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% Document Info
\author{Boko}
\pdfinfo{
  /Title (analysis-part4.pdf)
  /Creator (Cheatography)
  /Author (Boko)
  /Subject (Analysis Part4 Cheat Sheet)
}

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\fancyhead[L]{
\noindent
\begin{multicols}{3}
\begin{tabulary}{5.8cm}{C}
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    \vspace{-7pt}
    {\parbox{\dimexpr\textwidth-2\fboxsep\relax}{\noindent
        \hspace*{-6pt}\includegraphics[width=5.8cm]{/web/www.cheatography.com/public/images/cheatography_logo.pdf}}
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\columnbreak
\begin{tabulary}{11cm}{L}
    \vspace{-2pt}\large{\bf{\textcolor{DarkBackground}{\textrm{Analysis Part4 Cheat Sheet}}}} \\
    \normalsize{by \textcolor{DarkBackground}{Boko} via \textcolor{DarkBackground}{\uline{cheatography.com/55472/cs/15688/}}}
\end{tabulary}
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\fancyfoot[L]{ \footnotesize
\noindent
\begin{multicols}{3}
\begin{tabulary}{5.8cm}{LL}
  \SetRowColor{FootBackground}
  \mymulticolumn{2}{p{5.377cm}}{\bf\textcolor{white}{Cheatographer}}  \\
  \vspace{-2pt}Boko \\
  \uline{cheatography.com/boko} \\
  \end{tabulary}
\vfill
\columnbreak
\begin{tabulary}{5.8cm}{L}
  \SetRowColor{FootBackground}
  \mymulticolumn{1}{p{5.377cm}}{\bf\textcolor{white}{Cheat Sheet}}  \\
   \vspace{-2pt}Published 20th May, 2018.\\
   Updated 20th May, 2018.\\
   Page {\thepage} of \pageref{LastPage}.
\end{tabulary}
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  \mymulticolumn{1}{p{5.377cm}}{\bf\textcolor{white}{Sponsor}}  \\
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  \vspace{-5pt}
  %\includegraphics[width=48px,height=48px]{dave.jpeg}
  Measure your website readability!\\
  www.readability-score.com
\end{tabulary}
\end{multicols}}




\begin{document}
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\begin{multicols*}{3}

