using Laplace transform to solve heat equation

Along the whole positive x-axis, we have an heat-conducting rod, the surface of which is .  The initial temperature of the rod is 0 .  Determine the temperature functionu(x,t)  when at the time  t=0

(a) the head  x=0  of the rod is set permanently to the constant temperature;

(b) through the head  x=0  one directs a constant heat flux.

The heat equation in one dimension reads

uxx′′(x,t)=1c2ut(x,t). (1)

In this we have

(a) {boundary conditionsu(,t)=0,u(0,t)=u0,initial conditions  u(x, 0)=0,ut(x, 0)=0for x> 0


(b) {boundary conditionsu(,t)=0,ux(0,t)=-k,initial conditions  u(x, 0)=0,ut(x, 0)=0for x> 0.

For solving (1), we first form its Laplace transformMathworldPlanetmath (see the table of Laplace transforms)

Uxx′′(x,s)=1c2[sU(x,s)-u(x, 0)],

which is a ordinary linear differential equation


of order ( two.  Here, s is only a parametre, and the general solution of the equation is


(see this entry (  Since

U(,s)=0e-stu(,t)𝑑t=00𝑑t 0,

we must have  C1=0.  Thus the Laplace transform of the solution of (1) is in both cases (a) and (b)

U(x,s)=C2e-scx. (2)

For (a), the second boundary conditionMathworldPlanetmath implies   U(0,s)=u0s.  But by (2) we must have  U(0,s)=C21, whence we infer that  C2=u0s.  Accordingly,


which corresponds to the solution function

u(x,t):=u0 erfcx2ct

of the heat equation (1).

For (b), the second boundary condition says that  Ux(0,s)=-ks,  and since (2) implies that  Ux(x,s)=-scC2e-scx,  we can infer that now 




which corresponds to

u(x,t):=k[2ctπe-x24c2t-x erfcx2ct].

Title using Laplace transform to solve heat equation
Canonical name UsingLaplaceTransformToSolveHeatEquation
Date of creation 2015-05-30 6:55:05
Last modified on 2015-05-30 6:55:05
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 11
Author pahio (2872)
Entry type Example
Classification msc 35K20
Classification msc 35Q99
Classification msc 35K05
Synonym using Laplace transform to solve partial differential equation
Related topic LaplaceTransform