - A Gaussian bell.
%  bou0   - A thin spike.
%  bou2   - Similar to bou0 but with two peaks.
%  box0   - A function which looks like a square
%           box. This leads to a discontinous solution
%           of the sideways heat equation..
%  box1   - A smoothed box.
%  bran0  - A random signal starting and ending with zero.
%  bsin0  - A sine shaped wave.
%
%
% Sideways solvers:
%
%*  id0rk45  - Computes the solution to the sideways heat
%             equation given data at x=1. The data is filtered
%             and the time derivative is replaced by a 
%             central difference approximation. Space integration
%             is preformed using an explicit Runge-Kutta method.
%  ifftrk45 - Same as id0rk45 except that we calculate the time
%             derivative i Fourier space. Regularization is 
%             accomplished by cutting off high frequency parts.
%  imeyrk45 - Same as id0rk45 except that we use a Galerkin 
%             approximation of the time derivative, based on 
%             Meyer wavelets.
%  iwavrk45 - Same as imeyrk45 except that we use the different
%             wavetets available in wavelet toolbox. 
%  
%  fourier_solve - Explicit computation of the symbol of the 
%             solution operator using the Fourier transform.
%
% 
% Direct problem solvers:
% 
%  explhs     - Calculates the solution at x=1 given data at x=0,
%               using an explicit method.
%  crnhs      - Same as above except that we use the Crank-Nicholson
%               implicit scheme.
%  crnhs2     - Same as crnhs except that the Cauchy data [g,h] are 
%               computed at a point 0<x0<1, given boundary values 
%               u(0,t)=f(t) and u(1,t)=w(t).
%  expl_robin - Calculate the solution to the direct problem with
%               Robin conditions given at x=0, using an explicit 
%               method.
%  g_deriv    - Calculate the space derivative at x=1 given the 
%               temperature at x=1.
%
% Utility routines:
%  
%  regu_param - Determines the value of the regularization 
%               parameter, given some apriori information.
%  lcurve     - determines the value of the regularization 
%               parameter, based on the L-Curve criterion.
%  qp_err     - Compute an estimate of the error, when approximating
%               the solution domain with a quarter plane.
%  kappa_err  - Compute an estimate of the error, when using 
%               an inexact heat transfer coefficient..
%  rescale    - Transform a physical problem to a Mathematical 
%               standard problem.
%  diffmey    - Galerkin approximation of the derivative in the
%               Meyer wavelet base.
%  periodsm   - Extend a vector continiously to double
%               length using a cubic spline.
%  denoise    - Evaluate a grid function on a coarser 
%               grid and remove high frequencies.
%  kern       - A matrix representing the integral operator
%               in the quarter plane geometry.
%
% ODE-System functions:
%
%  d0syst   - ODE-system function used by id0rk45.
%  fftsyst  - ODE-system function used by ifftrk45.
%  meysyst  - ODE-system function used by imeyrk45.
%  wavsyst  - ODE-system function used by iwavrk45.
%
%
% Some of the above routines use the Spline toolbox and the
% wavelet toolbox. Also the WaveLab package is used. The Spline
% and Wavelet toolboxes are official Matlab products and the
% WaveLab package is freely available on the internet, see
% http://www-stat.stanford.edu/~wavelab/.
%