Fourier method for wave equation
by Aleksas Domarkas
Vilnius University, Faculty of Mathematics and Informatics,
Naugarduko 24, Vilnius, Lithuania
aleksas@ieva.mif.vu.lt
NOTE: In this session we find solutions of boundary value problems for wave equation using Fourier method.
1 Example
Problem
Solving eigenvalue problem
Eigenvalues and eigenfunctions:
Solving problem
Solution of this problem is sought in in the form:
For coefficients T[k](t) we have problem:
Solution
Checking the Solution
2 Example
Vlad. 20.14.1
Warning, the protected names norm and trace have been redefined and unprotected
3 Example
Particular solution
We find particular solution of problem:
We make change: u(x,t)=v(x,t)+psol. Then for v(x,t) we have problem:
4 Example
5 Example
[vlad] 20.16.7.
Solving
With not Fourier method . sol= sol1+sol2 .
6 Example
Membrane Problem
Solution representation formula
where - Zeros of Bessel J(0,x) function
Zeros of Bessel J(0,x) function
Note: see ?BesselJZeros
Plots
3d animation:
Other authors about Membrame Problem
See:
http://www.mapleapps.com/categories/engineering/engmathematics/html/K11.8-membrane.html by Alain Goriely:
or
Study of Membrane Oscillation Using Maple
by D. Frenkel , L. Golebiowski and R. Portugal
While every effort has been made to validate the solutions in this worksheet, Waterloo Maple Inc. and the conevibutors are not responsible for any errors contained and are not liable for any damages resulting from the use of this material.
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