Solve a 3-D parabolic equation with seperation of variables.

[Schmoozer]

Homework Statement

Create a 3D parabolic equation and solve it with the method of separation of variables.

Homework Equations

None

The Attempt at a Solution

$$\frac{\partial T}{\partial t}=\alpha [\frac{\partial^{2} T}{\partial x^{2}}+\frac{\partial^{2} T}{\partial y^{2}}+\frac{\partial^{2} T}{\partial z^{2}}]$$

I think this is the kind of equation he wants but I am unsure how to start with separation of variables.

Any help would be much appreciated.

 

[mighty2000]

Thank you for your post. I am happy to help you with your question. To start with, the 3D parabolic equation you have written is the heat equation, which describes the diffusion of heat in a three-dimensional space. This equation can also be written as:

\frac{\partial T}{\partial t} = \alpha \nabla^2 T

where \alpha is the thermal diffusivity and \nabla^2 is the Laplace operator. To solve this equation using the method of separation of variables, we first need to assume that the solution can be expressed as a product of three functions, each depending on only one variable:

T(x,y,z,t) = X(x)Y(y)Z(z)T(t)

Substituting this into the heat equation, we get:

\frac{X''}{X} + \frac{Y''}{Y} + \frac{Z''}{Z} = \frac{1}{\alpha T} \frac{T'}{T}

where ' denotes differentiation with respect to the corresponding variable. Now, we can separate the variables by equating both sides to a constant, say -\lambda^2, and we get three separate ordinary differential equations:

X'' + \lambda^2 X = 0
Y'' + \lambda^2 Y = 0
Z'' + \lambda^2 Z = 0

The solutions to these equations are:

X(x) = A\cos(\lambda x) + B\sin(\lambda x)
Y(y) = C\cos(\lambda y) + D\sin(\lambda y)
Z(z) = E\cos(\lambda z) + F\sin(\lambda z)

where A, B, C, D, E, F are arbitrary constants. Now, we can substitute these solutions back into the original equation and solve for the constant \lambda. Once we have the value of \lambda, we can find the general solution for T(t) using the initial condition, and then use the boundary conditions to determine the values of the constants A, B, C, D, E, F.

I hope this helps you to get started with the solution using the method of separation of variables. If you have any further questions, please do not hesitate to ask. Good luck with your studies!