%% Created by Maple 2015.2, Windows 8 %% Source Worksheet: Applying the taylor series method to the L1 model.mw %% Generated: Mon May 07 21:49:07 BST 2018 \documentclass{article} \usepackage{maplestd2e} \def\emptyline{\vspace{12pt}} \begin{document} \pagestyle{empty} \DefineParaStyle{Maple Heading 1} \DefineParaStyle{Maple Text Output} \DefineParaStyle{Maple Dash Item} \DefineParaStyle{Maple Bullet Item} \DefineParaStyle{Maple Normal} \DefineParaStyle{Maple Heading 4} \DefineParaStyle{Maple Heading 3} \DefineParaStyle{Maple Heading 2} \DefineParaStyle{Maple Warning} \DefineParaStyle{Maple Title} \DefineParaStyle{Maple Error} \DefineCharStyle{Maple Hyperlink} \DefineCharStyle{Maple 2D Math} \DefineCharStyle{Maple Maple Input} \DefineCharStyle{Maple 2D Output} \DefineCharStyle{Maple 2D Input} \section{\textbf{Apply the taylor series method to the L1 model}} \begin{Maple Normal}{ \begin{Maple Normal}{ \mapleinline{inert}{2d}{}{$\displaystyle$} }\end{Maple Normal} }\end{Maple Normal} \begin{maplegroup} \begin{mapleinput} \mapleinline{active}{2d}{read "LieDerProg.mpl"; -1}{} \end{mapleinput} \end{maplegroup} \begin{Maple Normal}{ \begin{Maple Normal}{ Define the L2 model statespace function and output function.}\end{Maple Normal} \begin{Maple Normal}{ Define the L2 model statespace function and output function.}\end{Maple Normal} }\end{Maple Normal} \begin{maplegroup} \begin{mapleinput} \mapleinline{active}{2d}{F := [k[a1]*C[T]*(R-x[1])-k[d1]*x[1]]; 1; H := alpha*x[1]; 1}{} \end{mapleinput} \mapleresult \begin{maplelatex} \mapleinline{inert}{2d}{F := [k[a1]*C[T]*(R-x[1])-k[d1]*x[1]]}{$\displaystyle F\, := \,[k_{{{\it a1}}}C_{{T}} \left( R-x_{{1}} \right) -k_{{{\it d1}}}x_{{1}}]$} \end{maplelatex} \mapleresult \begin{maplelatex} \mapleinline{inert}{2d}{H := alpha*x[1]}{$\displaystyle H\, := \,\alpha\,x_{{1}}$} \end{maplelatex} \end{maplegroup} \begin{Maple Normal}{ \begin{Maple Normal}{ }\end{Maple Normal} \begin{Maple Normal}{ Generate the first n derivatives of the output in either phase.}\end{Maple Normal} }\end{Maple Normal} \begin{maplegroup} \begin{mapleinput} \mapleinline{active}{2d}{n := 2; -1; La := subs([x[1] = 0, x[2] = 0, alpha = 1000], listLieDer(H, F, n)); -1; Ld := subs([C[T] = 0, alpha = 1000], listLieDer(H, F, n)); -1}{} \end{mapleinput} \end{maplegroup} Consider two parameter vectors giving equal the output in the dissociation. \begin{Maple Normal}{ \begin{Maple Normal}{ }\end{Maple Normal} }\end{Maple Normal} \begin{maplegroup} \begin{mapleinput} \mapleinline{active}{2d}{alt := C = Ch, R = Rh, k = kh, x = xh; -1; Sd := [solve([op(subs(alt, Ld)-Ld)])]; -1; i := [NonZeroSols(Sd)]; -1; Sd := Sd[i]}{} \end{mapleinput} \mapleresult \begin{maplelatex} \mapleinline{inert}{2d}{Sd := [{k[d1] = kh[d1], kh[d1] = kh[d1], x[1] = xh[1], xh[1] = xh[1]}]}{$\displaystyle {\it Sd}\, := \,[ \left\{ k_{{{\it d1}}}={\it kh}_{{{\it d1}}},{\it kh}_{{{\it d1}}}={\it kh}_{{{\it d1}}},x_{{1}}={\it xh}\\ \mbox{}_{{1}},{\it xh}\\ \mbox{}_{{1}}={\it xh}\\ \mbox{}_{{1}} \right\} ]$} \end{maplelatex} \end{maplegroup} \begin{Maple Normal}{ \begin{Maple Normal}{ }\end{Maple Normal} }\end{Maple Normal} \begin{Maple Normal}{ \begin{Maple Normal}{ Consider two parameter vectors giving equal the output in the Association.