PDEs 2

Question 1

what is the general semilinear first-order PDE for a scalar function u(x,y)

Answer 1

• a(x,y)du/dx + b(x,y)du/dy = f(x,y,u)

Question 2

what is the characteristic equation

Answer 2

• dy/dx = b(x,y) / a(x,y)

Question 3

what is the compatibility condition

Answer 3

• du/dx = f(x,y,u) / a(x,y)

Question 4

what is the general first-order quasilinear PDE for the function u(x,t)

Answer 4

• a(x,t,u)du/dt + b(x,t,u)du/dx = f(x,t,u)

Question 5

what is the general second-order linear PDE

Answer 5

• a(x,y)d^u/dx^2 + 2b(x,y)d^2u/dxdy + c(x,y)*d^2u/dy^2 + f(du/dx,du/dy,u,x,y) = 0

Question 6

for a(dn/dx)^2 + 2bdn/dxdn/dy + c(dn/dy)^2 (n and ξ are interchangeable), what are the characteristic equations

Answer 6

• dy/dx = [b +/- sqrt(b^2 - 4ac)] / a
• this applies to equations with u,x, and y too
• dx^2 corresponds to a, dy^2 corresponds to c and dydx corresponds to 2b

Question 7

what is the condition to ensure that the PDE has two real distinct families of characteristics

Answer 7

• if the discriminant, Δ = b^2 - 4ac > 0

Question 8

what primarily differentiates hyperbolic, parabolic and elliptic PDEs

Answer 8

• for hyperbolic, Δ > 0
• for parabolic, Δ = 0
• for elliptic, Δ < 0

Question 9

what is the expression for the greens function ∇^2(G)

Answer 9

• ∇^2(G) = δ(x)

Question 10

if G is a function of x and x0, ∇^2(G) = δ(x - x0), what is the other expression for δ(x - x0)

Answer 10

• δ(x - x0) = ∇^2[-1 / 4pi|x - x0|]