# paraheat_gaussian

paraheat_gaussian, a MATLAB program which sets up and solves a parameterized steady heat equation in a 2D spatial domain, with a gaussian diffusivity parameterized by (xc,yc), sc and vc.

The steady state heat equation to be solved is:

```        - del ( k(x,y) * grad u ) = f(x,y)
```
over the unit square 0 < x, y < 1.

Zero Dirichlet boundary conditions are applied. The right hand side function is set as:

```        f(x,y) = 1000 * x * ( 1 - x ) * y * ( 1 - y );
```

The diffusivity is represented by a gaussian function:

```        k(x,y) = 0.5 + vc * exp ( - 0.5 * ((x-xc)^2+(y-yc)^2)/sc^2 )
```
with parameters xc, yc, sc, vc.

After the solution is computed, the value of the solution is determined at a certain number of sensor locations.

This program is a step in studying the relationship between the parameters chosen to specify the diffusivity, and the subsequent sensor readings. In particular, it would be desirable to be able to estimate the parameters from the sensor data.

### Languages:

paraheat_gaussian is available in a MATLAB version.

### Related Data and Programs:

paraheat_gaussian_plot, a MATLAB program which calls paraheat_gaussian() to set up and solve a parameterized steady heat equation in a 2D spatial domain, with a gaussian diffusivity and then uses radial basis functions (RBF) to reconstruct the finite element solution from a set of sample values.

paraheat_gaussian_sample, a MATLAB program which repeatedly calls paraheat_gaussian() to set up and solve a parameterized steady heat equation in a 2D spatial domain, with a gaussian diffusivity, saving values of the finite element solution at selected points, and writing the saved data to a file.

paraheat_pwc, a MATLAB library which can set up and solve a parameterized steady heat equation in a 2D spatial domain, using diffusivity parameterized by vc, and reporting selected solution values vs.

paraheat_pwc_1d, a MATLAB program which sets up and solves a parameterized steady heat equation in a 1D region, with a piecewise constant diffusivity.

paraheat_pwc_plot, a MATLAB program which calls paraheat_pwc() to set up and solve a parameterized steady heat equation in a 2D spatial domain, with a piecewise constant diffusivity and then uses radial basis functions (RBF) to reconstruct the finite element solution from a set of sample values.

paraheat_pwc_sample, a MATLAB library which repeatedly calls paraheat_pwc() to set up and solve a parameterized steady heat equation in a 2D spatial domain, with a piecewise constant diffusivity, saving values of the finite element solution at selected points, and writing the saved data to a file.