Solve the following differential Equation using Galerkin Method.

$\hspace{2cm} \frac{d^2 \phi}{dx^2} + cos \hspace{0.1cm}\phi \hspace{0.1cm}x = 0 \hspace{2cm} 0 \lt x \lt 1.$

Boundary Conditions are : $\hspace{0.2cm} \phi(0) = 0 \hspace{0.2cm} \phi$(1) = 0

Find $\phi (0.25), \phi\hspace{0.2cm} (0.5) and\hspace{0.2cm} \phi\hspace{0.2cm} (0.75)$ Compare your answer with exact solution

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

Marks: 10M

Year: Dec2016

let the approximate solution,

$\phi = c_o + \in ci sin \frac{\pi}{2}(2i-1)x$

Boundary condition 1: $\phi (o)=0$

$\therefore c_o=0$

$\therefore \phi = \in c_i sin \frac{\pi}{2}(2i-1) x.$

=$c_i sin \frac{\pi x}{2}+ c_2 sin \frac{3 \pi x}{2}+ c_3 sin \frac{5 x}{2}$

Boundary condition 2 : $\phi$ (1)=0

$\therefore 0 = c_1-c_2+c_3$ …