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

02

22

2

2

gfuy

ue

x

ud

y

uc

yx

ub

x

ua

042 acb elliptic

042 acb parabolic

042 acb hyperbolic

Classification of PDEs

Examples of linear second-order PDEs

02

2

2

2

y

u

x

uLaplace’s equation elliptic

0 ,2

2

Kx

uK

t

uHeat equation parabolic

2

22

2

2

x

us

t

u

Wave equation hyperbolic

Initial and Boundary conditions

Diffusion of nutrient across a cell membrane

2

2

x

CD

t

C

C: the concentration of nutrient

D: the diffusivity of nutrient in the membrane

Initial and Boundary conditions

Diffusion of nutrient across a cell membrane

2

2

x

CD

t

C

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 variablesis 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