Fichas

B5.2

Ben Tidswell Part B uni Inglés 26 fichas

Fichas del conjunto (26)

1State the Cauchy-Kowalevski Theorem for the first order quasilinear equation a(x,y,u)u_x+b(x,y,u)u_y=c(x,y,u) with data curve Γ and show it is equivalent to Γ not being tangent to a characteristic projection
2Give a condition for a shock to occur
3state the general conservation form of a first order quasilinear PDE and the Rankine-Hugoniot condition at a shock
4Derive an expression satisfied by characteristics λ=dy/dx of the second order linear PDE a(x,y)u_xx+2b(x,y)u_xy+c(x,y)u_yy=f(x,y) By considering curves along which the second derivatives can be discontinuous
5state the eigenvalue problem satisfied by the characteristics of the first order system 𝗔(x,y,𝘂)𝘂_x+𝗕(x,y,𝘂)𝘂_y=𝗰(x,y,𝘂)
6State the left eigenvector problem for the first order system 𝗔(x,y,𝘂)𝘂_x+𝗕(x,y,𝘂)𝘂_y=𝗰(x,y,𝘂)
7State the quotient rule
8State the normal and divergence form of Green's theorem for 𝗙(x,y)=(P(x,y) Q(x,y))^t in a closed simply connected domain D
9Show that a solution, if it exists, is unique for Dirichlet problem ∇^2(u)=f(x,y), (x,y)∈D u=g(x,y), (x,y)∈∂Dphi=0 on dD gives phi=0 everywhere
10For the Dirichlet problem ∇^2(u)=f(x,y), (x,y)∈D u=g(x,y), (x,y)∈∂D derive the solution in terms of the Green's function, G, giving the problem it solves
11State the correct singularity for the Green's function for the Dirichlet problem ∇^2(u)=f(x,y), (x,y)∈D u=g(x,y), (x,y)∈∂D and show it works
12Show that, if the Cauchy-Kowalevski theorem condition is satisfied for a first order quasilinear equation a(x,y,u)u_x+b(x,y,u)u_y=c(x,y,u) with data curve Γ Then the second order derivatives of u are uniquely determined on some neighbourhood of Γ
13Derive Charpit's equations for the general first-order nonlinear PDE F(p,q,u,x,y)=0 where p=u_x, q=u_y
14State a necessary condition for p and q to be uniquely determined on the data curve Γ for the general first-order nonlinear PDE F(p,q,u,x,y)=0 where p=u_x, q=u_y
15Give the form of the similarity solution that may be sought for u(x,y) for some PDE
16Derive a problem solved by the Riemann function R for the linear hyperbolic PDE in canonical form 𝓛[u]=u_xy+a(x,y)u_x+b(x,y)u_y+c(x,y)u= f(x,y) with Cauchy data on a data curve Γ
17Derive the condition for a characteristic of the first order system: 𝗔(x,y,𝘂)𝘂_x+𝗕(x,y,𝘂)𝘂_y=𝗰(x,y,𝘂) by considering curves along which the first derivatives of 𝘂 may be continuous
18State two forms of an equation which can be solved to find the envelope for rays of the nonlinear PDE: F(p,q,u,x,y)=0 with data curve Γ:x=x_0(s),y=y_0(s),u=u_0(s)
19Define an envelope for the first order quasilinear equation: a(x,y,u)u_x+b(x,y,u)u_y=c(x,y,u)
20State the general form of: a first order quasilinear equation, a first order nonlinear equation, a first order quasilinear system, and a second order semi-linear equation
21Show that the Rankine-Hugoniot condition implies that first order semi-linear equations are only discontinuous on characteristics
22Describe the method of images
23State how to classify a second order semi-linear equation
24State the identities for cos(α±β) and sin(α±β)
25State the canonical form of a hyperbolic, elliptic, and parabolic second order semi-linear equation
26State the Rankine-Hugoniot condition for 𝗣_x+𝗤_y=𝗥 and determine the number of incoming characteristics required for the shock to be causal