Mathematica Package: ChebyshevSolver

This is a package for using the Pseudospectral method and a Chebyshev grid. I developed it during my PhD time at the University of Alabama.

Author: Marco Knipfer
Institution: University of Alabama
Year: 2020

Features and Restrictions

The package still has many restrictions, but it's enough for now

• solve first and second order ODEs for some intervall [x0,x1]
• boundary conditions:
  • have to be at boundary of intervall [x0,x1]
  • first order: one Dirichlet boundary condition or one Neumann boundary condition
  • second order:
    • two Dirichlet boundary condition
    • or one Dirichlet and one Neumann boundary condition
    • two Neumann boundary conditions not implemented!

Usage

The idea is that one can just replace NDSolve with ChebyNDSolve. Note that none of the options for NDSolve are implemented. For a working example see the file Tests/UseChebyshevSolver.wls

• Import the package: <<"pathToPackage`"
• Set up differential equation, an intervall [x0, x1] in which the DEQ is to be solved and boundary conditions (for second order: {bc1, bc2}. Currently the conditions have to be on the boundary of the intervall (at x0 or x1).

Example: Sin

DEQ =  f[x] + f''[x] == 0 ;
x0 = 0.;
x1 = 2 Pi;
bc1 = f[x0] == 0.;
bc2 = f'[x0] == 1.;
chebPoints = 50;

sol = ChebyshevSolver`ChebyNDSolve[{DEQ, bc1, bc2}, f, {x,x0,x1}, "GridPoints"->chebPoints];
plot = Plot[sol[x], {x,x0,x1}]

Some Plots

ToDo

• Setting of number of digits to keep in whole calculation
• boundary conditions at inner points