1540470959-Boundary_Value_Problems_and_Partial_Differential_Equations__Powers

1.8 Numerical Determination of Fourier Coefficients 105

Figure 12 Graph of the difference betweenf(x)=sin(x)/xandF(x), the sum of
the Fourier series using the approximate coefficientsaˆ 0 throughaˆ 6.

2. Express the Fourier cosine coefficients of the example in terms of integrals
   of the form

Si

(

(n+ 1 )π

)

=

∫(n+ 1 )π

0

sin(t)

t dt.

This is the sine integral function and is tabulated in many books, especially

Handbook of Mathematical Functions, Abramowitz and Stegun, 1972.

3. Each entry in the list that follows represents the depth of the water in Lake
   Ontario (minus the low-water datum of 242.8 feet) on the first of the cor-
   responding month. Assuming that the water level is a periodic function
   of period one year, and that the observations are taken at equal intervals,
   compute the Fourier coefficientsaˆ 0 ,aˆ 1 ,bˆ 1 ,aˆ 2 ,bˆ 2 , thus identifying the
   mean level, and fluctuations of period 12 months, 6 months, 4 months,
   and so forth. Takex 0 as January,...,x 11 as December, andx 12 as January
   again.

Jan. 0.75 July 2.35

Feb. 0.60 Aug. 2.15

Mar. 0.65 Sept. 1.75

Apr. 1.15 Oct. 1.05

May 1.80 Nov. 1.00

June 2.25 Dec. 0.90

4. The numbers in the table that follows represent the monthly precipi-
   tation(ininchesofwater)inLakePlacid,NY,averagedoverthepe-
   riod 1950–1959. Find the approximate Fourier coefficientsaˆ 0 ,...,ˆa 6 and
   bˆ 1 ,...,ˆb 5.