7 Heat Equation-2.pdf

Heat equation in Mathematics mathed of physics

1. Heat Equation

2. Introduction
Intheearly1800s,J.Fourierbeganamathematicalstudyof
heat.Adeeperunderstandingofheatflowhadsignificant
applicationsinscienceandwithinindustry.Abasicversionof
Fourier'seffortsistheproblem
??????
2
�
????????????=�
??????
BC
�0,�=0=�??????,�;�??????,0=�??????
where
�??????,�is the temperature at position x at time t
??????
2
a constant
f is a given function

3. Fourier's analysis resulted in the following solution form
�??????,�=෍
??????=1
∞
�
??????�
−??????Τ(????????????????????????)
2
sin
??????????????????
??????
Provided f could be written in the form
�??????=෍
??????=1
∞
�
??????sin
??????????????????
??????
ThispromptedFouriertoformamethodforexpressing
functionsasinfinitesumsofsines(and/orcosines),called
Fourierseries.

4. Derivation of Heat Equation in One
Dimension
Suppose we have a thin bar of length L wrapped around the x-
axis, so that x = 0 and x = L are the ends of the bar.
x = 0 x = L B
A
C
AtpointA,BandC:Temperatureuisfunctionofonlyxandt
andremainssameas
wehaveassumedthatBaristhin.

5. Assumptions
•barisofhomogeneousmaterial,isstraightandhasuniform
cross-sections.
•sidesofthebarareperfectlyinsulatedsothatnoheatpasses
throughthem.
•Sinceourbaristhin,thetemperatureucanbeconsideredas
constantonanygivencross-sectionandsodependsonthe
horizontalpositionalongthex-axis.Henceuisafunction
onlyofpositionxandtimet.

6. Let’s derive heat equation as given below
??????
2
�
????????????=�
??????(1)
where ??????
2
= k/ρcis called the thermal diffusivity (in (length)
2
/time). It
has the SI derived unit of m
2
/s.
k, ρ and c are positive constants that depend on the material of the bar.

7. Consider a section D of the bar, with ends at x
oand x
1.
x
0
x
1
D
**Thermaldiffusivityisthethermalconductivitydivided
bydensityandspecificheatcapacityatconstantpressure.It
measurestherateoftransferofheatofamaterialfromthehot
endtothecoldend.

8. Now, the total amount of heat H = H(t) in D (may be in joules, calories)
is
??????�=න
??????
0
??????1
ρc�??????,��??????
Differentiating we obtain
�??????�
��
=ρcන
??????0
??????1
�
????????????,��??????
Above, c is the specific heat of the material (it is the amount of heat that
must be added to one unit of mass of the substance in order to cause an
increase of one unit in temperature) and ρis the density of the material.

9. Now,sincethesidesofthebarareinsulated,theonlyway
heatcanflowintooroutofDisthroughtheendsatx
oandx
1.
Fourier'slawofheatflowstatesthatheatflowsfromhotter
regionstocolderregionsandtheflowrateisproportionalto
u
x.
Now,thenetratechangeofheatHinDisjusttherateat
whichheatentersDminustherateatwhichheatleavesD.i.e.,
�??????
��
=−k�
??????(??????
0,�)−(−k�
??????(??????
1,�))

10. The minus sign appears in the above two terms since there will be a
positive flow of heat from left to right only if the temperature is greater
to the left of x = xo than to the right (in this case, �
??????(??????
0,�)will be
negative).
Now. simplifying the above and applying the fundamental theorem of
calculus we obtain
�??????
��
=k�
??????(??????
1,�)−k�
??????(??????
0,�)
�??????�
��
=kන
??????
0
??????
1
�
??????????????????,��??????
The first fundamental theorem of calculus states that, if f is
continuous on the closed interval [a,b] and F is the indefinite
integral of f on [a,b], then
න
�
�
�??????�??????=??????�−??????(�)

11. Now, comparing our two expressions for dH/dt we form the
relationship
cρන
??????0
??????
1
�
????????????,��??????=kන
??????0
??????
1
�
??????????????????,��??????
Differentiating both sides with respect to x
lwe obtain
cρ�
??????=??????�
????????????(2)

12. Since the above arguments work for all intervals from x
oto x
1and for all t > 0 it
follows that the above PDE is satisfied for our interval of interest: from x = 0 to x =
L (and all t > 0)
The PDE (2) essentially describes a fundamental physical balance: the rate at which
heat flows into any portion of the bar is equal to the rate at which heat is absorbed
into that portion of the bar.
Hence the two terms in (2) are sometimes referred to as:
"absorption term" (cρ�
??????); and the "flux term” (??????�
????????????)
Interpretation of heat equation
What is 2D heat equation

17. The function will
satisfy the heat
equation and the
boundary condition of
zero temperature on
the ends of the bar.