}\end{Maple Normal} }\end{Maple Normal} \begin{maplegroup} \begin{mapleinput} \mapleinline{active}{2d}{Sa := [solve([op(subs(Sd[1], subs(alt, La))-La)])]; -1; i := [NonZeroSols(Sa)]; -1; Sa1 := Sa[i]}{} \end{mapleinput} \mapleresult \begin{maplelatex} \mapleinline{inert}{2d}{Sa1 := [{R = R, Rh = R*(Ch[T]*kh[a1]-k[d1]+kh[d1])/(Ch[T]*kh[a1]), C[T] = (Ch[T]*kh[a1]-k[d1]+kh[d1])/k[a1], Ch[T] = Ch[T], k[a1] = k[a1], k[d1] = k[d1], kh[a1] = kh[a1], kh[d1] = kh[d1]}]}{$\displaystyle {\it Sa1}\, := \,[ \left\{ R=R,{\it Rh}={\frac {R \left( {\it Ch}_{{T}}{\it kh}_{{{\it a1}}}-k_{{{\it d1}}}+{\it kh}_{{{\it d1}}} \right) \\ \mbox{}}{{\it Ch}_{{T}}{\it kh}_{{{\it a1}}}}},C_{{T}}={\frac {{\it Ch}_{{T}}{\it kh}_{{{\it a1}}}-k_{{{\it d1}}}+{\it kh}_{{{\it d1}}}}{k_{{{\it a1}}}}},{\it Ch}_{{T}}={\it Ch}_{{T}},k_{{{\it a1}}}=k_{{{\it a1}}},k_{{{\it d1}}}=k_{{{\it d1}}},{\it kh}_{{{\it a1}}}={\it kh}_{{{\it a1}}\\ \mbox{}},{\it kh}_{{{\it d1}}}={\it kh}_{{{\it d1}}} \right\} ]$} \end{maplelatex} \end{maplegroup} \begin{Maple Normal}{ \begin{Maple Normal}{ For a parameter vector to give equal output in both phases it must satisfy the following.}\end{Maple Normal} \begin{Maple Normal}{ For a parameter vector to give equal output in both phases it must satisfy the following.}\end{Maple Normal} \begin{Maple Normal}{ For a parameter vector to give equal output in both phases it must satisfy the following.}\end{Maple Normal} }\end{Maple Normal} \begin{maplegroup} \begin{mapleinput} \mapleinline{active}{2d}{list1 := solve(convert(union(Sd[1], Sa1[1]), list)); -1; ~[print](list1)[]; 1}{} \end{mapleinput} \mapleresult \begin{maplelatex} \mapleinline{inert}{2d}{R = Rh}{$\displaystyle R={\it Rh}$} \end{maplelatex} \mapleresult \begin{maplelatex} \mapleinline{inert}{2d}{Rh = Rh}{$\displaystyle {\it Rh}={\it Rh}$} \end{maplelatex} \mapleresult \begin{maplelatex} \mapleinline{inert}{2d}{C[T] = Ch[T]*kh[a1]/k[a1]}{$\displaystyle C_{{T}}={\frac {{\it Ch}_{{T}}{\it kh}_{{{\it a1}}}}{k_{{{\it a1}}}}}$} \end{maplelatex} \mapleresult \begin{maplelatex} \mapleinline{inert}{2d}{Ch[T] = Ch[T]}{$\displaystyle {\it Ch}_{{T}}={\it Ch}_{{T}}$} \end{maplelatex} \mapleresult \begin{maplelatex} \mapleinline{inert}{2d}{k[a1] = k[a1]}{$\displaystyle k_{{{\it a1}}}=k_{{{\it a1}}}$} \end{maplelatex} \mapleresult \begin{maplelatex} \mapleinline{inert}{2d}{k[d1] = kh[d1]}{$\displaystyle k_{{{\it d1}}}={\it kh}_{{{\it d1}}}$} \end{maplelatex} \mapleresult \begin{maplelatex} \mapleinline{inert}{2d}{kh[a1] = kh[a1]}{$\displaystyle {\it kh}_{{{\it a1}}}={\it kh}_{{{\it a1}}}$} \end{maplelatex} \mapleresult \begin{maplelatex} \mapleinline{inert}{2d}{kh[d1] = kh[d1]}{$\displaystyle {\it kh}_{{{\it d1}}}={\it kh}_{{{\it d1}}}$} \end{maplelatex} \mapleresult \begin{maplelatex} \mapleinline{inert}{2d}{x[1] = xh[1]}{$\displaystyle x_{{1}}={\it xh}_{{1}}$} \end{maplelatex} \mapleresult \begin{maplelatex} \mapleinline{inert}{2d}{xh[1] = xh[1]}{$\displaystyle {\it xh}_{{1}}={\it xh}_{{1}}$} \end{maplelatex} \end{maplegroup} \begin{maplegroup} \begin{mapleinput} \mapleinline{active}{2d}{}{} \end{mapleinput} \end{maplegroup} \begin{Maple Normal}{ \begin{Maple Normal}{ \mapleinline{inert}{2d}{}{$\displaystyle$} }\end{Maple Normal} }\end{Maple Normal} \end{document}