Flow Diagram for a numerical algorithm
Author
Mubarak Imam
Last Updated
8年前
License
Creative Commons CC BY 4.0
Abstract
Introducing Canonical Polynomial approximation methods for solving singularly perturbed bvps.
Introducing Canonical Polynomial approximation methods for solving singularly perturbed bvps.
\documentclass{article}
\usepackage{tikz,amsmath}
\usetikzlibrary{shapes.geometric, arrows}
\usepackage[paperheight=6.5in,paperwidth=4.8in,margin=0.1in]{geometry}
\tikzstyle{process} = [rectangle, minimum width=3em, minimum height=2em, text centered, draw=blue, fill=gray!10]
\tikzstyle{startend} = [ellipse, minimum width=2em, minimum height=1em, text centered, draw=red, fill=gray!10]
\tikzstyle{arrow} = [thick,->,>=stealth]
\pagestyle{plain}
\begin{document}
\begin{figure}
\centering
\hfill\\
{\scriptsize \begin{tikzpicture}[node distance=2em]
\node (S) [startend] {START};
\node (S1) [process, below of=S, yshift= -2em]{\parbox[t][][t]{10cm}{Put problem in standard form\newline \begin{math}\begin{aligned}
\varepsilon y^{(m)}(x)&+\alpha_1(x)y^{(m-1)}(x)+\alpha_2(x)y^{(m-2)}(x)+\cdots\\
&+\alpha_{m-1}(x)y'(x)+\alpha_m(x)y(x)=f(x), \qquad x\in[a,b]
\end{aligned}\end{math}}};
\node (S2) [process, below of=S1, yshift=-3em]{\parbox[t][][t]{10cm}{Assume an approximate solution $y_n(x)=\sum_{i=0}^nc_nQ_(x)$, where $n$ is the degree of approximation.}};
\node (S3) [process, below of=S2, yshift=-2em]{\parbox[t][][t]{10cm}{Obtain the canonical polynomials using the differential operator from the equation in STEP 1.}};
\node (S4) [process, below of=S3, yshift=-2em]{\parbox[t][][t]{10cm}{Apply the given boundary conditions on the assumed solution in STEP 2 to obtain $(m)$ linear algebraic equations.}};
\node (S5) [process, below of=S4, yshift=-2em]{\parbox[t][][t]{10cm}{Substitute the approximate solution from STEP 2 into the problem in STEP 1.}};
\node (S6) [process, below of=S5, yshift=-4em, xshift = -12em]{\parbox[t][][t]{5cm}{collocate the results from STEP 5 at $x_k=a+\frac{(b-a)k}{N+2-m}$, $k=0(1)N+1-m$ to obtain $(N-m+1)$ algebraic equations.}};
\node (S7) [process, below of=S5, yshift=-3em, xshift = 10em]{\parbox[t][][t]{5cm}{add a small perturbation term $H_n(x)$ into the results from STEP 5.}};
\node (S8) [process, below of=S6, yshift=-5em]{\parbox[t][][t]{5cm}{add the results from STEP 4 and STEP 6 to obtain two sets of $(n+1)$ system of equations.}};
\node (S9) [process, below of=S8, yshift=-4em]{\parbox[t][][t]{5cm}{solve the resulting equations from STEP 7 using the built-in Maple \texttt{solve} command.}};
\node (S10) [process, below of=S9, yshift=-4em]{\parbox[t][][t]{5cm}{Substitute the values of the constant coefficients into the approximate solution in STEP 2 for SCM}};
\node (S11) [process, below of=S7, yshift=-4em]{\parbox[t][][t]{5cm}{collocate the results from STEP 6 at $x_l=a+\frac{(b-a)l}{N+2}$, $l=0(1)N+1$ to obtain $(n+1)$ algebraic equations.}};
\node (S12) [process, below of=S11, yshift=-4em]{\parbox[t][][t]{5cm}{add the results from STEP 4 and STEP 7 to obtain two sets $(n+m+1)$ system of equations.}};
\node (S13) [process, below of=S12, yshift=-4em]{\parbox[t][][t]{5cm}{solve the resulting equations from STEP 8 using the built-in Maple \texttt{solve} command.}};
\node (S14) [process, below of=S13, yshift=-4em]{\parbox[t][][t]{5cm}{Substitute the values of the constant coefficients into the approximate solution in STEP 2 PCM.}};
\node (S15) [startend, below of=S, yshift=-52em]{STOP};
\draw [arrow] (S) -- (S1);
\draw [arrow] (S1) -- (S2);
\draw [arrow] (S2) -- (S3);
\draw [arrow] (S3) -- (S4);
\draw [arrow] (S4) -- (S5);
\draw [arrow] (S5) -- node[anchor=east]{SCM}(S6);
\draw [arrow] (S5) -- node[anchor=west]{PCM}(S7);
\draw [arrow] (S6) -- (S8);
\draw [arrow] (S8) -- (S9);
\draw [arrow] (S9) -- (S10);
\draw [arrow] (S7) -- (S11);
\draw [arrow] (S11) -- (S12);
\draw [arrow] (S12) -- (S13);
\draw [arrow] (S13) -- (S14);
\draw [arrow] (S10) |- (S15);
\draw [arrow] (S14) |- (S15);
\end{tikzpicture}}
\end{figure}
\end{document}