ece602 bme i ordinary differential equations in biomedical engineering
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ECE602 BME I
Ordinary Differential Equations in Biomedical Engineering
Classification of ODEs Canonical Form of ODE Linear ODEs Nonlinear ODEs Steady-State Solutions and Stability Analysis BME Example 1- The dynamics of Drug
Absorption BME example 2 – Hodgkin-Huxley Model for
Dynamics of Nerve Cell Potentials
Classification of ODEs
General Form of ODE
)()()(...)()( 011
1
1 tRytbdt
dytb
dt
ydtb
dt
ydtb
n
n
nn
n
n
• The order of an ODE: the order of the highest derivative
• R(t)=0: Homogeneous; R(t)0: Nonhomogeneous
• Nonlinear: an ODE contains powers of the dependent variable, powers of the derivatives, or products of the dependent variable with the derivatives
Classification of ODEs
Examples
0 ydt
dyFirst-order, linear, homogeneous
ktydt
dy First-order, linear, nonhomogeneous
ktydt
dy 2 First-order, nonlinear, nonhomogeneous
teydt
dy
dt
yd
2
2
Second-order, linear, nonhomogeneous
)cos(2
2
tydt
dy
dt
ydy Second-order, nonlinear, nonhomogeneous
)sin()(2 22
2
3
3
tydt
dy
dt
yd
dt
yd Third-order, nonlinear, nonhomogeneous
Canonical Form of ODEs
Canonical form
• A set of n simultaneous first-order ODEs
• Required for methods for integrating ODEs
0,021
0,2022122
0,1012111
)( ),,,(
)( ),,,(
)( ),,,(
nnnnn
n
n
ytyyyytfdt
dy
ytyyyytfdt
dy
ytyyyytfdt
dy
00 )( ),( yyyfy
ttdt
d
Vector format
Canonical Form of ODEs
Transformation to Canonical form
),...,,,,(1
1
2
2
n
n
n
n
dt
zd
dt
zd
dt
dzztG
dt
zddt
dy
dt
zd
ydt
dy
dt
zd
ydt
dy
dt
zd
ydt
dy
dt
dz
yz
nn
n
nn
n
n
1
1
1
32
2
2
21
1
.
.
.
),...,,(
.
.
.
21
32
21
nn yyytGdt
dy
ydt
dy
ydt
dy
Linear ODEs
0)0(
'
yy
Ayy
0yy Ate
...!3!2
3322
tt
te t AAAIA
Matrix Exponential Method
Linear ODEs
EXPM Matrix exponential. EXPM(A) is the matrix exponential of A.
>> syms t>> >> A=[1 1;-1 1]; y0=[1;1];>> y=expm(A*t)*y0y = exp(t)*cos(t)+exp(t)*sin(t) -exp(t)*sin(t)+exp(t)*cos(t)
Linear ODEs
0)0(
'
yy
Ayy
Method using eigenvalues and eigenvectors
01][ yXXy Λ te
XΛAX
Eigenvector matrix Eigenvalue matrix
Linear ODEs
EIG Eigenvalues and eigenvectors. [X,D] = EIG(A) produces a diagonal matrix D of eigenvalues and a full matrix X whose columns are the corresponding eigenvectors so that A*X = X*D.
>> syms t>> A=[1 1;-1 1]; y0=[1;1];>> [X,D]=eig(A);>> y=(X*expm(D*t)*inv(X))*y0 y = exp(t)*cos(t)-1/2*i*(exp(t)*cos(t)+i*exp(t)*sin(t))+1/2*i*(exp(t)*cos(t)-i*exp(t)*sin(t)) 1/2*i*(exp(t)*cos(t)+i*exp(t)*sin(t))-1/2*i*(exp(t)*cos(t)-i*exp(t)*sin(t))+exp(t)*cos(t)
>> y=simplify(y) y = exp(t)*(cos(t)+sin(t)) exp(t)*(-sin(t)+cos(t))
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