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Solve differential Equation using Galerkin Method.

Solve the following differential Equation using Galerkin Method.

d2ϕdx2+cosϕx=00<x<1.

Boundary Conditions are : ϕ(0)=0ϕ(1) = 0

Find ϕ(0.25),ϕ(0.5)andϕ(0.75) Compare your answer with exact solution

Mumbai University > Mechanical Engineering > Sem 6 > Finite Element Analysis

Marks: 10M

Year: Dec2016

1 Answer
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let the approximate solution,

ϕ=co+cisinπ2(2i1)x

Boundary condition 1: ϕ(o)=0

co=0

ϕ=∈cisinπ2(2i1)x.

=cisinπx2+c2sin3πx2+c3sin5x2

Boundary condition 2 : ϕ (1)=0

0=c1c2+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

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