APM4813
Assignment 03
Unique number: 696941
Closing date: 28 August 2025
, Finite Element Analysis for Ordinary Differential
Equations
Question 1: Finite-element analysis for Ordinary Dif-
ferential Equations
Consider the differential equation:
d2 y
2
+ y = 3x2 ,
dx
with boundary conditions:
y(0) = 0, y(2) = 4.
1.1 Rayleigh-Ritz Method
Suppose y is approximated by:
u = 7x4 + c2 x(x − 2) + c3 x2 (x − 2).
Note: u does not satisfy the boundary condition at x = 2 exactly. However, the
Rayleigh-Ritz method seeks to minimize the residual over the interval.
Step 1: Define the Functional
2 2
d2 u
Z
I= 2
+ u − 3x2 dx.
0 dx
1
Assignment 03
Unique number: 696941
Closing date: 28 August 2025
, Finite Element Analysis for Ordinary Differential
Equations
Question 1: Finite-element analysis for Ordinary Dif-
ferential Equations
Consider the differential equation:
d2 y
2
+ y = 3x2 ,
dx
with boundary conditions:
y(0) = 0, y(2) = 4.
1.1 Rayleigh-Ritz Method
Suppose y is approximated by:
u = 7x4 + c2 x(x − 2) + c3 x2 (x − 2).
Note: u does not satisfy the boundary condition at x = 2 exactly. However, the
Rayleigh-Ritz method seeks to minimize the residual over the interval.
Step 1: Define the Functional
2 2
d2 u
Z
I= 2
+ u − 3x2 dx.
0 dx
1
