ece602 bme i partial differential equations in biomedical engineering
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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) - PowerPoint PPT PresentationTRANSCRIPT
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