let the approximate solution,
ϕ=co+∈cisinπ2(2i−1)x
Boundary condition 1: ϕ(o)=0
∴co=0
∴ϕ=∈cisinπ2(2i−1)x.
=cisinπx2+c2sin3πx2+c3sin5x2
Boundary condition 2 : ϕ (1)=0
∴0=c1−c2+c3
∴c2=c1+c3
ϕ=c1[sinπx2+sin3πx2]+c3[sin3πx2+sin5πx2]
ϕ′=c1π2[cosπx2+3cos3πx2]+c3π2[3cos3πx2+5cos5πx2]
ϕ″=−c1π24[sinπx2+9sin3πx2]−c3π24[9sin3πx2+25sin5πx2]
\therefore Residue, R,
R = \phi''-cos \pi x
=\frac{\pi^2}{4}\bigg[c_1\bigg[sin \frac{\pi x}{2}+9 sin\frac{3 \pi x}{2}\bigg]+c_3\bigg[9sin \frac{3 \pi x}{2}+25 sin \frac{5x \pi}{2}\bigg]-cos\hspace{0.2cm} \pi x
W_i= coefficients of ci in u .
for i = 1 \Rightarrow W_1= sin \frac{\pi}{2}x+ sin \frac{3 \pi}{2}x.
for i = 2 \Rightarrow W_2= sin \frac{3 \pi}{2}x + sin \frac{5\pi}{2}x.
for i = 1
\int\limits_o^1 w_1 R dx=0
\frac{\pi^2}{4} \int\limits_0^1 \bigg[sin \frac{\pi}{2}x+sin\frac{3 \pi}{2}x\bigg]\bigg[c_1\bigg(sin \frac{\pi}{2}x+9 sin \frac{3\pi}{3}x\bigg)
+ c_3\bigg(9 sin \frac{3\pi x}{2}+25sin \frac{3pi x}{2}\bigg)-cos \pi x\bigg] dx=c
\therefore c_1 \int\limits_0^1\bigg(sin\frac{\pi}{2}+sin \frac{3 \pi}{2}x\bigg) \bigg(sin\frac{\pi}{2}x+9 sin \frac{3\pi}{2}x\bigg)dx
c_3\int\limits_0^1\bigg(sin\frac{\pi}{2}x+sin\frac{3\pi}{2}x\bigg)\bigg(9sin\frac{3\pi}{2}x+25 sin \frac{5\pi}{2}x)
- \int\limits_0^1\bigg(sin\frac{\pi}{2}x+sin\frac{3\pi}{2}x\bigg)cos \hspace{0.2cm}\pi x dx \frac{4}{\pi^2=0}
5 c_1+2.84c_3-0.069=0
5 c_1+2.84 c_3 = 0.069 \hspace{2cm}------(1)
Exact solution :
\frac{d^2\phi}{dx^2}+cos \pi x=0
\frac{d^2 \phi}{dx^2}=-cos \pi x
\frac{d \phi}{dx}=\frac{sin \pi x}{\pi}+c^1
\phi=\frac{cos \pi x}{\pi^2}+c_1x+c_2
for x = 0 : \phi = 0
0 = \frac{1}{\pi^2}+c_2 \hspace{0.6cm} \therefore c_2=-\frac{1}{\pi ^2}
\therefore \phi = \frac{cos \pi x}{\pi ^2}-\frac{1}{\pi^2}+c_1x
for x = 1 : \phi = 0
0 = -\frac{1}{\pi ^2}-\frac{1}{\pi^2}+c_1 \hspace{0.6cm} \therefore c_1=+\frac{2}{\pi^2}
\therefore u =\frac{cos \pi x}{\pi^2}+\frac{2x}{\pi^2}-\frac{1}{\pi^2}
u (0.25) = 0.02098
u (0.5) = 0
u (0.75) = -0.02098
for i= 2
\int\limits_0^1 w_2R dx=0
\frac{\pi^2}{4} \int\limits_0^1\bigg(sin\frac{3 \pi x}{2}+ sin \frac{5 \pi x}{2}\bigg)\bigg[c_1\bigg(sin\frac{\pi x}{2}+9 sin \frac{3 \pi x}{2}\bigg)+c_3\bigg(9sin\frac{3 \pi x}{2}+25sin\frac{5 \pi x}{2}\bigg)-cos \pi x \bigg] -dx=c
c_1 \int\limits_0^1\bigg(sin\frac{3 \pi x}{2}+sin \frac{5 \pi x}{2}\bigg)\bigg(sin \frac{\pi x}{2} +9 sin\frac{3 \pi x}{2}\bigg)dx
+c_3\int\limits_0^1\bigg(sin\frac{3 \pi x}{2}+ sin \frac{5 \pi x}{2}\bigg)\bigg(9 sin \frac{3 \pi x}{2}+25 sin \frac{5 \pi x}{2}\bigg) dx
- \int\limits_0^1\bigg(sin\frac{3 \pi x}{2}+sin \frac{5 \pi x}{2}\bigg)cos \pi x dx.\frac{4}{\pi^2}=0
3.9 c_1+14.75 c_3=0.0122\hspace{2cm}------(2)
from (1)& (2)
c_1=0.0069 & c_3=0.0122.
\therefore \phi = 0.0069 \bigg(sin\frac{\pi x}{2}+sin\frac{3 \pi x}{2}\bigg)+0.0122\bigg(sin\frac{3\pi x}{2}+sin\frac{5\pi x}{2}\bigg)
u (0.25) = 0.03156
u(0.5) = 9.7581\times10^{-3} = 0.0097581
u (0.75) = -5.6033 \times10^{-3}=-0.0056