Background for Hyperbolic Equations

Wave Equation

    As an example of a hyperbolic partial differential equation, we consider the wave equation  

           
        
for    0 < x < a   and   0 < t < b,  with the boundary conditions  

        

The wave equation models the displacement u of a vibrating elastic string with fixed ends at  x=0  and  x=a.  Although analytic solutions to the wave equation can be obtained with Fourier series, we use the problem as a prototype of a hyperbolic
equation.

Proof  Finite Difference Method for PDE's  Finite Difference Method for PDE's  

Computer Programs  Finite Difference Method for PDE's  Finite Difference Method for PDE's  

Program (Finite-Difference Solution for the Wave Equation)  To approximate the solution of the wave equation    over the rectangle    with  ,  for  ,  and  ,  for  .  

Example 1.  Consider the wave equation where  .  The string at rest has length  .  Assume that the initial position is

        .  

Use the finite difference method to solve the wave equation over the rectangle  .  Compare the solution with the exact solution:  

        .  

Solution 1.

Example 2.  Consider the wave equation where  .  The string at rest has length  .  Assume that the initial position

          

Use the finite difference method to solve the wave equation over the rectangle  .  
Solution 2.

Example 3.  Consider the wave equation where  .  The string at rest has length  .  Assume that the initial position

          

Use the finite difference method to solve the wave equation over the rectangle  .  
Solution 3.

Research Experience for Undergraduates

Finite Difference Method for PDE's  Finite Difference Method for PDE's  Internet hyperlinks to web sites and a bibliography of articles.  

Download this Mathematica Notebook Hyperbolic P.D.E's

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(c) John H. Mathews 2004