PDE theory

The exercise was created 2023-01-13 by nathaliepettersson. Question count: 6.

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  • Definition 2.1 Characteristics of the PDE The solution of X'(s)=V(X(s), s) is called the characteristics of the PDE
  • Definition 2.2 Breaking time of the burger equation The time Tb(u0)≥0 is called the breaking time (in the future) of the burger equation with initial data u0
  • Definition 2.3 Global weak solution of the burger equation A bounded function u:Rn x R -> R is said to be a global weak solution to the burger equation/conservation law ∂tu+u∂xu=0 if u(x, t) satisfies ∫Rn∫R(∂tv(x, t)u(x, t) + divxv(x, t)*J(u(x, t)))dtdx = 0 for all v€Cc'(Rn x R)
  • Definition 3.1 + 3.2 Spherically symmetric and spherical wave A function u:Rn->R, u=(u(x), is said to be spherically symmetric if there exists a function û:[0, inf) -> R such that u(x) = û(|x|), we call û the radial representation of u. A spherical wave is a strong solution u(x, t) of the 3-dimentional wave equation ∂tu-c^2∆xu=0, x€R3, t€R, such that u(x, t) is spherically symmetric for all t€R
  • Definition 4.1 Fundamental solution of the heat equation (or the heat Kernel) The function K(x, y, t) given by K(x, y, t) = 1/((4πt)^(n/2))*exp(-(x-y)^2/4t), is called the fundamental solution of the heat equation or heat kernel
  • Definition 5.2 Weak/variational formulation of poisson equatino The problem "Find u€C0'(Ω) such that -∫Ωdivv(x)*divu(x) dx = ∫Ωv(x)f(x)dx, for all v€C0'(Ω)" is called the weak (or variational) formulation of the problem ∆u = f(x), x€Ω, u(x) = 0, x€∂Ω in the space C0'(Ω)

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