AdvancedTopic 75 of 104Engineering Mathematics
Separation of variables for PDEs
Assumes product solutions and reduces a PDE to ODEs with eigenvalue conditions.
In depth
Assume u(x,t)=X(x)T(t), substitute, and separate to get two ODEs linked by a constant λ. Boundary conditions make λ discrete (eigenvalues). The full solution is a sum of products found with Fourier coefficients from the initial condition.
Key points
- Works on simple geometries
- Sturm-Liouville theory
- Eigenfunctions are orthogonal
ExampleFor a heated rod, X''+λX=0 with X(0)=X(L)=0 gives X=sin(nπx/L).
Separation of variables turns a PDE into simpler ODEs that can be solved exactly, which works for regular geometries such as rods, plates and cylinders.
What is Separation of variables for PDEs?
Assumes product solutions and reduces a PDE to ODEs with eigenvalue conditions.
Why is it important to understand separation of variables for PDEs?
Separation of variables turns a PDE into simpler ODEs that can be solved exactly, which works for regular geometries such as rods, plates and cylinders.
When does separation of variables work?
It works when the equation and boundary conditions are linear and the boundaries align with coordinate surfaces, such as rectangles, cylinders or spheres.
Why do the solutions form an infinite series?
Each separated eigenfunction satisfies the boundary conditions, and the full solution is a sum of these modes weighted to match the initial state.
How is Separation of variables for PDEs used in practice?
Typical applications include finding the temperature distribution in a long rod with fixed end temperatures, computing the natural modes of a vibrating membrane and solving the potential inside a rectangular box with a voltage applied to one face.
What should I learn before Separation of variables for PDEs?
Study "Wave equation" first (topic 74 of 104 in Engineering Mathematics). It covers the groundwork this topic builds on.
What are the key points of Separation of variables for PDEs?
Works on simple geometries. Sturm-Liouville theory. Eigenfunctions are orthogonal.
Can you give an example of Separation of variables for PDEs?
For a heated rod, X''+λX=0 with X(0)=X(L)=0 gives X=sin(nπx/L).
What comes after Separation of variables for PDEs?
The next topic is "Boundary and initial conditions", which builds directly on this one.
How difficult is Separation of variables for PDEs?
It is classed as Advanced in Engineering Mathematics (topic 75 of 104). It is an advanced topic. It assumes you are comfortable with the intermediate topics in Engineering Mathematics.