Download Partial Differential Equations: Introduction and Solutions - Prof. Hans Lindblad and more Study notes Differential Equations in PDF only on Docsity!
Math 231 Partial Differential Equations
Lecture 1: Introduction.
1.1 Definition. A Partial Differential Equation (PDE) of order k for a function u(x) of x ∈ Rn^ is an equation involving u and its derivatives up to order k
(1.1) F (x, u(x), ∂u(x),... , ∂ku(x)) = 0
Here ∂ku stands for the jet of all partial derivatives ∂αu = ∂ xα 11 · · · ∂α xnn u, of order k = | α| = α 1 + · · · + αn. The functions u and F may also be vector valued in which case its called a system of partial differential equations. A PDE is called linear if it has the form
|α|≤k
aα(x)∂αu(x) = f (x)
1.2 Examples. Partial Differential equations arise in e.g. physics and geometry:
Linear. Laplace equation:
4 u =
∑^ n
i=
∂ x^2 i u = 0
Heat equations ∂tu − 4u = 0
Wave equation §u = ∂ t^2 u − 4u = 0
Schroedinger equation i∂tu + 4 u = 0
Transport equation ∂tu + bi∂xi u = 0
Ordinary differential equation ∂tu + Au = 0
Nonlinear equations Burgers’ equation
∂tu + u∂xu = 0
Minimal surface equation
∑^ n
i=
∂xi
xi u (1 + |∂u|^2 )^1 /^2
Linear Systems Maxwell’s equations
Et = curl B Bt = −curl E div B = div E = 0 1
2
Nonlinear systems Euler’s equations of an incompressible fluid
{ ∂tui +
∑n ∑ k=1^ uk∂xk^ ui^ =^ −∂ip n i=1 ∂xi^ ui^ = 0
Einstein’s vacuum equations of general relativity for the metric tensor gαβ , α, β = 0 , 1 , 2 , 3, of space time is that the Ricci curvature vanishes:
Rμν (g) = 0
which in harmonic coordinates becomes a system of nonlinear wave equations
§g gμν = Fμν (g, ∂g), §g =
α,β=0, 1 , 2 , 3
gαβ^ ∂xα^ ∂xβ
Evolution equations. The wave, heat, Schroedinger, transport equations and the ordinary differential equations are evolution equations describing evolving phenom- ena. For evolution equations we want to find a solution for future times from the knowledge of initial conditions.
Stationary equations. Laplace equation is a stationary equation. For stationary equations we want to find a solution in the interior of a domain from boundary conditions.
1.3 Strategies for Solving PDE’s. Linear PDEs can be solved more or less explicitly, in particular if the coefficients aα are constants. For nonlinear equations we can in general not find an explicit solution but instead we just ask if the problem is well posed, i.e. if: (a) the problem has a solution, (b) the solution is unique, (c) the solution depends continuously on data in a certain class. For nonlinear equations one can usually prove local existence of a solution but the solution might but the solution might blow up after some time.
