Heat Conduction Equation

Author

Eric W. Weisstein
February 6, 2005

This notebook downloaded from http://mathworld.wolfram.com/notebooks/DifferentialEquations/HeatConductionEquation.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/HeatConductionEquation.html.

For a list of Eric's math packages that may be needed to evaluate this notebook, see http://mathworld.wolfram.com/packages/.  An archive of these packages may be downloaded from http://mathworld.wolfram.com/packages/MathWorldPackages.zip, as well as from the Mathematica Information Center's MathSource item 4775.  A list of Eric's utility packages that may be needed to evaluate this notebook may be downloaded from MathSource item 5087.

©2005 Wolfram Research, Inc. except for portions noted otherwise

1-D Solution

eqn=D[T[x,t],t]==κ D[T[x,t],{x,2}];

DSolve[eqn,T[x,t],{x,t}]

eqn/.T:>Function[{x,t},Exp[-κ t/(λ^2)](𝒟 Cos[(x/λ)]+ℰ Sin[(x/λ)])]//Simplify

DSolve[{eqn,T[0,t]==0,T[L,t]==0},T[x,t],{x,t}]

Orthogonality

Simplify[Integrate[Sin[n π x/L]^2,{x,0,L}],n∈Integers]

FullSimplify[Integrate[Sin[n π x/L]Sin[m π x/L],{x,0,L}],(m|n)∈Integers]