Chem 302 Math 252

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Chem 302 - Math 252

• Chapter 6Differential Equations

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Differential Equations

• Many problems in physical chemistry (eg.
  kinetics, dynamics, theoretical chemistry)
  require solution to a differential equation
• Many can not be solved analytically
• Deal only with first order ODE
• Higher order equations can be reduced to a system
  of 1st order DE

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Differential Equations

• Simplest form

• Can integrate analytically or numerically (using
  techniques of Chapter 4)

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Differential Equations

• General case

• Many simpler problems can be solved analytically
• Many involve ex
• However, in chemistry (physics engineering)
  many problems have to be solved numerically (or
  approximately)

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Picard Method

• Can not integrate exactly because integrand
  involves y
• Approximate iteratively by using approximations
  for y

• Continue to iterate until a desire level of
  accuracy is obtained in y
• Often gives a power series solution

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Picard Method Example

• Continue to iterate until a desire level of
  accuracy is obtained in y

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Picard Method Example 2
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Euler Method

• Assume linear between 2 consecutive points
• Between initial point and 1st (calculated) point

• User selects Dx
• Need to be careful - too big or too small can
  cause problems

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Euler Method Example
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Taylor Method

• Based on Taylor expansion

Euler method is Taylor method of order 1
Use chain rule
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Taylor Method Example
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Improved Euler (Heuns) Method

• Euler Method
• Use constant derivative between points i i1
• calculated at xi
• Better to use average derivative across the
  interval

• yi1 is not known

Predict Correct(can repeat)
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Improved Euler Method Example
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Modified Euler Method

• Modified Euler Method
• Use derivative halfway between points i i1

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Modified Euler Method Example
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Runge-Kutta Methods

• Improved and Modified Euler Methods are special
  cases
• 2nd order Runge-Kutta
• 4th order Runge-Kutta
• Runge
• Kutta
• Runge-Kutta-Gill

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Runge Methods
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Kutta Methods
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Runge-Kutta-Gill Methods
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Systems of Equations

• All the previous methods can be applied to
  systems of differential equations
• Only illustrate the Runge method

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Systems of Equations Example 1
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Systems of Equations Example 2
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Systems of Equations Example 3
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Systems of Equations Example 4
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Systems of Equations Example 5