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Diffstat (limited to 'src/input/heat.input.pamphlet')
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diff --git a/src/input/heat.input.pamphlet b/src/input/heat.input.pamphlet new file mode 100644 index 00000000..d1bf86c9 --- /dev/null +++ b/src/input/heat.input.pamphlet @@ -0,0 +1,47 @@ +\documentclass{article} +\usepackage{axiom} +\begin{document} +\title{\$SPAD/src/input heat.input} +\author{The Axiom Team} +\maketitle +\begin{abstract} +\end{abstract} +\eject +\tableofcontents +\eject +\section{License} +<<license>>= +--Copyright The Numerical Algorithms Group Limited 1996. +@ +<<*>>= +<<license>> +-- ----------[ A x i o m ]---------- +-- ---------- Initialization ---------- +)set messages autoload off +)set quit unprotected +-- This is the heat equation +u:= operator('u); +heat:= D(u(x, t), t) - D(u(x, t), x, 2) = 0 +-- This is the similarity form of the proposed solution +f:= operator('f); +s:= rule(u(x, t) == f(x/sqrt(t))/sqrt(t)) +-- Apply s to the heat equation +s(lhs(heat)) = 0 +-- Change to the similarity variable z = x/sqrt(t) +subst(lhs(%), x = z*sqrt(t)) = 0 +-- Eliminate the denominator +% * denom(lhs(%)) +-- Now, solve the ordinary differential equation +eq:=% +solve(%, f, z=0,[k1,k2]) +-- Finally, transform back to the original variables +subst(%, z = x/sqrt(t))/sqrt(t) +-- If we set k2 = 0 and k1 = 1/(2*sqrt(%pi)) in the previous expression, +-- we will obtain the usual fundamental solution of the heat equation +subst(%, [k2 = 0, k1 = 1/(2*sqrt(%pi))]) +@ +\eject +\begin{thebibliography}{99} +\bibitem{1} nothing +\end{thebibliography} +\end{document} |