Part Seven: Ordinary Differential Equations

Mech300 Numerical Methods, Hong Kong University of Science and Technology.

Part Seven Ordinary Differential Equations

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Mech300 Numerical Methods, Hong Kong University of Science and Technology.

Basics Differential equation: an equation composed of an unknown function and its derivatives Example: the falling parachutist

dv c =g− v dt m

v: dependent variable (function) t: independent variable

Ordinary differential equation:

if there is only one independent variable

Partial differential equation:

if there are two or more independent variables

Order of ODE:

the order of the highest derivative in the equation d 2x dx m + c + kx = 0 Example: second order ODE 2 dt dt

Reduction of order: higher-order ODE can be reduced to a system of 1st-order ODE dx dx  y= 2 y = d x dx  dt dt m + c + kx = 0 dt 2

dt

 dy m + cy + kx = 0  dt

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Mech300 Numerical Methods, Hong Kong University of Science and Technology.

Why Study Differential Equations? Many physical phenomena are best formulated mathematically in terms of their rate of change (which is derivative)!

Example: motion of a swinging pendulum

d 2θ

g + sinθ = 0 2 l dt

dθ : rate of change of θ dt d 2θ dt 2

: rate of change of

dθ ( rate of change of rate of change of θ ) dt 3

Mech300 Numerical Methods, Hong Kong University of Science and Technology.

ODE and Engineering Practice Empirical observations

Fundamental laws

Sequence of the application of ODEs for engineering problems

ODE

Analytical/numerical methods

Solutions

Independent variable: spatial and temporal 4

Mech300 Numerical Methods, Hong Kong University of Science and Technology.

Noncomputer Methods for Solving ODEs Differential equation

conversion

dv c =g− v dt m

Integration

c   v = ∫  g − v  dt m  

Analytical integration techniques

v( t ) =

Solution

(

gm 1 − e −( c/m ) t c

)

One particular useful analytical integration technique: linearization an(x)y(n) + an-1(x)y(n-1) + … +a1(x)y’+ a0(x)y = f(x) This can be solved analytically!

d 2θ

g + sinθ = 0 2 l dt (non-linear)

Sinθ ≈ θ if θ is small

d 2θ dt 2

+

g θ =0 l

(linear)

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Mech300 Numerical Methods, Hong Kong University of Science and Technology.

Solution by Integration y = −0.5 x 4 + 4 x 3 − 10 x 2 + 8.5 x + 1 differentiation

dy = −2 x 3 + 12 x 2 − 20 x + 8.5 x dx integration

(

)

y = ∫ − 2 x 3 + 12 x 2 − 20 x + 8.5 dx

y = −0.5 x 4 + 4 x 3 − 10 x 2 + 8.5 x + C

Multiple solutions

For an nth-order ODE, n conditions are required to obtain a unique solution All n conditions are specified at a same value of x

Initial-value problem

n conditions occur at different x

Boundary-value problem

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Mech300 Numerical Methods, Hong Kong University of Science and Technology.

Overall Structure

Initial-value problem

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