MAST20029 · Engineering Mathematics
Second-Order PDEs and Separation of Variables
The final section classifies second-order PDEs by the sign of B² − 4AC — elliptic (Laplace), parabolic (diffusion/heat) or hyperbolic (wave) — and solves them on a rectangle by separation of variables. You write u = X(x)Y(y), split into two ODEs sharing a separation constant, apply the homogeneous boundary conditions to find the eigenvalues, superpose, and match the inhomogeneous boundary data by Fourier coefficients. This is the signature long question (Q11, 16–18 marks).
What this chapter covers
- 011. Classification by B² − 4AC: elliptic (<0), parabolic (=0), hyperbolic (>0)
- 022. The three model equations: Laplace u_xx+u_yy=0, wave u_tt=c²u_xx, diffusion u_t=k u_xx
- 033. Separation u(x,y) = X(x)Y(y) and the separation constant λ
- 044. The two resulting ODEs (e.g. X''+λX=0, Y''−λY=0 for Laplace)
- 055. Homogeneous boundary conditions fixing the admissible eigenvalues λₙ
- 066. The cases λ>0, λ=0, λ<0 and discarding trivial ones
- 077. Superposition u = Σ cₙ Xₙ Yₙ over the eigenfunctions
- 088. Matching the inhomogeneous boundary condition via Fourier coefficients
Laplace's equation on a square by separation of variables
- 1 markSeparate u = X(x)Y(y): X''/X = −Y''/Y = −λ, giving X'' + λX = 0 and Y'' − λY = 0.
- 2 marksApply the homogeneous BCs on X: X(0) = X(1) = 0 forces the eigenvalues λₙ = (nπ)² with Xₙ = sin(nπx) (the λ ≤ 0 cases give only the trivial solution).
- 1 markSolve Y with the homogeneous BC Y(0) = 0: Y'' − (nπ)²Y = 0 gives Yₙ = sinh(nπy), chosen to vanish at y = 0.
- 1 markSuperpose: u(x,y) = Σ (n≥1) cₙ sin(nπx) sinh(nπy).
- 2 marksFit the inhomogeneous BC u(x,1): Σ cₙ sinh(nπ) sin(nπx) = sin(2πx) + 3 sin(5πx). Match coefficients: c₂ sinh(2π) = 1, c₅ sinh(5π) = 3, all other cₙ = 0.
- 1 markSolution: u(x,y) = [sin(2πx) sinh(2πy)]/sinh(2π) + 3[sin(5πx) sinh(5πy)]/sinh(5π).
Key terms
- PDE classification
- A second-order PDE A u_xx + B u_xy + C u_yy + ... = 0 is elliptic if B²−4AC < 0 (Laplace), parabolic if = 0 (diffusion), hyperbolic if > 0 (wave).
- Model equations
- Laplace's equation u_xx + u_yy = 0 (steady state), the wave equation u_tt = c² u_xx (vibration), and the diffusion/heat equation u_t = k u_xx (spreading) — the three canonical types.
- Separation of variables
- Seeking u(x,y) = X(x)Y(y), substituting, and dividing so each side depends on one variable; both must equal a constant, the separation constant λ, giving two ODEs.
- Separation constant λ
- The shared constant from separation; the homogeneous boundary conditions restrict it to a discrete set of eigenvalues λₙ with corresponding eigenfunctions Xₙ.
- Eigenvalue cases
- One must examine λ > 0, λ = 0 and λ < 0; usually only one sign yields non-trivial solutions satisfying the BCs (e.g. λ = (nπ)² for X(0)=X(1)=0).
- Superposition and matching
- Summing the eigen-solutions u = Σ cₙ Xₙ Yₙ and fixing the coefficients cₙ from the inhomogeneous boundary condition, by direct matching or Fourier coefficients.
Second-Order PDEs and Separation of Variables FAQ
How do I classify a second-order PDE?
Compute B² − 4AC from the coefficients of u_xx, u_xy and u_yy. If it is negative the PDE is elliptic (like Laplace's equation, steady states); if zero it is parabolic (like the diffusion/heat equation); if positive it is hyperbolic (like the wave equation). The class tells you the physical character and the appropriate boundary/initial data.
What are the steps of separation of variables?
Write u = X(x)Y(y), substitute into the PDE, and rearrange so one side depends only on x and the other only on y; both equal a constant λ. Solve the two ODEs, apply the homogeneous boundary conditions to find the eigenvalues λₙ and eigenfunctions, superpose them as u = Σ cₙ Xₙ Yₙ, and finally match the inhomogeneous condition to fix the cₙ.
Why do I have to consider λ > 0, λ = 0 and λ < 0?
Each sign of the separation constant gives a different form of solution to the ODE (oscillatory, linear, or exponential). The homogeneous boundary conditions usually kill all but one case — for X(0) = X(1) = 0, only λ = (nπ)² > 0 gives non-trivial sine eigenfunctions. You must show the other cases give only the trivial solution.
How do I find the coefficients cₙ?
Impose the inhomogeneous boundary condition on the superposed solution. If that data is a finite sum of the eigenfunctions (a few sine terms), match coefficients directly. If it is a general function, expand it as a Fourier series over the eigenfunctions and read off each cₙ from the corresponding Fourier coefficient (dividing by the sinh or other factor).
Exam move
Approach the Q11 PDE question as a fixed pipeline: classify by B²−4AC, separate into two ODEs with constant λ, apply the homogeneous boundary conditions to extract the eigenvalues and eigenfunctions (showing the trivial cases die), superpose, then match the inhomogeneous data. Drill the eigenvalue-case analysis so you can quickly discard λ ≤ 0 when the BCs demand sines. Pick the second factor (e.g. sinh) so it already satisfies its homogeneous BC, simplifying the matching. Practise both the finite-sum matching and the full Fourier-coefficient route, and lay the work out in clear stages — this long question carries the most marks of any on the paper.
