ME565 Lecture 7: Canonical Linear PDEs: Wave equation, Heat equation, and Laplace's equation

ME565 Lecture 7
Engineering Mathematics at the University of Washington
Canonical Linear PDEs: Wave equation, Heat equation, and Laplace's equation
Notes: faculty.washington.edu/sbrunto...
Course Website: faculty.washington.edu/sbrunto...
faculty.washington.edu/sbrunton/

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0:00-14:50 Meaning, Use of PDEs. (How spatial distribution evolves in time for eg.)
16:50. Three Canonical PDEs (Dim 1)
The Heat Equation. u_t=uxx. (Parabolic) The Wave Equation. u_tt_u_xx. (Hyperbolic) Laplace’s Equation uxx=0
Some combination of sines is the solution to the Wave Equation.
The Heat Equation. In response to a unit impulse, there is Gaussian diffusion (from the sines and cosines which are the eigenfunctions of the Laplacian).
The Heat equation which has a stationary in space, Heat distribution satisfies Laplace’s Equation. There is cancellation of the curvature changes in each dimension.
These are homogenous equations. (No direct dependence of derivatives on the underlying parameter.)
Linear PDEs are constructed from Linear Operators acting on the state variable.
L(αg+βf)=αL(g)+βL(f)
D(αg+βf)=αD(g)+βD(f).
Quasilinearity.
Eg. L(u)=u+7, L(u1+u2)=u1+u2+7, L(u1)+L(u2)=u1+u2+14.
45:00 Nonlinear PDEs.
N(u)=uu_x
N(u)=u^2
Burger’s Equation. u_t+uu_x=u_xx
Model of Shockwaves.
uu_x. Nonlinear convection. Viscous term spreads shockwaves.
u_xx is a diffusion term.

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[@anchalbhandari5901]

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[@obaroolamide3276]

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[@shahryarhabibi7187]

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