1. Boyce/DiPrima 9th ed, Ch 10.8 Appendix A: Derivation of the Heat Conduction Equation Elementary Differential Equations and Boundary Value Problems, 9th edition, by William E. Boyce and Richard C. DiPrima, ©2009 by John Wiley & Sons, Inc.
In this section we derive the differential equation that, to a first approximation at least, governs heat conduction in solids.
It is important to understand that the mathematical analysis of a physical situation or process such as this ultimately rests on a foundation of empirical knowledge of the phenomenon.
Consider the uniform rod insulated on the lateral surfaces so that heat can flow only in the axial direction.
It has been demonstrated many times if two parallel cross sections of same area A and different temperatures T1 and T2 are separated by a small distance d, an amount of heat per unit time will pass from the warmer section to the cooler one.

2. Fourier’s Law of Heat Conduction
The amount of heat per unit time that passes from the warmer section to the cooler one is proportional to the area A and the temperature difference |T2 – T1|, and is inversely proportional to the distance d. Thus
Amount of heat per unit time =  A |T2 – T1| / d,
where the proportionality constant  > 0 is called the thermal conductivity and depends primarily on the material of the rod.
The relation above is often called Fourier’s law of heat conduction, and is an empirical result, not a theoretical one.
However, has been verified by careful experiment many times. It is the basis of the mathematical theory of heat conduction.

3. Assumptions (1 of 3)
Now consider a straight rod of uniform cross section and homogeneous material.
Let the x-axis be chosen to lie along the axis of the bar, and let x = 0 and x = L denote the ends of the bar. See figure below.
We assume the sides of the bar are perfectly insulated so that no heat passes through them.

4. Assumptions (2 of 3)
Assume that u is a function only of the axial coordinate x and time t, and not on the lateral coordinates y and z.
In other words, we assume that the temperature remains spatially constant on any cross section of the bar.
This assumption is usually satisfactory when the lateral dimensions of the rod are small compared to its length.

5. Assumptions (3 of 3)
The differential equation governing the temperature in the bar is an expression of a fundamental physical balance.
That is, we assume that the rate at which heat flows into any portion of the bar is equal to the rate at which heat is absorbed in that portion of the bar.
The terms in the equation are called the flux (flow) term and the absorption term, respectively.

6. Flux Term (1 of 3)
Consider an element of the rod between the cross sections at x = x0 and x = x0 + x, where x0 is arbitrary and x is small.
Recall Fourier’s law of heat conduction:
Amount of heat per unit time =  A |T2 – T1| / d
The instantaneous rate of heat transfer H(x0,t) from left to right across the cross section x = x0 is given by

7. Flux Term (2 of 3)
The heat transfer H(x0,t) rate from left to right at x = x0 is
The negative sign appears because there will be a positive flow of heat from left to right only if the temperature is greater on left of x = x0 than on the right. In this case ux(x0, t) is negative.
Similarly, the rate at which heat passes from left to right through the cross section at x = x0 + x is given by

8. Flux Term (3 of 3) From the previous slide, we have
The net rate at which heat flows into the segment of the bar between x = x0 and x = x0 + x thus is given by
The amount of heat entering this bar element in time t is thus

9. Absorption Term (1 of 2)
Consider again the element of the rod between x0 and x0 + x.
The average change in temperature, u, in the time interval t, is proportional to the amount of heat Qt introduced and inversely proportional to the mass m of the element.
Thus where the constant of proportionality s is the specific heat of the material of the bar, and  is its density.

10. Absorption Term (2 of 2)
The average change in temperature u in the bar element under consideration is equal to the actual temperature change at some intermediate point x0 + x, where 0 <  < 1.
Thus or

11. Heat Conduction Equation
From flux and absorption derivations, we have, respectively,
Balancing the flux and absorption terms, we have
Dividing this equation by xt, and then letting x  0 and t  0, we obtain the heat conduction or diffusion equation
where 2 =  / s is the thermal diffusivity, and is a parameter that depends on the material in the bar.

12. Boundary Conditions (1 of 2)
Some relatively simple boundary conditions may be imposed at the ends of the bar.
For example, the temperature at an end may be maintained at some constant value T.
This might be accomplished by placing the end of the bar in thermal contact with some reservoir of sufficient size so that any heat that flows between the bar and the reservoir does not appreciably alter the temperature of the reservoir.
At an end where this is done the boundary condition is u = T.
If the end of the bar is insulated so that no heat passes through it, then the boundary condition is ux = 0.

13. Boundary Conditions (2 of 2)
A more general type of boundary condition occurs if the rate of heat through an end of the bar is proportional to the temperature there.
Let us consider the end x = 0, where the rate of flow of heat from left to right is given by H(0,t) = - Aux(0,t).
Hence the rate of heat flow out of the bar (from right to left) at x = 0 is  Aux(0,t).
If this quantity is proportional to the temperature u(0,t), then we obtain the boundary condition
Similarly, if heat flow is taking place at x = L, then

14. Initial Condition
To determine completely the flow of heat in the bar it is necessary to state the temperature distribution at one fixed instant, usually taken as the initial time t = 0.
This initial condition is of the form
The problem is then to solve the differential equation
subject to one of the boundary conditions at each end and to the initial condition above at t = 0.

15. Generalizations of the Heat Equation (1 of 7)
Several generalizations of the heat equation
also occur in practice.
First, the bar material may be nonuniform and the cross section may not be constant along the length of the bar.
In this case the parameters , , s, and A may depend on the axial variable x.
Then the rate of heat transfer equations becomes

16. Axially Dependent Dimensions (2 of 7)
As before, if we substitute
into
and eventually into
and proceeding as before, we obtain the partial differential equation

17. Heat Equation for Nonuniform Bar (3 of 7)
The equation
is usually written in the form
where p(x) = (x)A(x) > 0 and r(x) = s(x)(x)A(x) > 0.

18. Source Term (4 of 7)
Another generalization occurs if there are other ways in which heat enters or leaves the bar.
Suppose that there is a source that adds heat to the bar at a rate G(x,t,u) per unit time per unit length, where G(x,t,u) > 0.
Then we must add the term G(x,t,u)xt to the flux term, or left side, of the balance equation
This leads to the differential equation

19. Generalized Heat Equation (5 of 7)
Our heat equation is
If G(x,t,u) < 0, then we speak of a sink that removes heat from the bar at the rate G(x,t,u) per unit time per unit length.
To make the problem tractable, we must restrict the form of the function G.
In particular, we assume that G is linear in u and that the coefficient of u does not depend on t. Thus we write
and hence
This equation is sometimes called the generalized heat conduction equation.

20. Multidimensional Heat Equation (6 of 7)
Finally, if instead of a one-dimensional bar, we consider a body with more than one significant space dimension, then the temperature is a function of two or three space coordinates rather than x alone.
Considerations similar to those leading up to the one-dimensional heat equation
can be employed to yield the two dimensional equation
or in three dimensions,

21. Multidimensional Heat Equation and Boundary Conditions (7 of 7)
Thus the two and three dimensions, respectively, we have
The boundary conditions corresponding to u = T and ux = 0 for multidimensional problems correspond to a prescribed temperature distribution on the boundary, or to an insulated boundary.
Similarly, the initial temperature distribution will in general be a function of x and y, or x, y and z, for the above two and three dimensional heat equations, respectively.