Page 120 - Algorithm Collections for Digital Signal Processing Applications using MATLAB

108 Chapter 3

9. SOLVING DIFFERENTIAL EQUATION USING
EIGEN DECOMPOSITION

Consider the problem of solving the third order differential equation as
given below.

3
2
2
3
∂ u / ∂t +2 ∂ u / ∂t - ∂u / ∂t +u = 0 with the initial condition given
U’’(0)=3, U’(0)=1, U(0)= -1.

Representing the above equation in the form of matrix representation.

V = U’’
U’
U

V’ = U’’’
U’’
U’

V’= -2 1 1 V
1 0 0
0 1 0

V = c1 e λ1 t E 1 +c2 e λ2 t E 2 + c3 e λ3 t E 3

Where c1, c2 and c3 are the constants obtained using initial conditions.
E1, E2 and E3 are the eigen vectors of the matrix A corresponding to the
eigen values λ 1,λ 2,λ 3 respectively.
The Eigen vectors and the eigen values of the matrix A are displayed
below.

λ 1 = -2.2470 λ 2 = 0.8019 λ 3= -0.5550

T
E1= [-0.8990 0.4001 -0.1781]