A partial differential equation (PDE) connects an unknown function of several variables with one or more of its partial derivatives. PDEs describe how quantities such as temperature, pressure, and wave amplitude vary across space and time.
A function of several variables can change at different rates in different directions. For example, temperature u(x,y,t) may depend on two spatial coordinates and time. Its partial derivatives describe changes in one variable while the others are held fixed.
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How is the order of a PDE determined?
It is the highest derivative order appearing in any term.
What characterizes a semilinear PDE?
Highest-order derivatives appear linearly with coefficients depending only on independent variables; lower-order terms may be nonlinear.
How does the discriminant classify a second-order PDE in two variables?
For principal part Auxx+2Buxy+Cuyy,B2−AC<0 is elliptic, =0 parabolic, and >0 hyperbolic.
What is a fundamental solution?
A solution describing an equation’s response to a point source, often represented by a delta distribution.
What is a weak solution?
A solution satisfying an integral or distributional formulation, without necessarily satisfying the PDE pointwise.
Quiz questions
For a second-order PDE with principal part Auxx+2Buxy+Cuyy, what type corresponds to B2−AC>0?Answer: Hyperbolic
What is the main step in separation of variables for a PDE in x and t?Answer: Assume a product form such as u(x,t)=X(x)T(t), then derive ODEs when the equation permits.
For certain first-order PDEs, what happens along characteristic curves?Answer: The PDE reduces to an ordinary differential equation along those curves.
What feature of the finite volume method supports conservation?Answer: It balances fluxes across control-volume boundaries, with shared fluxes canceling between neighbors.
When is a PDE problem commonly called well-posed?Answer: When a solution exists, is unique, and depends continuously on the prescribed data.