$\frac{d}{dx}(a\frac{du}{dx})+bu+c=0; 0 \leq x \leq 1$ Solve BCs are u(0)=u0 and $a(\frac{du}{dx})|_{x=L}=0$

Use Lagrange's linear shape function

Given $\frac{d}{dx}(a\frac{du}{dx})+bu+c=0; 0 \leq x \leq 1$

Boundary Condition

u[0]=u_{o} a$a\frac{du}{dx}|_{x=l}$=0

1) Discretization: Dividing the domain into 3 elements

2) Take a general element of length he and set its local co-ordinate

Differentiating above eq, we get,

dx=$d\bar{x}$

3) Converting governing differential equation into local co-ordinate

dx=$d\bar{x}$

$\frac{d}{dx}(a\frac{du}{dx})+cu+q$=0

Local boundaries are

5) Weighted Integral form

$\int^{1}_{0}w_{i}R dx=0$

$\int _{o}^{he} w_{i}$ $[\frac{d}{d\bar{x}}(a \frac{du}{d\bar{x}})+cu+q]d\bar{x}$=0

$\int_{0}^{he} w_{i}(\frac{du}{d\bar{x}})d\bar{x}+\int_{0}^{he}w_{i}cu d{\bar{x}}+\int_{0}^{he}w_{i}qd\bar{x}$........(A)

Consider $\int_{0}^{he} w_{i}\frac{d}{d\bar{x}(a\frac{d}{d{\bar{x}}})}d\bar{x}$

Integrating by parts, above equation becomes,