# fd1d_bvp

fd1d_bvp, a Python code which applies the finite difference method (FDM) to solve a two point boundary value problem (BVP) in one spatial dimension.

The boundary value problem (BVP) that is to be solved has the form:

```        - d/dx ( a(x) * du/dx ) + c(x) * u(x) = f(x)
```
in the interval X(1) < x < X(N). The functions a(x), c(x), and f(x) are given functions, and a formula for a'(x) is also available.

Boundary conditions are applied at the endpoints, and in this case, these are assumed to have the form:

```        u(X(1)) = 0.0;
u(X(N)) = 0.0.
```

To compute a finite difference approximation, a set of n nodes is defined over the interval, and, at each interior node, a discretized version of the BVP is written, with u''(x) and u'(x) approximated by central differences.

### Usage:

u = fd1d_bvp ( n, a, aprime, c, f, x )
where
• n the number of nodes.
• a the function which evaluates a(x);
• aprime the function which evaluates a'(x);
• c the function which evaluates c(x);
• f the function which evaluates f(x).
• x the vector of n nodes, which may be nonuniformly spaced.
• u the output vector of n finite difference values.

### Licensing:

The computer code and data files described and made available on this web page are distributed under the MIT license

### Languages:

fd1d_bvp is available in a C version and a C++ version and a FORTRAN90 version and a MATLAB version and a Python version.

### Related Data and codes:

fd1d_advection_lax_wendroff, a Python code which applies the finite difference method (FDM) to solve the time-dependent advection equation ut = - c * ux in one spatial dimension, with a constant velocity, using the Lax-Wendroff method to treat the time derivative.

fd1d_heat_explicit, a Python code which uses the finite difference method (FDM) and explicit time stepping to solve the time dependent heat equation in 1D.

fd1d_heat_implicit, a Python code which uses the finite difference method (FDM) and implicit time stepping to solve the time dependent heat equation in 1D.

### Reference:

1. Dianne O'Leary,
Finite Differences and Finite Elements: Getting to Know You,
Computing in Science and Engineering,
Volume 7, Number 3, May/June 2005.
2. Dianne O'Leary,
Scientific Computing with Case Studies,
SIAM, 2008,
ISBN13: 978-0-898716-66-5,
LC: QA401.O44.
3. Hans Rudolf Schwarz,
Finite Element Methods,
Academic Press, 1988,
ISBN: 0126330107,
LC: TA347.F5.S3313..
4. Gilbert Strang, George Fix,
An Analysis of the Finite Element Method,
Cambridge, 1973,
ISBN: 096140888X,
LC: TA335.S77.
5. Olgierd Zienkiewicz,
The Finite Element Method,
Sixth Edition,
Butterworth-Heinemann, 2005,
ISBN: 0750663200,
LC: TA640.2.Z54

### Source Code:

Last revised on 26 May 2019.