Math for ME · Chapter 16 of 19 · Advanced
Partial Differential Equations
Heat spreading through a wall, a string vibrating, a steady temperature field: when a quantity changes across both space and time, it obeys a partial differential equation. This is the math beneath heat transfer, vibrations, and every field solver.
The thread: Until now things changed in time alone. Heat in a wall and a vibrating string change across space and time together, which is what a partial differential equation captures.
Readiness check
From Multivariable Calculus, ODEs, and Fourier Series. Tick only what you can do closed-notes.
- Take partial derivatives and say what is held constant.
- Solve a separable and a second-order linear ODE.
- Read A sin(nπx/L) as a spatial mode shape.
- Explain a Fourier series as a sum of sine and cosine modes.
- State a time constant from a decay rate.
The core idea
A PDE relates a field to its rates of change in space and time. The boundary and initial conditions select the one real solution.
∂T/∂t = α ∂²T/∂x²∂²y/∂t² = c² ∂²y/∂x²Three equations cover most of mechanical engineering: the heat equation (left) smooths and decays, the wave equation (right) travels and oscillates, and Laplace's equation describes a steady field at rest. A PDE on its own has infinitely many solutions; the conditions on the boundary and the starting state are what make it a real engineering problem.
The skills, taught in order
16.1 What a PDE is
A partial differential equation relates an unknown field, such as T(x, t), to its partial derivatives in more than one independent variable. Because it lives over a region of space and (often) time, it needs conditions all around that region's boundary, plus a starting state for time problems, before it has a single answer.
16.2 The three canonical equations
Almost every PDE a mechanical engineer meets is one of three, and each has a distinct character.
| Equation | Form | Behaviour | Where it appears |
|---|---|---|---|
| Heat / diffusion | ∂T/∂t = α ∂²T/∂x² | smooths out, decays to steady | transient conduction |
| Wave | ∂²y/∂t² = c² ∂²y/∂x² | travels and oscillates | vibrating strings, beams, acoustics |
| Laplace | ∂²φ/∂x² + ∂²φ/∂y² = 0 | steady, smooth equilibrium | steady temperature, potential flow |
16.3 Boundary and initial conditions
Conditions are what turn a PDE into a real problem. A Dirichlet condition fixes the value on a boundary (an end held at 0 °C); a Neumann condition fixes the gradient (an insulated end, where ∂T/∂x = 0). Time-dependent problems also need an initial condition, the full field at t = 0. The boundary conditions set the allowed spatial shapes; the initial condition sets how much of each is present.
16.4 Separation of variables: the main hand method
The workhorse technique assumes the solution is a product of a space part and a time part:
T(x, t) = X(x) · G(t)Substituting splits the PDE into two ordinary differential equations, one in x and one in t, linked by a shared constant. The spatial ODE plus the boundary conditions is an eigenvalue problem (Eigenvalues and Modes): it admits only certain mode shapes, Xn = sin(nπx/L), each with its own time behaviour. Summing the modes with a Fourier series (Fourier Series) then matches any initial condition.
16.5 Classification and what it predicts
The three types, parabolic (heat), hyperbolic (wave), and elliptic (Laplace), are not just labels: the type tells you the physics. Parabolic problems erase detail over time, hyperbolic ones carry it along at finite speed, and elliptic ones have no time at all, only a balanced steady state. Knowing the type tells you what answer to expect before you compute it.
Engineering connection: Heat Transfer, Vibrations, Fluid Mechanics, FEA and CFD.
Worked example: how fast does a hot rod even out?
A thin rod of length L = 0.5 m has both ends held at 0 °C. Its initial temperature is the single mode T(x, 0) = 80 sin(πx/L) °C, and its thermal diffusivity is α = 1.2 × 10⁻⁵ m²/s. Find T(x, t) and the time for the midpoint to cool from 80 °C to 40 °C.
- ProblemSolve the heat equation for the rod in Figure 1 and find the half-cooling time at the midpoint.
