Background:
We consider a very simple partial differential equation, the heat equation
describing the heat transfer in a solid rod. In this case, the
structure of a system of partial differential algebraic equations is to be
outlined.
The Mathematical Model:
The one-dimensional diffusion equation is given in the form
u(x,t)_{t} = u(x,t)_{xx}
The subindices 't' and 'xx', respectively, denote the partial differentiation with respect to time variable t and spatial variable x varying between 0 and 1. We suppose inhomogeneous initial value u(x,0) = sin(p x) and homogeneous Dirichlet boundary conditions x(0,t) = x(1,t) = 0. c is the unknown parameter to be estimated. Fitting criteria for which measurements are generated, are u(0.25,t) , u(0.5,t) , and u(0.75,t). In this simple situation, the explicit solution is known,
u(x,t) = exp(-t p^{2}) sin(x p )
Now we introduce another, so-called algebraic state variable v(x,t), and replace above equation by
u(x,t)_{t} = v(x,t)_{x}
v(x,t) = u(x,t)_{x}
An additional homogeneous initial value v(x,0) = 0 is defined, but only for formal reasons. Consistent initial values are computed internally by PDEFIT at the grid points. Now we introduce a heat transfer coefficient c and consider the equation u(x,t)_{t} = c v(x,t)_{x} with initial value u(x,0) = c sin(p x). c is the unknown parameter to be estimated with true value c=1. Fitting criterion for which measurements are generated, is u(0.5,t).
Literature:
Schittkowski (2002):
Numerical Data Fitting in Dynamical Systems - A Practical Introduction with
Applications and Software,
Kluwer
Academic Publishers
Implementation:
The complete solution of a data fitting problem is described
in six
steps:
Results:
Then you would like to take a look at reports and graphs:
- parameter values
- experimental data versus fitting criterion
- surface plot of state variable
Model equations (or use your own favorite editor):
Measurement data (or use import function for text file or Excel):
Parameters, tolerances, and start of a data fitting run:
Numerical results (computed by the least squares code DFNLP):
Report about parameter values, residuals, performance, etc. (or export to Word):
Experimental data versus fitting criterion (also available for Gnuplot):
Surface plot of state variable (also available for Gnuplot):