APM4813
Assignment 03
Unique number: 696941
Closing date: 28 August 2025
, Finite-Element Analysis for Ordinary Differential
Equations
Problem Statement
Consider the differential equation:
d2 y
+ y = 3x2 ,
dx2
with boundary conditions:
y(0) = 0, y(2) = 4.
1.1 Rayleigh-Ritz Method
Assume an approximate solution of the form:
u(x) = 7x4 + c2 x(x − 2) + c3 x2 (x − 2),
which automatically satisfies the boundary condition at x = 0. However, at x = 2,
u(2) = 7(16) = 112 ̸= 4.
Thus, the approximation does not satisfy both boundary conditions exactly.
Functional Formulation
Define the functional to minimize:
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
Problem Statement
Consider the differential equation:
d2 y
+ y = 3x2 ,
dx2
with boundary conditions:
y(0) = 0, y(2) = 4.
1.1 Rayleigh-Ritz Method
Assume an approximate solution of the form:
u(x) = 7x4 + c2 x(x − 2) + c3 x2 (x − 2),
which automatically satisfies the boundary condition at x = 0. However, at x = 2,
u(2) = 7(16) = 112 ̸= 4.
Thus, the approximation does not satisfy both boundary conditions exactly.
Functional Formulation
Define the functional to minimize:
2 2
d2 u
Z
I= 2
+ u − 3x2 dx.
0 dx
1
