The Classical Separation Of Variables Method

The Classical Separation of Variables Method

The Separation of Variables (SOV) method can only be applied directly to linear homogeneous problems with homogeneous boundary conditions. The idea is to assume that the original function of two variables can be written as a product of two functions, each of which is only dependent upon a single independent.
The separated form of the solution isinserted into the original linear PDE and, after some manipulation, one obtains two homogeneous ODEs that can be solved by traditional means. If the original boundary conditions (BCs) for the problem are homogeneous, one of the ODEs will give a Sturm-Liouville type problem, which leads to a set of orthogonal eigenfunctions as solutions.
Because the original PDE is linear, its final solution isformed as a linear combination of the individual solutions. A final condition imposed on the problem determines the expansion coefficients in the infinite series solution. The analytical solution is complete once the coefficients have been determined.

Treatment of Inhomogeneous Equations and Boundary Conditions
If the the original linear PDE or the BCs have inhomogeneous terms, the classicalSeparation of Variables (SOV) method will not work. Then it is often successful is to break the desired solution into two components - one that is a function of both independent variables and one that is only a function of one of the variables. The original PDE breaks into two problems; a homogeneous PDE with homogeneous boundary conditions (BCs) solved by the SOV method, and an ODE that can be solvedby traditional methods.
As an example, consider the transient heat conduction in a 1-D laterally insulated bar:
[pic]
where Q(x) is an internal heat generation. If, in addition, fixed endpoint temperatures are imposed (i.e. inhomogeneous BCs), we have BCs of the form
[pic]
where L represents the length of the bar and uL and uR are the left and right endpoint temperatures, respectively.Example: Heat Equation 1/4
Finally, specifying some initial temperature distribution,
[pic]
completes the mathematical description of a particular heat transfer problem.
Now, we try separating the desired solution, u(x,t), into two components - the transient solution, v(x,t), and the steady state solution, w(x):
[pic]
With this assumed solution, we have
[pic]
and substitution into eqn. (10.3)gives
[pic]


Example: Heat Equation 2/4
However, since we desire a homogeneous PDE, we let
[pic]
and, with this condition, the remaining PDE for v(x,t) is homogeneous, or
[pic]
We also desire homogeneous boundary conditions for the PDE. Performing similar operations as above, we substitute eqn. (10.6) into the original statement for the BCs [eqn. (10.4)] giving
[pic]
and
[pic]
asdesired.

Example: Heat Equation 3/4
As a last step, eqn. [pic] is also inserted into eqn. [pic] which describes the initial temperature distribution, or
[pic]
which gives the initial condition for the v(x,t) problem as
[pic]
Thus, the conversion process is now complete - the original problem which included an inhomogeneous PDE with inhomogeneous BCs has been converted into two problems, eachof which is straightforward to solve. The transient solution, v(x,t), is completely defined by
[pic]
This problem includes a homogeneous PDE with homogeneous boundary conditions. Therefore, we can use the classical Separation of Variables method to determine v(x,t).

Example: Heat Equation 4/4

The steady state problem is completely defined by
[pic]
which is a relatively simple ODE thatcan be solved by standard methods.
Finally, the solutions to the two separate problems given by eqns. (10.7) and (10.8) are substituted into eqn. (10.6) to give the desired space-time temperature distribution, u(x,t); thus completing the original problem of interest.

The Eigenfunction Expansion Method
Unfortunately, one can not always remove the inhomogeneous terms from the PDE or BCs as...