MTH2032 Chap.6 Classifying PDEs and Deriving Physical Models
Classifying PDEs and Deriving Physical Models
Week 6 opens Part II with vocabulary and models. A PDE is classified by order, by grade of linearity from linear through semilinear and quasilinear to fully nonlinear, by homogeneity when it is linear, and by type when it is second-order semilinear.
The same week derives the advection equation from conservation of mass, the wave equation from a chain of masses and springs, the heat equation from Fourier's law, and Laplace's equation as the steady state of heat. Each classification is given with its reason, the form in which markers expect it.
What this chapter covers
- 01
Order of a PDE
- 02
Linear, semilinear, quasilinear, fully nonlinear
- 03
Homogeneity and superposition
- 04
Hyperbolic, parabolic and elliptic type
- 05
Arbitrary functions in general solutions
- 06
Deriving the advection, wave, heat and Laplace equations
Worked example · free
Integrating a mixed derivative twice
- 1Integrate in y with x fixed: u_x = 2xy + f(x), where f is an arbitrary function of x.
- 1Integrate in x with y fixed: u = x²y + F(x) + G(y), where F′ = f and G is arbitrary.
- 1Check: u_x = 2xy + F′(x) and then u_xy = 2x.
Key terms
- Order of a PDE
- The order of the highest partial derivative appearing in the equation.
- Semilinear PDE
- A PDE whose highest-order terms are linear with coefficients depending only on the independent variables.
- Advection Equation
- The conservation law u_t + (vu)_x = 0 for a density carried by a velocity field.
- Mixed-Type PDE
- A second-order equation whose type changes between hyperbolic, parabolic and elliptic across its domain.
Classifying PDEs and Deriving Physical Models FAQ
How do I find the most specific linearity class?
Start from the highest-order terms. If they are linear with coefficients depending only on the independent variables, check the remaining terms: all linear means linear, otherwise semilinear. If the highest-order coefficients involve u or lower derivatives, the equation is quasilinear. If the highest derivatives appear non-linearly, for example squared, it is fully nonlinear.
Does the type of a PDE depend on position?
Yes. Type is decided at each point by the sign of B² − 4AC. With constant coefficients the type is the same everywhere, but an equation such as x u_xx + u_yy = 0 is hyperbolic for negative x, parabolic on the axis and elliptic for positive x, so it is called mixed.
Why does the heat equation need one initial condition but the wave equation two?
The heat equation is first order in time, so its time factor needs one starting value, the initial temperature. The wave equation is second order in time, so both the initial displacement and the initial velocity are required, in the same way that a second-order ODE needs two initial values.
What does a Neumann condition mean physically?
It fixes the spatial derivative at a boundary. For heat it fixes the heat flow through the end, and a zero value means the end is insulated. For a vibrating rod or string it fixes the force at the end, and a zero value means the end is free. The lecture tables pair each condition with its meaning.
How do I justify superposition for a PDE?
Superposition needs a linear, homogeneous equation: if two functions each satisfy L(u) = 0, so does any combination of them, because the operator distributes over sums and constants. Say that the equation is linear and homogeneous before adding solutions, since the step fails for an inhomogeneous or non-linear equation.
Exam move
Classification is quick marks if it is drilled. Write ten PDEs of your own covering every grade of linearity and every type, then classify each in one sentence that includes the reason. Practise putting equations in the lecture standard form before computing B² − 4AC, and watch for variable coefficients that make the type change across the domain.
Next, work through general solutions of PDEs with derivatives in one variable: solve the ODE in that variable and replace each constant with an arbitrary function of the others, then try a successive-substitution example with mixed derivatives. For the models, do not memorise the derivations line by line.
Instead, be able to explain the conservation idea behind each, state the resulting equation, and say what each constant means: the speed c² = E/ρ, the diffusivity D = k/(ρC). Finally, practise writing complete boundary value problems from word descriptions, with a sentence for each condition that says what it means physically and whether it is homogeneous.
Finally, attach a physical meaning to each boundary condition you write, since modelling questions reward the explanation as much as the equation.
Working through Classifying PDEs and Deriving Physical Models in MTH2032? Sia is AskSia’s AI Mathematics tutor — ask any MTH2032 Classifying PDEs and Deriving Physical Models question and get a clear, step-by-step explanation grounded in how MTH2032 is taught and assessed. Read this chapter free, then take your hardest questions to Sia.
