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