- Given / findL = 0.5 m, α = 1.2 × 10⁻⁵ m²/s, T(x,0) = 80 sin(πx/L), ends at 0 °C. Find T(x, t) and the time for T(L/2) to reach 40 °C.
- AssumptionsOne-dimensional conduction, constant α, ends held exactly at 0 °C.
- ModelSeparation of variables gives modes sin(nπx/L) e−α(nπ/L)²t. The initial condition is already a single mode (n = 1), so only that one term survives.
- EquationsT(x, t) = 80 sin(πx/L) e−λt λ = α(π/L)²
- Decay rateλ = 1.2 × 10⁻⁵ × (π/0.5)² = 1.2 × 10⁻⁵ × 39.5 = 4.74 × 10⁻⁴ s⁻¹, a time constant of 1/λ = 2110 s.
- Half-cooling timeAt the midpoint sin(π/2) = 1, so T(L/2, t) = 80 e−λt. Set it to 40: e−λt = 0.5, so t = ln 2 / λ = 0.693 / 4.74 × 10⁻⁴ = 1463 s ≈ 24 min.
- CheckUnits: α(1/L)² has units (m²/s)(1/m²) = 1/s, correct for a rate. The shape never changes, only its height, which is exactly what a single separated mode does. A finer initial wiggle (n = 2) would carry λ four times larger and vanish in a quarter of the time.
- ConclusionA PDE became one ODE in time once the spatial shape was fixed by the boundaries. The n² in the decay rate is why sharp temperature features smooth out almost instantly while the broad profile lingers, the defining behaviour of diffusion.
Worked example 2: the note a string plays
A steel string of length L = 0.65 m is stretched to a tension T = 80 N and has a linear density μ = 5.0 g/m. Using the wave equation, find the wave speed and the fundamental frequency, and write the first three natural frequencies.
- Given / findL = 0.65 m, T = 80 N, μ = 0.0050 kg/m. Find the wave speed c, the fundamental f₁, and f₁, f₂, f₃.
- ModelThe string obeys the wave equation ∂²y/∂t² = c² ∂²y/∂x² with wave speed c = √(T/μ). Fixed at both ends, it supports standing waves y = sin(nπx/L) cos(ωt).
- Wave speedc = √(T/μ) = √(80/0.0050) = √16 000 = 126.5 m/s.
- Mode frequenciesThe boundary conditions allow only half-wavelengths that fit the length, giving fn = n c / (2L).
- Fundamentalf₁ = 126.5 / (2 × 0.65) = 126.5 / 1.30 = 97.3 Hz.
- Harmonicsf₂ = 2 × 97.3 = 194.6 Hz; f₃ = 3 × 97.3 = 291.9 Hz: integer multiples of the fundamental.
- CheckUnits: √(N / (kg/m)) = √((kg·m/s²)/(kg/m)) = √(m²/s²) = m/s, correct. Tightening the string (larger T) raises c and every frequency, exactly how a musician tunes up.
- ConclusionThe same separation-of-variables idea that decayed the heat mode here makes the wave mode oscillate instead, because the wave equation has a second time derivative. The boundary conditions, not the material, set which frequencies are allowed.
Misconceptions and diagnostics
| Mistake | Symptom | Diagnostic question | Correction |
|---|---|---|---|
| Forgetting boundary or initial conditions | A solution with leftover free constants | "Have I pinned the value all around the boundary and the state at t = 0?" | The boundary conditions fix the spatial shapes; the initial condition fixes how much of each. Without both, there is no single answer. |
| Wrong equation type for the physics | Oscillation predicted for a cooling problem | "Does this field settle, travel, or rest?" | Heat smooths and decays, the wave equation travels and oscillates, Laplace just rests. Match the type to the behaviour. |
| Separation of variables forced on the wrong problem | A product solution that cannot meet the boundaries | "Are the boundary conditions homogeneous and the domain simple?" | Separation needs clean boundaries on a regular shape. Otherwise discretise and use FEA or CFD. |
| Treating every mode as equal | Expecting all wiggles to fade at one rate | "How does the decay rate depend on the mode number?" | Heat modes decay like e−α(nπ/L)²t: the n² means fine features vanish far faster than broad ones. |
Practice ladder
Classify each equation: (a) ∂u/∂t = α ∂²u/∂x² and (b) ∂²u/∂t² = c² ∂²u/∂x².
