Flashcards

PDE's stack 1

Elliot Bryce Inglês 20 flashcards

Flashcards do conjunto (20)

1When is the method of characteristics applied?Used to solve the wave equation and can be applied to PDEs of the form (note that they are non -linear).
D(ro)/D(t) + c*D(ro)/D(x) = Q
When c is dependent on ro, then Q is not linear and superposition not valid. However it is linear in the differentials in x and t.
This is a quasilinear pde.
2What is a quasilinear pde?A type of pde where the coefficient of the highest order derivative DOES NOT depend on the derivative of the unknown function (i.e ro)
3What is a characteristic? (definition)A characteristic is a curve along which a PDE reduces to an ODE. So in the method of characterisitcs, the idea is you start from the non linear pde,
D(ro)/D(t)+c*D....
and you can replace its description with two ODEs i.e.
D(x)/D(t) = c
D(w)/D(t) = 0
4What is the notation you'll use for these cards for the total derivative (ordinary derivative) and the partial derivative.D(x/t) means
ODE(x)/ODE(t)
d(x/t) means
PDE(x)/PDE(t)
These are just for your cards.
5What is the equation for the family of characteristics, how is it found? What can it tell us?Found by integrating the characteristics.
E.g. Int(D(x))=Int(c*D(t))
== x=ct+x(0)
&
Int(D(w))=Int(0*D(t))
== 0
This gives the graph where you have multiple lines/curves. They are curves along which a PDE reduces to an ODE.
6What is the 1D wave equation?
7What is a 'characteristic'?A curve along a PDE that reduces to an ODE.
8From a wave equation how do you derive to find the family of characteristics?
9what is the general solution of dw/dt +c*dw/dx
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11What is D'almbert's formula and how is it derived?
12How does a vibrating string behave at rest? (i.e. its vibrating but its at one of its peaks or troughs)
13How do you find the left and right parts of a 1D wave? (useing d'alembert)
14How do you sketch the solution u(x,t)?
15What is the general solution of D'almbert?
16What is the range of influence?The region effected by the initial comditions at a point x,t.
E.g. Its the region that has been affected by the wave, and so it will grow as time increases, since wave has propogated further.
17What is the domain of dependence?The region between the x+ct and x-ct and so is the domain of the intial condition that affects your solution at the point (x,t)
e.g think like (x,t) is the mapped solution, from the domain of values the solution came from in previous time
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20What is the transport equation?It's simply just the equation describing a wave moving with constant shape in 1D