1540470959-Boundary_Value_Problems_and_Partial_Differential_Equations__Powers

172 Chapter 2 The Heat Equation

A

n 0. 2500 0. 5000 1. 0000 2. 0000 4. 0000

12. 5704 2. 2889 2. 0288 1. 8366 1. 7155

25. 3540 5. 0870 4. 9132 4. 8158 4. 7648

38. 3029 8. 0962 7. 9787 7. 9171 7. 8857

411. 3348 11. 1727 11. 0855 11. 0408 11. 0183

514. 4080 14. 2764 14. 2074 14. 1724 14. 1548

Table 2 First five positive solutions of the equation tan(x)=−Ax

Figure 6 Graphs of tan(λa)and−λκ/h. The points of intersection are solutions
of tan(λa)=−λκ/h, eigenvalues of the problem Eqs. (9)–(10). The intersection
atλ=0 corresponds to the trivial solution.

From sketches of the graphs of tan(λa)and−κλ/h(Fig. 6), we see that there is
an infinite number of solutions,λ 1 ,λ 2 ,λ 3 ,..., and that, for very largen,λnis
givenapproximatelyby

λn∼=^2 n 2 −^1 πa.

Table 2 shows the first five values of the productλafor several different values
of the dimensionless parameterκ/ha. (More solutions are tabulated inHand-
book of Mathematical Functionsby Abramowitz and Stegun.)
Thus we have for eachn= 1 , 2 ,...an eigenvalueλ^2 nand an eigenfunction
φn(x), which satisfies the eigenvalue problem Eqs. (9) and (10). Accompanying
φn(x)is the function

Tn(t)=exp

(

−λ^2 nkt

)