MTH2032 Chap.11 Separation of Variables with Non-Periodic Conditions
Separation of Variables with Non-Periodic Conditions
Week 11 completes separation of variables. When initial data are not already sines or cosines, they are expanded in the eigenfunctions with the Euler formulae, and wave problems with two non-zero initial conditions need two sets of coefficients.
When boundary values are not zero, a heat problem is split into a steady state carrying those values plus a transient with homogeneous conditions, and a Laplace problem with data on every edge is split into two simpler problems. Each worked solution states its coefficient integral in general before reading off cases.
What this chapter covers
- 01
Fourier coefficients for initial data
- 02
Two inhomogeneous initial conditions
- 03
Steady state and transient
- 04
Theorem 28.3 for non-zero boundaries
- 05
Splitting Laplace problems as u_v + u_h
- 06
Checking the boundary of the sum
Worked example · free
Heat in a rod with a parabolic starting profile
- 1Separation with Dirichlet ends gives eigensolutions sin(nπx)e^(−n²π²t).
- 1So u = Σ b_n sin(nπx)e^(−n²π²t) with b_n = 2∫₀¹ x(1 − x) sin nπx dx.
- 1Integrating by parts twice gives b_n = 4(1 − (−1)^n)/(nπ)³.
- 1So b_n = 8/(nπ)³ when n is odd and b_n vanishes when n is even.
- 1Hence u = Σ over odd n of 8 sin(nπx)e^(−n²π²t)/(nπ)³.
Key terms
- Homogenising Boundary Data
- Subtracting a function that carries the boundary values so the remainder has zero boundary conditions.
- Steady-State Shift
- The decomposition of an evolution problem with non-zero boundary data into its steady state plus a transient with zero boundary values.
- Coefficient Integral
- The integral that matches a series of eigenfunctions to a given initial or boundary function.
- Inhomogeneous Condition
- A boundary or initial condition containing a term that does not involve the unknown function.
- Shifted Initial Data
- The initial profile minus the steady state, which is the data the transient part of a heat problem must match.
Separation of Variables with Non-Periodic Conditions FAQ
How do I handle a heat problem whose ends are held at different temperatures?
Find the steady state first: for the plain heat equation it is the straight line joining the two end temperatures. Subtract it from u. The remainder satisfies the heat equation with zero boundary values and a shifted initial profile, so ordinary separation of variables applies. Add the steady state back at the end.
What changes when a wave problem has two non-zero initial conditions?
Each eigensolution then contains both cos and sin in time, so there are two sets of constants. Fit the initial displacement to the cosine coefficients and the initial velocity to the sine coefficients, remembering that differentiating in time brings down a factor nπc/L that must be divided out.
How do I solve Laplace's equation with data on all four sides?
Split the solution into two parts that each satisfy Laplace's equation. One keeps the data on the vertical edges and is zero on the horizontal ones; the other does the opposite. Each part then has two homogeneous opposite edges and separates as in Week 9, and adding them restores all four boundary values.
How many terms of a series should I write in an answer?
Write the general term with its coefficient formula and state which values of n contribute, then simplify. Listing two or three terms can help a reader but is not a substitute for the general term. If a question asks for a numerical value, say how many terms you used and why they suffice.
Exam move
Put Week 9 and Week 10 together on full problems. For each separation question, first list the conditions and mark which are homogeneous. If two opposite boundaries are homogeneous, separate directly; if a heat problem has non-zero boundary values, find the steady state and shift the problem; if a Laplace problem has data on every side, split it.
Then write the eigensolutions and compute the coefficient integrals in general form before reading off odd and even cases. Practise at least one problem of each kind: a wave with a polynomial initial displacement, a wave with both displacement and velocity given, a heat problem with unequal fixed end temperatures, and a Laplace problem with one polynomial edge.
After each, check that your formula meets every condition, including the corners for Laplace. Finally, time yourself: the final emphasises this material, so a complete, checked solution in about twenty minutes is a good target, with the coefficient integral as the step to practise most.
When time is short, make sure you can at least set up the decomposition and write the coefficient integral correctly, as these steps carry much of the credit.
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