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Solving a partial differential equation by separation of variables

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Section 9-4 : Separation Of Variables

Okay, it is finally time to at least start discussing one of the more common methods for solving basic partial differential equations. The method of Separation of Variables cannot always be used and even when it can be used it will not always be possible to get much past the first step in the method. However, it can be used to easily solve the 1-D heat equation with no sources, the 1-D wave equation, and the 2-D version of Laplace’s Equation, ∇2u=0∇2u=0.
In order to use the method of separation of variables we must be working with a linear homogenous partial differential equations with linear homogeneous boundary conditions. At this point we’re not going to worry about the initial condition(s) because the solution that we initially get will rarely satisfy the initial condition(s). As we’ll see however there are ways to generate a solution that will satisfy initial condition(s) provided they meet some fairly simple requirements.
The method of separation of variables relies upon the assumption that a function of the form,
u(x,t)=φ(x)G(t)(1)(1)u(x,t)=φ(x)G(t)
will be a solution to a linear homogeneous partial differential equation in xx and tt. This is called a product solution and provided the boundary conditions are also linear and homogeneous this will also satisfy the boundary conditions. However, as noted above this will only rarely satisfy the initial condition, but that is something for us to worry about in the next section.
Now, before we get started on some examples there is probably a question that we should ask at this point and that is : Why? Why did we choose this solution and how do we know that it will work? This seems like a very strange assumption to make. After all there really isn’t any reason to believe that a solution to a partial differential equation will in fact be a product of a function of only xx’s and a function of only tt’s. This seems more like a hope than a good assumption/guess.
Unfortunately, the best answer is that we chose it because it will work. As we’ll see it works because it will reduce our partial differential equation down to two ordinary differential equations and provided we can solve those then we’re in business and the method will allow us to get a solution to the partial differential equations.
So, let’s do a couple of examples to see how this method will reduce a partial differential equation down to two ordinary differential equations.