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ECE602 BME I Ordinary Differential Equations in Biomedical Engineering

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

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ECE602 BME I Ordinary Differential Equations in Biomedical Engineering

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  1. ECE602 BME I Ordinary Differential Equations in Biomedical Engineering

  2. 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

  3. 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

  4. Classification of ODEs • Examples First-order, linear, homogeneous First-order, linear, nonhomogeneous First-order, nonlinear, nonhomogeneous Second-order, linear, nonhomogeneous Second-order, nonlinear, nonhomogeneous Third-order, nonlinear, nonhomogeneous

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

  6. Canonical Form of ODEs • Transformation to Canonical form

  7. Linear ODEs Matrix Exponential Method

  8. 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)*y0 y = exp(t)*cos(t)+exp(t)*sin(t) -exp(t)*sin(t)+exp(t)*cos(t)

  9. Linear ODEs Eigenvector matrix Eigenvalue matrix Method using eigenvalues and eigenvectors

  10. 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. Linear ODEs >> 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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