MTH2032 Chap.9 Separation of Variables with Homogeneous Boundaries
Separation of Variables with Homogeneous Boundaries
Week 9 introduces separation of variables for the wave, heat and Laplace equations when two boundary conditions are homogeneous and the remaining data are already sines or cosines. Assuming a product solution splits the PDE into two ODEs linked by a separation constant.
The spatial ODE with its boundary conditions is an eigenvalue problem, the other factor oscillates, decays or grows like cosh and sinh, and superposition assembles the eigensolutions into the answer. Every example finishes with a check of all boundary and initial conditions.
What this chapter covers
- 01
When separation of variables applies
- 02
Product solutions and the separation constant
- 03
Spatial eigenvalue problem
- 04
Time factors for wave and heat
- 05
Laplace equation on a rectangle
- 06
Superposition with boundary conditions
Worked example · free
A string released from a sine-shaped displacement
- 1Separate u = XT: T″/(9T) = X″/X = K, with X(0) = X(1) = 0 and T′(0) = 0.
- 1The eigenvalue problem gives K_n = −(nπ)² and X_n = sin nπx.
- 1Then T″ + 9n²π²T = 0 with T′(0) = 0 gives T_n = cos 3nπt.
- 1Superpose u = Σ a_n sin(nπx) cos(3nπt); the data need only n = 1 with a_1 = 2.
- 1So u = 2 sin(πx) cos(3πt), which satisfies every condition.
Key terms
- Separation Constant
- The common constant K that both sides of a separated equation must equal.
- Eigensolution
- A product X_n T_n that satisfies the PDE and all its homogeneous conditions.
- Dirichlet Condition
- A boundary condition that fixes the value of the solution on a boundary.
- Neumann Condition
- A boundary condition that fixes the normal derivative of the solution, such as zero heat flow through an insulated end.
Separation of Variables with Homogeneous Boundaries FAQ
Why can I set both sides of the separated equation equal to one constant?
After separation the left side depends only on t and the right side only on x, yet they are equal for every x and t. Changing t alone cannot change the right side, so the left side must be constant, and likewise the other way round. That shared constant is the separation constant K.
Where do the boundary conditions go in separation of variables?
Homogeneous conditions are converted into conditions on the single-variable factors at the start, for example u(0, t) = 0 becomes X(0) = 0. Those conditions create the eigenvalue problem. Non-homogeneous conditions are imposed only at the end, after the eigensolutions have been added together by superposition.
Why does heat decay while waves oscillate?
The heat equation is first order in time, so each mode satisfies T′ = DKT and decays exponentially when K is negative. The wave equation is second order in time, so T″ = c²KT and each mode oscillates as a cosine or sine. The spatial eigenfunctions are the same in both problems.
What if K = 0 gives a non-zero solution?
That happens with Neumann conditions at both ends, where a constant satisfies X′ = 0 at each end. The constant is a genuine eigenfunction, its time factor is constant for heat and linear or constant for waves, and it usually produces the constant term in the final series. Leaving it out is a common way to lose marks.
Exam move
Work through the eight-step algorithm until you can write it without notes, then apply it to one wave, one heat and one Laplace problem with each combination of Dirichlet and Neumann ends. Each time, write the separation step with the constant on the side that keeps the spatial problem familiar, and list which conditions are homogeneous before you start.
For the eigenvalue problem, reuse your Week 5 working, writing the three cases separately or using the −ω² shortcut with its explanation. For time factors, check the order in time: one condition for heat, two for waves, with the homogeneous one used early. For Laplace, decide first which pair of opposite edges is homogeneous, because that variable carries the sines and the other carries cosh and sinh.
Finish every problem by checking all boundary and initial conditions against your formula. Because the final examination emphasises Part II, aim to complete a full separation problem in about fifteen minutes under timed conditions. Keep one page of eigenvalue results for the four combinations of Dirichlet and Neumann ends, since these are the building blocks reused in Week 11.
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