Show answer
(a) is the heat (diffusion) equation, parabolic: it smooths and decays. (b) is the wave equation, hyperbolic: it propagates and oscillates.
Then reduce the 1D heat equation to its steady state (∂T/∂t = 0) and give the temperature profile in a wall whose faces are held at T₁ and T₂.
Show answer
Steady state gives d²T/dx² = 0, so T is linear: T(x) = T₁ + (T₂ − T₁)x/L. Steady conduction through a plane wall is just a straight line.
A string fixed at both ends has L = 1.0 m and wave speed c = 200 m/s. Find its first three natural frequencies.
Show answer
fn = n c / (2L) = n × 100 Hz, so 100, 200, and 300 Hz.
For a heat-equation rod, how does the decay rate of the n = 3 mode compare with the fundamental?
Show answer
The rate scales as n², so the n = 3 mode decays 9 times faster than the n = 1 mode. Sharp detail disappears almost at once.
A rod of length L = 0.4 m with α = 1.0 × 10⁻⁵ m²/s starts in the single mode sin(πx/L). Find the time constant of its decay.
Show answer
λ = α(π/L)² = 1.0 × 10⁻⁵ × (π/0.4)² = 1.0 × 10⁻⁵ × 61.7 = 6.17 × 10⁻⁴ s⁻¹, so τ = 1/λ = 1621 s, about 27 minutes.
Pick a real object that is heating, cooling, or vibrating (a quenched part, a heat sink, a guitar string, a vibrating panel). Name which canonical PDE governs it, state realistic boundary and initial conditions, and predict from the type alone whether the field will settle, travel, or oscillate.
What good work looks like
The correct equation named, boundary conditions identified as fixed-value or insulated, an initial state described, and a behaviour prediction (decay, oscillation, or steady) justified by the equation type rather than by intuition.
Working with AI, and proving it yourself
Use AI as an examiner, not a solver
Portfolio task
Write a one-page "Three Equations" card: for the heat, wave, and Laplace equations give the form, the physical behaviour, one mechanical engineering use, and one worked mode you computed yourself (the rod's decay and the string's frequency both count).
Retrieval and spaced review
Closed notes. Answer out loud, then reveal.
1. What does a PDE need that an ODE does not, in spirit?
Several independent variables, and therefore conditions around the whole spatial boundary, plus an initial state for time problems.
2. Name the three canonical PDEs and one mechanical engineering use of each.
Heat or diffusion (transient conduction), wave (a vibrating string or beam), Laplace (steady temperature or potential flow).
3. Outline separation of variables in three steps.
Assume a product X(x)·G(t); substitute to split into a space ODE (an eigenvalue problem set by the boundaries) and a time ODE; sum the modes with a Fourier series to match the initial condition.
4. How fast does the nth heat mode decay relative to the first?
As n²: the n = 3 mode decays 9 times faster. Fine features vanish first.
5. For a string of length L and wave speed c, what are the natural frequencies?
fn = n c / (2L), for n = 1, 2, 3, and so on.
Textbook mapping
| Item | Mapping |
|---|---|
| Main source | Kreyszig, Advanced Engineering Mathematics, Ch 12 (partial differential equations) |
| Core topics | 16.1 What a PDE is · 16.2 Heat, wave, Laplace · 16.3 Boundary and initial conditions · 16.4 Separation of variables · 16.5 Classification and applications |
| Engineering connection | Heat Transfer (transient conduction), Vibrations (continuous systems), Fluid Mechanics (potential flow), FEA and CFD, which discretise these equations. |
| Skip on first pass | Bessel and spherical-harmonic expansions, Green's functions, the method of characteristics for nonlinear PDEs. |
| Read next | Numerical Methods for Mechanical Engineers. |