Partial Differential Equations

PDEs are use to model many kinds of problems. Their solutions give evolution of a function as a function of time and space. Boundary conditions involving time and space are used as initial conditions.

A method of separation of variables is used for solving them, where it is assumed that . Two other auxiliary ODE results are also needed:

Another auxillary ODE are needed for some situations

The general process for solving PDEs:

  • Apply separation of variables
  • Make an appropriate choice of constant
    • Nearly always
  • Solve resulting ODEs
  • Combine ODE solutions to form general PDE solution
  • Apply boundary conditions to obtain particular PDE solution
    • Work out values for the arbitrary constants

Laplace's Equation

Laplace's equation described many problems involving flow in a plane:

Find the solution with the following boundary conditions:

  • and
  • as

Starting with separation of variables:

Substituting back into the original PDE:

We have transformed the PDE into an ODE, where each side is a function of / only. The only circumstances under which the two sides can be equal for all values of and is if both sides independent and equal to a constant. Since the constant is arbitrary, let it be . Now we have two ODEs and their solutions from the auxiliary results earlier:

Substituting the solutions back into , we have a general solution to our PDE in terms of 4 arbitrary constants:

We can now apply boundary conditions:

  • Substituting in gives
  • Substituting in gives
  • Using the two together gives , so either:
  • If , then , so
    • This is the trivial solution and is of no interest
  • If , then
    • This also implies that , so is useless too

The issue is that we selected our arbitrary constant badly. If we use instead, then our solutions are the other way round:

Checking the boundary conditions again:

  • First condition,
    • Gives
  • Second condition
    • Gives
      • Either (not interested)
    • is an integer,

We now have:

Where is any integer. Using the other boundary conditions:

  • as
  • If is positive, then (otherwise )
  • If is negative, then (otherwise )

Taking as positive, the form of the solutions is:

The most general form is the sum of these:

Applying the final boundary condition:

  • for all other

The complete solution is therefore:

The Heat Equation

The heat equation describe diffusion of energy or matter. With a diffusion coefficient :

Solving with the following boundary conditions:

Separating variables, , and substituting, exactly the same as Laplace's equation, we have:

Setting both sides again equal to a constant , we have two ODEs (one 2nd order, one 1st):

The general solution is therefore:

Tidying up a bit, let , , :

Applying the first boundary condition:

  • Gives
  • Since for all ,

We now have . The second boundary condition:

  • , so
  • For the non trivial solution ,and since ,
  • Therefore, for

Substituting this in gives:

The above equation is valid for any , so summing these gives the most general solution:

The last boundary condition is :

This is in the form of the a Fourier series:

We have:

Substituting this into , and letting :

The Wave Equation

The wave equation is used to describe vibrational problems:

Solving the equation with the boundary conditions:

Doing the usual separation of variables and substitution, and choosing a constant :

Solving both ODEs:

This is the general solution. Start applying boundary conditions:

  • implies that
    • As this is true for all ,
  • implies that
    • This is also true for all , so
    • Required that , so
    • for

We now have:

Applying the third boundary condition, :

As this is for all , , so . We now have:

The general solution is then:

Applying the final boundary condition of , gives , else . The particular solution is therefore: