Problem-solving recipes

A step-by-step procedure for every problem type this course teaches — what to do first, what to do next, and how to tell which case you're in.

What are you being asked to do?

Start with what you're actually being asked to produce.

Readiness review — diagonalize a matrix, using the same recipe linear algebra taught, before it gets reused for a PDE system further down.

A physical conservation statement, need the PDE it implies — the conservation-law derivation recipe.

A first-order linear or quasilinear PDE, need to solve it — the variable-coefficient recipe for linear equations, the quasilinear recipe when the coefficient on $u$ 's derivative depends on $u$ itself.

A boundary value problem with nonhomogeneous boundary data, need separation of variables to apply — shift to homogeneous first.

A vibrating string, membrane, or drumhead, need its normal modes, frequencies, or energy — the matching normal-mode/frequency/energy recipe for that geometry.

A linear PDE on the whole line, need to solve it by transform — the Fourier-transform recipe.

A Legendre polynomial, need to compute it, or check an orthogonality or normalization integral — the Rodrigues-formula recipe, or the orthogonality-integral recipe.

Two spherical harmonics, need to know if they're orthogonal — the spherical-harmonics recipe.

The wave equation's Green's function, need its value at a point — the Green's-function recipe.

A hyperbolic system of first-order PDEs, need canonical form, or a transmission-line/acoustics system's impedance and speed — the matching systems recipe.

Burgers' equation or a similar conservation law, need a shock speed, or need to evaluate inside a rarefaction fan — the two conservation- laws recipes, kept independent since they solve genuinely different problems.

Readiness

Diagonalize a matrix

Use when: you have a square matrix $A$ and need $P$ , $D$ with $A=PDP^{-1}$

Goal: the eigenvalues and a matching set of independent eigenvectors, assembled into $P$ and $D$

Eigenvalues, eigenvectors, and diagonalization, revisited

Where PDEs come from

Derive a PDE from a conservation law

Use when: you're given a conserved quantity, a flux law tying its flow rate to a gradient, and asked what PDE it satisfies

Goal: a PDE for $u(x,t)$ , built from a segment-level balance carried to the limit $\Delta x\to0$

Physical derivation: the heat equation

First-order PDEs and the method of characteristics

Solve a variable-coefficient linear first-order PDE

Use when: the equation is $a(x,y)u_x+b(x,y)u_y=0$ with $a,b$ depending on $x,y$ but not on $u$

Goal: the general solution $u(x,y)=f(\varphi(x,y))$ for the curve-invariant $\varphi$ , or a specific $u$ once initial data pins down $f$

Variable-coefficient characteristics

Solve a quasilinear first-order PDE

Use when: the equation is $a(x,y,u)u_x+b(x,y,u)u_y=c(x,y,u)$ , with a coefficient depending on $u$ itself

Goal: $u(x,y)$ , built from the three coupled characteristic ODEs plus the prescribed initial data

Quasilinear first-order PDEs

Separation of variables and the heat equation

Shift a nonhomogeneous boundary value problem to a homogeneous one

Use when: the heat equation's boundary values are fixed constants that aren't both zero, such as $u(0,t)=T_1$ , $u(L,t)=T_2$

Goal: a complete $u(x,t)$ , built from a linear reference profile plus the usual zero-boundary sine series

Nonhomogeneous boundary data: shifting to homogeneous

The wave equation on bounded domains

Find the normal modes of a vibrating string

Use when: you have $u_{tt}=c^2u_{xx}$ on a finite string with both ends held fixed

Goal: every normal mode $u_n(x,t)$ and its natural frequency $\omega_n$ , ready to superpose into the general solution

Separation of variables for the vibrating string

Compute the energy of a vibrating string

Use when: you're given $u(x,t)$ , or its mode coefficients, for a string satisfying $u_{tt}=c^2u_{xx}$ , and asked for $E(t)$

Goal: the total energy $E(t)=\frac12\int_0^L(u_t^2+c^2u_x^2)\,dx$ at the time you need it

Energy of a vibrating string

Find the natural frequencies of a rectangular membrane

Use when: you have $u_{tt}=c^2(u_{xx}+u_{yy})$ on a rectangle with $u=0$ on all four edges

Goal: every natural frequency $\omega_{mn}$ , and (if needed) the coefficients for the full series solution

The vibrating rectangular membrane

Whole-line and transform methods

Solve a linear PDE by Fourier transform

Use when: the PDE holds for all $x$ on the whole line and its coefficients don't depend on $x$

Goal: $u(x,t)$ , recovered by inverting a transformed equation that was solved as an ordinary differential equation in $t$

Solving the heat equation by Fourier transform

Special functions in curvilinear domains

Find the modes of a circular drumhead

Use when: you have $u_{tt}=c^2\nabla^2u$ on a disk of radius $a$ with $u(a,\theta,t)=0$

Goal: the natural frequency $\omega_{m,n}$ of any mode, and (if needed) that mode's own coefficient from the initial data

The vibrating circular drumhead

Compute a Legendre polynomial by Rodrigues' formula

Use when: you need $P_n(x)$ itself, or its value at one point, for a specific nonnegative integer $n$

Goal: the polynomial $P_n(x)$ , or its value at the point you need, matching the standard normalization

Legendre's equation and Legendre polynomials

Evaluate a Legendre orthogonality or normalization integral

Use when: you need $\int_{-1}^1P_n(x)P_k(x)\,dx$ for two specific degrees $n$ and $k$

Goal: the integral's exact value, either 0 or $2/(2n+1)$ , without re-deriving Rodrigues' formula from scratch

Orthogonality of Legendre polynomials, two ways

Check whether two spherical harmonics are orthogonal

Use when: you have $Y_{\ell_1}^{m_1}$ and $Y_{\ell_2}^{m_2}$ and need to know whether $\iint Y_{\ell_1}^{m_1}\overline{Y_{\ell_2}^{m_2}}\,d\Omega$ vanishes

Goal: a yes or no verdict on orthogonality, without carrying out the double integral by hand

Spherical harmonics

Green's functions and integral representations

Evaluate the wave equation's Green's function at a point

Use when: you have wave speed $c$ , a source at $(x_0,t_0)$ , and a target point $(x,t)$ , and need $G(x,t;x_0,t_0)$

Goal: the value $1/(2c)$ or $0$ , decided from the wedge condition alone, with no delta-function integration by hand

The Green's function for the wave equation

Systems of first-order PDEs

Diagonalize a hyperbolic system to canonical form

Use when: you have a strictly hyperbolic system $\mathbf u_t+A\mathbf u_x=\mathbf 0$ , constant matrix $A$ , and want it as decoupled scalar transport equations

Goal: the canonical variables $\mathbf r$ , the decoupled equations they satisfy, and the way back to $\mathbf u$

Diagonalizing to canonical form

Find a transmission-line or acoustics system's impedance and speed

Use when: you have $L,C$ for a transmission line, or $\rho_0,c$ for acoustics, and need the characteristic impedance and propagation speed

Goal: the impedance ( $Z_0$ or $\rho_0c$ ) and the wave speed, correctly matched to the physical system given

Linear transmission-line and acoustics examples

Nonlinear PDEs and conservation laws

Compute a shock speed via the Rankine-Hugoniot condition

Use when: you have a conservation law $u_t+f(u)_x=0$ with a jump between states $u_L$ and $u_R$ , and need the shock's own speed $\sigma$

Goal: the numeric shock speed $\sigma$ , confirmed by the general formula

The Rankine-Hugoniot condition, revisited in full

Evaluate a rarefaction fan at a point

Use when: you have a rarefaction connecting $u_L$ to $u_R$ (Lax-inadmissible ordering $f'(u_L)<f'(u_R)$ ) and need $u(x,t)$ at a specific point

Goal: the correct piece of the fan's piecewise formula, $u_L$ , $u_R$ , or the self-similar profile, for this exact $(x,t)$

Rarefaction waves