ECE602 BME I Partial Differential Equations in Biomedical Engineering.

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Presentation transcript:

ECE602 BME I Partial Differential Equations in Biomedical Engineering

 Classification of PDEs  Initial and Boundary Conditions  Numerical solution of PDEs  BME Examples

Classification of PDEs Classification according to order (the highest-order partial derivative present in the equation) linearity

Classification of PDEs Classification of linear second-order PDEs elliptic parabolic hyperbolic

Classification of PDEs Examples of linear second-order PDEs Laplace’s equation elliptic Heat equation parabolic Wave equation hyperbolic

Initial and Boundary conditions Diffusion of nutrient across a cell membrane C: the concentration of nutrient D: the diffusivity of nutrient in the membrane

Initial and Boundary conditions Diffusion of nutrient across a cell membrane C: the concentration of nutrient D: the diffusivity of nutrient in the membrane

Initial and Boundary conditions Dirichlet conditions (first kind): the values of the dependent variables are given at fixed values of the independent variables

Initial and Boundary conditions Nuemann conditions (second kind): the derivative of the dependent variables is given as a constant or as a function of the independent variable.

Initial and Boundary conditions Cauchy conditions: a problem that combines both Dirichlet and Neumann conditions

Initial and Boundary conditions Robins conditions: the derivative of the dependent variables is given as a function of the dependent variable itself.

Initial and Boundary conditions PDE can be classified into initial-value problem: at least one of the independent variables has an open region boundary-value problem: the region is closed for all independent variables, and conditions are specified at all boundaries.

Numerical Solutions of PDEs Finite Difference Central Difference Forward Difference Backward Difference