**Numerical Analysis of PDEs: A Finite Difference Method for the Wave Equation, Damped Wave Equation and Free Vibration of Beam**
# Numerical results

Some simulation results will be presented.
Other results can be computed using MaimGui.m
In all cases the plucked string was analyzed 10 percent off the center at x=0.4

## Wave Equation

Plucked String

*dx=0.001, dt=0.001, T*_{final}=10

The method is very accurate for this case as can be seen in the next figure. Only after zoom it is possible to see difference between two lines. The exact solution is green line and Finite Difference results are blue line.

From the next figure can be seen that maximum error is 0.0015.

`Error = FD method - Exact Solution.`

The next figures show the same problem with sampling decreased 10 times:

*dx=0.01, dt=0.01, T*_{final}=10

In this case the method is accurate enough, as can be seen in the next two figures. The exact solution is green line and Finite Difference results are blue line.

The maximum error is 0.015. The result is less accurate, as expected.

## Damped Wave Equation

Plucked String

*dx=0.001, dt=0.001, T*_{final}=10

The method is very accurate for this case as can be seen in the next figures. Only after zoom it is possible to see difference between two lines. The exact solution is green line and Finite Difference results are blue line.

### Effect of different damping coefficients

In the case of damped wave equation higher frequencies die out faster than lower harmonics. As can be seen the oscillations depend upon the amount of damping the greater damping causing the oscillations to die out sooner. The height and broadness of the peak in frequency domain indicate the amount of damping.

*dx=0.01, dt=0.01, T*_{final}=10