Solution of the Wave Eqn.

The ODE’s for the wave equation were solved for the following boundary conditions:

The spatial boundary conditions represented by the partials can be converted into ODE’s using the forward and backward difference formulations:

  

The ODE’s were solved using the Rosenbrock method with the GRK4T coefficients, and the result is plotted below.

The Wave Equation

The PDE that defines the wave equation falls under the class of Hyperbolic partial differential equations.  The equation is:

 

 

 
This 2^nd order PDE with the boundary value conditions can be numerically solved by first converting it into two 1^st order PDE’s, described by:

 

 

 

The two PDE’s can then be converted into ODE’s using the Method of Lines:

 

 

By dividing the spatial domain into N elements over the total length L, the 2^nd order PDE has now reduced to a set of ODE’s that can be quite easily integrated numerically.