ECE 602 BME I Ordinary Differential Equations in

ECE 602 BME I Ordinary Differential Equations in Biomedical Engineering

Ø Classification of ODEs Ø Canonical Form of ODE Ø Linear ODEs Ø Nonlinear ODEs Ø Steady-State Solutions and Stability Analysis Ø BME Example 1 - The dynamics of Drug Absorption Ø BME example 2 – Hodgkin-Huxley Model for Dynamics of Nerve Cell Potentials

Classification of ODEs General Form of ODE • 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 First-order, linear, homogeneous First-order, linear, nonhomogeneous First-order, nonlinear, nonhomogeneous Second-order, nonlinear, nonhomogeneous Third-order, nonlinear, nonhomogeneous

Canonical Form of ODEs Canonical form • A set of n simultaneous first-order ODEs • Required for methods for integrating ODEs Vector format

Canonical Form of ODEs Transformation to Canonical form

Linear ODEs Matrix Exponential Method

Linear ODEs EXPM Matrix exponential. EXPM(A) is the matrix exponential of A. >> syms t >> >> A=[1 1; -1 1]; y 0=[1; 1]; >> y=expm(A*t)*y 0 y= exp(t)*cos(t)+exp(t)*sin(t) -exp(t)*sin(t)+exp(t)*cos(t)

Linear ODEs Eigenvector matrix Eigenvalue matrix Method using eigenvalues and eigenvectors

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]; y 0=[1; 1]; >> [X, D]=eig(A); >> y=(X*expm(D*t)*inv(X))*y 0 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))