1
1. Using Least Squares weighted residual method obtain an approximate solution of the
following boundary value problem
2 3 0 1 3
1 1 Essential boundary condition
3 1 Natural boundary condition
u x u x
u
u
(a) Assume a quadratic polynomial satisfying the essential boundary condition as a trial
solution.
(b) Assume a cubic polynomial satisfying the essential boundary condition as a trial
solution. Compare graphically the two solutions and their first derivatives. A good
approximation to the exact solution is given by
0 29986 1 2247 2 1166 1 2247u x . sin . x . cos . x
.
2. Solve Problem 1 using the Collocation method.
3. Solve Problem 1 using the Galerkin method (basic form).
4. Using modified Galerkin method obtain an approximate solution of the following boundary
value problem
0 0 1
0 0 1 0
u x u x x x
uu
(a) Assume a quadratic polynomial as a trial solution.
(b) Assume a cubic polynomial as a trial solution. Compare graphically the two
solutions and their first derivatives with the exact solution
10
sin x
u x x sin .
.
5. Using modified Galerkin method obtain an approximate solution of the following boundary
value problem
2 3 0 1 3
1 1 Essential boundary condition
3 1 Natural boundary condition
u x u x x
u
u
(a) Assume a quadratic polynomial satisfying the essential boundary condition as a trial
solution.
(b) Assume a cubic polynomial satisfying the essential boundary condition as a trial
solution. Compare graphically the two solutions and their first derivatives. A good
approximation to the exact solution is given by
0 29986 1 2247 2 1166 1 2247u x . sin . x . cos . x
.
6. Solve Problem 4 using the Rayleigh-Ritz method.
7. Solve Problem 5 using the Rayleigh-Ritz method.
Finite Element Method
Assignment 1 (Due Date: 12 - 02 - 2024)
Jan - May 2024
