Question

Ex. 3.9. Develop Fourier-Legendre series for f(x) = sin(21x). Graph on common axes the partial sums Sm, m = 1,2, ...,M up to that SM whose graph visually coincides with that of f(x) within the graphical accuracy. State the value of M you determined and list the exact values of all expansion coefficients from ao to am. Hint: to solve this problem you need to use one of the available sym- bolic algebra packages MAPLE or MATHEMATICA to evaluate the re- quired integrals and plot the resulting partial sums. Search function LegendreP and examples of its syntax and application in the built-in software help.

Answer 1

SOLUTION:

Given That

f(x)=sin(2πx) with m=1,2,.......M

So

The Legendre polynomials form

a complete Orthogonal system over the interval [-1,1]

with respect to the weight function==>
WI) = 1
,

The function may be expanded interms of them as

                          
(1) - Σα, ΡΑ(α)

To obtain, the coefficient $a_n$ in the expansion multiply both sides by
Pm (2)
and integrate,

                 
1 pcm)f(x)dır Əa | Pa(w)Pm(a)dr n=0 J-1

where the coefficient

an = + | Pn(z)f(x)dx J-1

Here   

  
f(1) = sin(272)

To find the coefficients
do, d1, 02, ...
:

The Legendre's Polynomial is as follows:

Pol.2) = 1

P1(x) = 2

P2(x) = 5(3.- 1)

$P_3(x)=\frac{1}{2}(5x^3-3x)$

$P_4(x)=\frac{1}{8}\left ( 35x^4-30x^2+3 \right )$

P5(x) == (6325 – 70x3 + 152)

To find $a_0:$

       $n=0\Rightarrow P_0=1\Rightarrow a_0=\frac{1}{2}\int_{-1}^{1}1sin(2\pi x)dx$

                                                    $a_0=0$

To find $a_1:$

      $n=1\Rightarrow P_1=x\Rightarrow a_1=\frac{3}{2}\int_{-1}^{1}xsin\left ( 2\pi x \right )dx$

                                                    $a_1=\frac{-3}{2\pi }$

To find $a_2:$

     $n=2\Rightarrow P_2=\frac{1}{2}(3x^2-1)$

                   $\Rightarrow a_2=\frac{5}{2}\int_{-1}^{1}\frac{1}{2}(3x^2-1)sin(2\pi x)dx=0$

                     $a_2=0$

To find $a_3:$

    $n=3\Rightarrow P_3(x)=\frac{1}{2}(5x^3-3x)$

   $a_3=\frac{7}{2}\int_{-1}^{1}\frac{1}{2}(5x^3-3x)sin(2\pi x)dx$

    $a_3=\frac{7}{4}\left [ \frac{15-4\pi^2}{2\pi^3} \right ]$

To find $a_4:$

     $n=4\Rightarrow P_4=\frac{1}{8}\left ( 35x^4-30x^2+3 \right )$

     $a_4=\frac{9}{2}\int_{-1}^{1}\frac{1}{8}\left ( 35x^4-30x^2+3 \right )sin(2\pi x)dx$

       $a_4=0$

To find $a_5:$

n = 5= P3(x) = (63x5 – 70x} + 152)

$a_5=-\frac{(945-480\pi^2+16\pi^4)}{16\pi^5}$

From the above calculations,

we observe that $a_0=a_2=a_4=....=0$ ,

That is , all even $a_n$ vanish. $sin(2\pi x)$ is an odd function.

   $a_1=\frac{-3}{2\pi }$ ,        $a_3=\frac{7}{4}\left [ \frac{15-4\pi^2}{2\pi^3} \right ]$ ,          $a_5=-\frac{(945-480\pi^2+16\pi^4)}{16\pi^5}$