Numerical Differentiation

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Numerical Differentiation
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• Consider a particle whose position as a function
  of time is recorded in a table. How do we
  determine its velocity v(t)dx/dt?

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• Consider a particle whose position as a function
  of time is recorded in a table. How do we
  determine its velocity v(t)dx/dt?
• We differentiate!

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• Consider a particle whose position as a function
  of time is recorded in a table. How do we
  determine its velocity v(t)dx/dt?
• We differentiate!
• This form is not usefully as it is plagued by
  subtractive cancellation h oscillates between 0
  and as

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• Alternative methods are the forward difference,
  central difference and extrapolated difference.

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• Alternative methods are the forward difference,
  central difference and extrapolated difference.
• For completeness we will look very briefly at the
  forward difference (the other methods will be
  left as research).

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Forward Difference

• This the most direct method and starts with
  expanding the function as a Taylor series.

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Forward Difference

• This the most direct method and starts with
  expanding the function as a Taylor series.
• The series advances the function one small step
  forward,
• h is the step size.

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Forward Difference

• The forward difference algorithm is obtained by
  solving for f prime.
• Where c represents the compute value.
• Ie an approx. using two points to rep the
  function by a straight line in the interval x to
  xh

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Forward Difference

• The forward difference algorithm is obtained by
  solving for f prime.
• Where c represents the compute value.
• Ie an approx. using two points to rep the
  function by a straight line in the interval x to
  xh

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Diagram illustrating the forward difference
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Forward Difference

• Consider the case where,

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Forward Difference

• Consider the case where,
• The exact derivative is
• The computed derivative is
• For small h the approximation becomes good. ie

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Numerical Solutions to Differential equations
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• Most real world problems are written as (modelled
  with) differential equations.

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• Most real world problems are written as (modelled
  with) differential equations.
• in chemistry, physics, and engineering.
• more recently in models for medicine, biology.

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• Most real world problems are written as (modelled
  with) differential equations.
• in chemistry, physics, and engineering.
• more recently in models for medicine, biology.
• Eg. Oscillators (harmonic motion).
• Assume we have a mass m attached to a spring
  forced by an external force.

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• A problem may require you to solve for the motion
  of the mass as a function of time.

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• From classical mechanics we use Newtons 2nd law,
  which can be written in differential form.

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• From classical mechanics we use Newtons 2nd law,
  which can be written in differential form.
• may be used to determine v(t) and x(t) given
  initial conditions.

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• A differential equation is an equation involving
  an unknown function and one or more of its
  derivatives.

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• A differential equation is an equation involving
  an unknown function and one or more of its
  derivatives.
• The equation is an ordinary differential equation
  (ODE) if the unknown function depends on only one
  independent variable.

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• Types of ordinary differential equations include
  initial-value problems (IVP) and boundary-value
  problems (BVP).

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• Types of ordinary differential equations include
  initial-value problems (IVP) and boundary-value
  problems (BVP).
• Examples of ODEs

Growth equation
Harmonic Oscillator
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Differential equations and Oscillators
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• The exciting feature of computational work is
  highlighted in that we can solve for problems of
  this type very easily.
• We are not restricted to linear differential
  equations or nearly linear. There is no need for
  the usual common assumptions.

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• Linear Driven Oscillator
• Where is the force exerted by the spring and
  is the external force.

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• Linear Driven Oscillator
• Where is the force exerted by the spring and
  is the external force.
• Nonlinear Oscillator
• PE is an arbitrary power p of displacement x
  from equilibrium.

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