\begin{tabularx}{5.377cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{ODE}}  \tn
% Row 0
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{Boundary Value Problem} \tn 
% Row Count 1 (+ 1)
% Row 1
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{values of dependent variable at more than one value of the independent variable} \tn 
% Row Count 3 (+ 2)
% Row 2
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{Turn it into an initial condition problem} \tn 
% Row Count 4 (+ 1)
% Row 3
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{{\bf{Shooting Method}}} \tn 
% Row Count 5 (+ 1)
% Row 4
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{guess initial condition for the arbitrary variable} \tn 
% Row Count 6 (+ 1)
% Row 5
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{z= dT/dx} \tn 
% Row Count 7 (+ 1)
% Row 6
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{linear interpolation if you can} \tn 
% Row Count 8 (+ 1)
% Row 7
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{initial conditions satisfy boundary values} \tn 
% Row Count 9 (+ 1)
% Row 8
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{{\bf{Finite Methods}}} \tn 
% Row Count 10 (+ 1)
% Row 9
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{approximate the derivative using one of the finite methods} \tn 
% Row Count 12 (+ 2)
% Row 10
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{reduce it to a system of linear equations} \tn 
% Row Count 13 (+ 1)
% Row 11
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{more computationally efficient than shooting method} \tn 
% Row Count 15 (+ 2)
% Row 12
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{shooting method: 1- solve RK4 multiple times 2- interpolate} \tn 
% Row Count 17 (+ 2)
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{PDE}}  \tn
% Row 0
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{more than 1 independent variable} \tn 
% Row Count 1 (+ 1)
% Row 1
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{{\bf{Elliptic Model}}} \tn 
% Row Count 2 (+ 1)
% Row 2
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{Laplace equation if equal to 0} \tn 
% Row Count 3 (+ 1)
% Row 3
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{Poisson's equation if not equal to 0} \tn 
% Row Count 4 (+ 1)
% Row 4
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{d\textasciicircum{}2\textasciicircum{}T/dx\textasciicircum{}2\textasciicircum{} + d\textasciicircum{}2\textasciicircum{}T/dy\textasciicircum{}2\textasciicircum{} =0} \tn 
% Row Count 5 (+ 1)
% Row 5
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{not affected by time -{}-x,y independent} \tn 
% Row Count 6 (+ 1)
% Row 6
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{{\bf{Parabolic Model}}} \tn 
% Row Count 7 (+ 1)
% Row 7
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{dT/dt= K' (d\textasciicircum{}2\textasciicircum{}T/dx\textasciicircum{}2\textasciicircum{})} \tn 
% Row Count 8 (+ 1)
% Row 8
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{time is a factor -{}-x,t independent} \tn 
% Row Count 9 (+ 1)
% Row 9
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{{\bf{Hyperbolic Model}}} \tn 
% Row Count 10 (+ 1)
% Row 10
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{d\textasciicircum{}2\textasciicircum{}y/dx\textasciicircum{}2\textasciicircum{} = (1/c\textasciicircum{}2\textasciicircum{}) (d\textasciicircum{}2\textasciicircum{}y/ dt\textasciicircum{}2\textasciicircum{})} \tn 
% Row Count 11 (+ 1)
% Row 11
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{waveform -{}-x, t are independent} \tn 
% Row Count 12 (+ 1)
% Row 12
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\mymulticolumn{1}{x{5.377cm}}{{\bf{Elliptic Model}}} \tn 
% Row Count 13 (+ 1)
% Row 13
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{{\bf{1}} boundary values -{}-\textgreater{} closed system or {\bf{2}} secondary variable} \tn 
% Row Count 15 (+ 2)
% Row 14
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{maximum of 5 non-zeros per equation} \tn 
% Row Count 16 (+ 1)
% Row 15
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{Gauss siedel: does not take into account zeros  + DDS} \tn 
% Row Count 18 (+ 2)
% Row 16
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{centered difference} \tn 
% Row Count 19 (+ 1)
% Row 17
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{Can i find one independent of other values? NO} \tn 
% Row Count 20 (+ 1)
% Row 18
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{without borders unknowns increase} \tn 
% Row Count 21 (+ 1)
% Row 19
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{centered difference equations (depends on order)} \tn 
% Row Count 22 (+ 1)
% Row 20
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{flux : derivative  - insulated (=0)} \tn 
% Row Count 23 (+ 1)
\hhline{>{\arrayrulecolor{DarkBackground}}-}
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\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{Interpolation}}  \tn
% Row 0
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{fit 1 function to all points} \tn 
% Row Count 1 (+ 1)
% Row 1
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{given points without function} \tn 
% Row Count 2 (+ 1)
% Row 2
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{increase accuracy, decrease step size or increase order} \tn 
% Row Count 4 (+ 2)
% Row 3
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{3 pts f(x)=ax\textasciicircum{}2+bx+c  - substitute points - system of linear equations (GE- GJ- inverse)} \tn 
% Row Count 6 (+ 2)
% Row 4
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{Alternative function representation} \tn 
% Row Count 7 (+ 1)
% Row 5
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{f(x)= b0 + b1(x- x0) +b2(x-x0)(x-x1)+ b3(x-x0)(x-x1)(x-x2) +...} \tn 
% Row Count 9 (+ 2)
% Row 6
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{based on Taylor series} \tn 
% Row Count 10 (+ 1)
% Row 7
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{bs represent the slopes} \tn 
% Row Count 11 (+ 1)
% Row 8
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{efficient - quick} \tn 
% Row Count 12 (+ 1)
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{Splines}}  \tn
% Row 0
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{fit a function to each interval} \tn 
% Row Count 1 (+ 1)
% Row 1
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{used for large datapoints-{}-to avoid kinks} \tn 
% Row Count 2 (+ 1)
% Row 2
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{{\bf{Linear Splines}}} \tn 
% Row Count 3 (+ 1)
% Row 3
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{f(x)= f(xo) + m(x-xo)} \tn 
% Row Count 4 (+ 1)
% Row 4
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{interval surrounds point} \tn 
% Row Count 5 (+ 1)
% Row 5
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{{\emph{issues}}:} \tn 
% Row Count 6 (+ 1)
% Row 6
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{linearizing a non-linear function, oversimplifies behavior} \tn 
% Row Count 8 (+ 2)
% Row 7
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{discontinuity at the intermediate points - slope is no the same on either side} \tn 
% Row Count 10 (+ 2)
% Row 8
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{{\bf{Quadratic Splines}}} \tn 
% Row Count 11 (+ 1)
% Row 9
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{minimum of 2 intervals or 3 points} \tn 
% Row Count 12 (+ 1)
% Row 10
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{f(x)= a1x\textasciicircum{}2 +b1x+c1} \tn 
% Row Count 13 (+ 1)
% Row 11
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{3n unknowns -{}- n is \# of intervals} \tn 
% Row Count 14 (+ 1)
% Row 12
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{(2n equations ) substitute points in formulas} \tn 
% Row Count 15 (+ 1)
% Row 13
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{(n-1 equations) establish continuity with the slope at the intermediate points} \tn 
% Row Count 17 (+ 2)
% Row 14
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{assumption: a1=0} \tn 
% Row Count 18 (+ 1)
% Row 15
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{minimal effect on other intervals} \tn 
% Row Count 19 (+ 1)
% Row 16
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{under determined system by 1 equation} \tn 
% Row Count 20 (+ 1)
% Row 17
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{intermediate points are not independent} \tn 
% Row Count 21 (+ 1)
% Row 18
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{System of linear equations -{}-do not use iterative methods (not DDS)} \tn 
% Row Count 23 (+ 2)
% Row 19
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{all functions are dependent} \tn 
% Row Count 24 (+ 1)
% Row 20
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{{\bf{Cubic Splines}}} \tn 
% Row Count 25 (+ 1)
% Row 21
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{most popular method} \tn 
% Row Count 26 (+ 1)
% Row 22
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{minimum of 3 intervals or 4 points} \tn 
% Row Count 27 (+ 1)
% Row 23
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{4n equations - undetermined by 2 equations} \tn 
% Row Count 28 (+ 1)
% Row 24
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{assume 2nd derivative of outer points is 0} \tn 
% Row Count 29 (+ 1)
% Row 25
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{Alternative - Lagrange} \tn 
% Row Count 30 (+ 1)
\end{tabularx}
\par\addvspace{1.3em}

\vfill
\columnbreak
\begin{tabularx}{5.377cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{Splines (cont)}}  \tn
% Row 26
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{(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){]}} \tn 
% Row Count 3 (+ 3)
% Row 27
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{f(x)= (f''(xi-1)/6(xi -xi-1)) (xi -x)\textasciicircum{}3 + (f''(xi)/6(xi- xi-1)) (x- xi-1)\textasciicircum{}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)} \tn 
% Row Count 7 (+ 4)
% Row 28
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{solve all second derivatives first} \tn 
% Row Count 8 (+ 1)
% Row 29
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{all related by continuity} \tn 
% Row Count 9 (+ 1)
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}


% That's all folks
\end{multicols*}

\end{document}