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

The finite difference method proceeds by replacing the derivatives in the differential equations with finite difference approximation. This gives a large but finite algebraic system of equations to be solved in place of the differential equations, something that can be done on a computer.

## Wave Equation

In dimensionless form *u*_{xx} = u_{tt}. We use following derivative approximation:

The finite difference scheme for wave equation then becomes

This simplifies to the difference equation:

### Matrix form

In the matrix form numerical solution of the wave equation is a two-dimensional array defined by

The collection of points *x*_{i}, *t*_{j} is called the computation grid. From matrix-grid approximation to the wave equation we obtain the following iteration.

In order to guarantee the stability of algorithm, the condition must be satisfied.

### Getting the iteration started

The initial conditions are used to compute the first iteration:

Now the finite difference approximation can be used to first-order derivative (central difference).

In order to reduce complexity of the problem initial vertical velocity of the string equals to zero, i.e. *G=0*.
A well-defined iteration can be written

## Damped Wave Equation

This simplifies to the difference equation:

### Matrix form

Damping term tends to smooth and stabilize the iterated solution.
In order to guarantee the stability of algorithm, the condition must be satisfied.

### Getting the iteration started

The initial conditions are used to compute the first iteration:

Now the finite difference approximation can be used to first-order derivative (central difference).

A well-defined iteration can be written

## Free Vibration of a Beam

This equation can be approximated by

This simplifies to the difference equation:

The fourth derivative involves the value of u at distances up to spatial units away from the point where the derivative is centered. Now, when the program approaches the end of the string, we need to know the displacement at the end and also the displacement one spatial unit past the end.

#### Pinned Beam

Zero deflection and zero moment at x=0 and x=1.

The simplest way to deal with this problem is to take the displacement at each end to be zero and assume that there are phantom locations one unit beyond the ends, which have displacements that are opposite the displacements at locations one unit inside the ends.

##### In the matrix form

In order to guarantee the stability of algorithm, the condition must be satisfied.
A well-defined iteration can be written