Week 8 Partial Differential equations Flashcards
What is a linear PDE
Has the dependent variable, and all its derivatives, in linear form.
This means
there are no products of dependent variables, their derivatives, and they are not arguments of
non-linear functions. This is the same definition as for linear ordinary differential equations.
What is a homogenous PDE?
when only dependent (TOP) variable terms are present on the left side of
the equation,
the right hand side is equal to zero. This is the same definition as for linear
ordinary differential equations.
Second order linear partial differential equations in terms of: PHOTO
general form?
Implicitly defined form?
im:
Ax^x + Bxy + Cy^2 + Dx + Ey + F = 0
What can PDE’s be classified into ?
what form does the diffusion type PDE’s take?
In 1-dimension
- second-order
- homogeneous
- linear partial
- it’s parabolic
differential equation
what form does the wave type PDE’s take?
In 1 dimension
- second-order
- linear
- homogeneous partial
differential equation.
-hyperbolic
what form does the laplace type PDE’s take?
2-dimension
- second order
- linear homogeneous partial differential equation
-elliptic
what form does the poisson’s type PDE’s take?
2-dimension
-similar to Laplace equation, but has some function of x/y instead of 0
How does the separation of variables work?
- assumption that solution can be expressed as: photo
- separated solution into spatial and temporal part (the product = solution to PDE)
- IF works, splits PDE into 2 ODEs (1 for temporal, 1 for spatial)
- If solution can be separated in 2, overall shape of solution remains same
What does this mean?
2nd derivative of x, with respect to x
initial condition in mathematical form?
for dy/dt
dy/dt
(y,t)
form= (y,0)
t=0
Solve this
diffusion type
what does 0 integrate into?
constant, c
Solve this
diffusion type
Solve this
wave type
Solve this
wave type
Solve this
laplace type
what does this represent?
first derivative of u with respect to t
first derivative= differentiating once
