Construct the weak form and find the two$-$parameter $(N=2)$ Rayleigh$-$Ritz

approximation of the differential equation.

$u(0)=0,u(1)=1(N=1)u(0)=1,u(1)=0\frac{O}{{?}_0}\frac{O}{{?}_i}\frac{O}{{?}_0}\frac{O}{{?}_j}{\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}{}_{i}Rdx=0,{}_i={}_j-2u\frac{d^{2}u}{d{x}^2}+(\frac{du}{dx}{)}^2=4,$ for $0$

$\frac{w}{b}$oundary condition

$u(0)=1,u(1)=0$

Note:

Ritz requirement:

$\frac{O}{{?}_0}$ should satisfy the specified EBC

$\frac{O}{{?}_i}$ should satisfy $at$ least the homogenous form $of$ EBC

Weighted residual requirement:

$\frac{O}{{?}_0}$ should satisfy all the specified boundary conditions

$\frac{O}{{?}_j}$ should satisfy the homogenous form $of$ all specified boundary condition

Galerkin: ${\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}{}_{i}Rdx=0,{}_i={}_j-\frac{d}{dx}[(1+x)\frac{du}{dx}]=0$ for $,0$

$\frac{w}{b}$oundary condition:

$u(0)=0,u(1)=1$

Find a one$-$parameter $(N=1)$ approximate solution $of$ the nonlinear differential

equation using Galerkin method.

$-2u\frac{d^{2}u}{d{x}^2}+(\frac{du}{dx}{)}^2=4,$ for $0$

$\frac{w}{b}$oundary condition

$u(0)=1,u(1)=0$

Note:

Ritz requirement:

$\frac{O}{{?}_0}$ should satisfy the specified EBC

$\frac{O}{{?}_i}$ should satisfy $at$ least the homogenous form $of$ EBC

Weighted residual requirement:

$\frac{O}{{?}_0}$ should satisfy all the specified boundary conditions

$\frac{O}{{?}_j}$ should satisfy the homogenous form $of$ all specified boundary condition

Galerkin: ${\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}{}_{i}Rdx=0,{}_i={}_j$