Cheatography
https://cheatography.com
ODE
Boundary Value Problem |
values of dependent variable at more than one value of the independent variable |
Turn it into an initial condition problem |
Shooting Method |
guess initial condition for the arbitrary variable |
z= dT/dx |
linear interpolation if you can |
initial conditions satisfy boundary values |
Finite Methods |
approximate the derivative using one of the finite methods |
reduce it to a system of linear equations |
more computationally efficient than shooting method |
shooting method: 1- solve RK4 multiple times 2- interpolate |
|
|
PDE
more than 1 independent variable |
Elliptic Model |
Laplace equation if equal to 0 |
Poisson's equation if not equal to 0 |
d2T/dx2 + d2T/dy2 =0 |
not affected by time --x,y independent |
Parabolic Model |
dT/dt= K' (d2T/dx2) |
time is a factor --x,t independent |
Hyperbolic Model |
d2y/dx2 = (1/c2) (d2y/ dt2) |
waveform --x, t are independent |
Elliptic Model |
1 boundary values --> closed system or 2 secondary variable |
maximum of 5 non-zeros per equation |
Gauss siedel: does not take into account zeros + DDS |
centered difference |
Can i find one independent of other values? NO |
without borders unknowns increase |
centered difference equations (depends on order) |
flux : derivative - insulated (=0) |
Interpolation
fit 1 function to all points |
given points without function |
increase accuracy, decrease step size or increase order |
3 pts f(x)=ax^2+bx+c - substitute points - system of linear equations (GE- GJ- inverse) |
Alternative function representation |
f(x)= b0 + b1(x- x0) +b2(x-x0)(x-x1)+ b3(x-x0)(x-x1)(x-x2) +... |
based on Taylor series |
bs represent the slopes |
efficient - quick |
|
|
Splines
fit a function to each interval |
used for large datapoints--to avoid kinks |
Linear Splines |
f(x)= f(xo) + m(x-xo) |
interval surrounds point |
issues: |
linearizing a non-linear function, oversimplifies behavior |
discontinuity at the intermediate points - slope is no the same on either side |
Quadratic Splines |
minimum of 2 intervals or 3 points |
f(x)= a1x^2 +b1x+c1 |
3n unknowns -- n is # of intervals |
(2n equations ) substitute points in formulas |
(n-1 equations) establish continuity with the slope at the intermediate points |
assumption: a1=0 |
minimal effect on other intervals |
under determined system by 1 equation |
intermediate points are not independent |
System of linear equations --do not use iterative methods (not DDS) |
all functions are dependent |
Cubic Splines |
most popular method |
minimum of 3 intervals or 4 points |
4n equations - undetermined by 2 equations |
assume 2nd derivative of outer points is 0 |
Alternative - Lagrange |
(xi - xi-1) f''(xi-1)+ 2(xi+1 -xi-1) f''(xi) +(xi+1 -xi) f''(xi+1) = (6/(xi+1 -xi)) [f(xi+1)-f(xi)] + (6/(xi - xi-1)) [f(xi-1) -f(xi)] |
f(x)= (f''(xi-1)/6(xi -xi-1)) (xi -x)^3 + (f''(xi)/6(xi- xi-1)) (x- xi-1)^3 +[ (f(xi-1)/(xi - xi-1)) - (f''(xi-1)(xi -xi-1)/6) ] (xi - x)+ [ (f(xi)/(xi -xi-1)) - (f''(xi)(xi- xi-1)/6) ] (x-xi-1) |
solve all second derivatives first |
all related by continuity |
|
Created By
Metadata
Comments
No comments yet. Add yours below!
Add a Comment