Introduction
This chapter extends the toolkit from Chapter IV in two connected directions: representing periodic functions as sums of simple sine and cosine waves (Fourier series), and solving equations involving functions of more than one variable (partial differential equations, or PDEs) — together, these are the standard tools for analysing systems that vary in both space and time, such as heat flow or vibrating strings.
1. Fourier coefficients and series
A Fourier series represents a periodic function as an infinite sum of sines and cosines of increasing frequency: f(x) = a₀/2 + Σ[aₙcos(nx) + bₙsin(nx)]. The coefficients aₙ and bₙ are found using specific integral formulas that exploit the orthogonality of sine and cosine functions over one period (the integral of the product of two different such functions over a full period is zero) — this orthogonality is precisely what allows each coefficient to be isolated and computed independently of all the others.
2. Even/odd functions and half-range expansions
If a function is even (f(−x) = f(x)), its Fourier series contains only cosine terms (and the constant term); if odd (f(−x) = −f(x)), it contains only sine terms — recognising this symmetry in advance saves substantial calculation, since roughly half the coefficients are automatically zero. A half-range expansion extends a function defined only on half an interval to either an even or odd function on the full interval (whichever is more convenient), allowing a pure cosine or pure sine series to represent it on the original half-interval.
3. Formation and classification of basic PDEs
A PDE involves partial derivatives with respect to more than one independent variable (commonly space and time). PDEs are classified (by analogy with conic sections, based on the coefficients of the second-derivative terms) as elliptic (e.g., Laplace’s equation, describing steady-state/equilibrium situations), parabolic (e.g., the heat/diffusion equation, describing time-evolving diffusive processes), or hyperbolic (e.g., the wave equation, describing oscillatory/propagating phenomena) — this classification predicts the equation’s general qualitative behaviour and the type of boundary/initial conditions it requires for a well-posed problem.
4. Separation of variables
Separation of variables is the primary technique for solving linear PDEs with appropriate boundary conditions: assume the solution can be written as a product of functions, each depending on only one variable (e.g., u(x,t) = X(x)T(t)), substitute into the PDE, and algebraically manipulate it so that all terms depending on x are on one side and all terms depending on t are on the other — since these sides depend on different independent variables yet must be equal, both must equal the same constant (the separation constant), splitting the single PDE into two separate, more tractable ODEs (each solvable by Chapter IV’s methods).
5. Standard heat, wave, and Laplace equations
The heat equation (∂u/∂t = k∂²u/∂x²) describes how temperature diffuses over time; the wave equation (∂²u/∂t² = c²∂²u/∂x²) describes propagating disturbances like a vibrating string; Laplace’s equation (∇²u = 0, already met physically in Physics Chapter I of Paper II) describes steady-state distributions with no time-dependence. Solving each with given boundary conditions (via separation of variables) typically produces a solution as an infinite Fourier series — directly connecting the two halves of this chapter: the separation-of-variables solution’s spatial part is expanded in a Fourier series to match the specific initial or boundary shape of the problem.
Key Takeaways
- Fourier series decompose periodic functions into sine/cosine components, with coefficients isolated via the orthogonality of sines and cosines over one period.
- Even/odd symmetry immediately determines whether a Fourier series needs only cosine or only sine terms, substantially simplifying calculation.
- PDEs are classified as elliptic, parabolic, or hyperbolic, mirroring conic section classification and predicting the equation’s qualitative behaviour and appropriate conditions.
- Separation of variables converts a PDE into a pair of ODEs by introducing a separation constant, exploiting the fact that expressions depending on different independent variables can only be equal if both equal the same constant.
- The heat, wave, and Laplace equations are the three canonical PDEs of applied mathematics, and their solutions with realistic boundary conditions typically emerge as Fourier series.