The following differential equation is available for a physical phenomenon \(\dfrac{d^{2}u}{dx^{2}}+u+x=0\)

,\(0≤ x ≤1\), \(u(0)=u(1)=0\). Solve above equation,using subdomain method and Galerkin method.

Derive the cubic shape function of Largrange's family.what are the characteristic of the shape function? Plot the shape function along of the element. State the difference shape function and interpolation function?

Explain the following:
1.Global Local and Natural Co-ordinate system
2.Boundary condition and its types.

Evaluate the following integral \(I=\int_{0}^{4} x^{3}dx\\\)
Using (a) trapezoidal rule
(b) Simpsons one third rule
Compare the solution with the exact solution.

The governing differential equation for a rod loaded axial force is
\(\frac{d}{dx}(AE\frac{du}{dx})+q =0\), Where E is Young's modulus of elasticity, A is the area of cross section. q is the load intensity and u is the axial displacement. Obtain the variational form for this equation
Assume that the boundary condition are
At \(x=0,u =0\)
at , \(x=l, EA \dfrac{du}{dx}=P\)
where \(l\) is the length of the rod.