Polymath Tutorial: Matrix and ODE

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Polymath TutorialMatrix and ODE

• By Kelvin WONG
• CENG 364 _at_ HKUST
• Feb, 2009

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Using Polymath to Solve Matrix(Linear Equations)

• An example, based on the TV program ?????, Jan
  31, 2009
• Mr. A, Mr. B, Mr. C and Mr. D have some money.
  They have 112 in total, i.e., A B C D
  112. And we also know that
• A 3 B - 3
• A 3 3 x C
• D 3 x (A 3).
• How much do Mr. A, Mr. B, Mr. C and Mr. D have?
• We can solve it by substitute the last three
  equations into the first equation.
• On the other hand, we can solve it by using
  matrix(linear algebra).

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Matrix Example (1)

• Rearrange those four equations as
• A B C D 112 (eq. 1)
• A - B -6 (eq. 2)
• A - 3C -3 (eq. 3)
• 3A - D -9 (eq. 4)
• We have 4 equations and we need to solve for 4
  unknowns. This system is determined.
• If of equations gt of unknowns ?
  overdetermined.
• If of equations lt of unknowns ?
  underdetermined.

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Matrix Example (2)

• Construct the matrix

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Matrix Example (3)

• In Polymath, click the Linear Equation Problem
  button
• Since we have only 4 equations and 4 unknowns, we
  need to change the Number of Equations as 4

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Matrix Example (4)

• Then start to input the matrix. Let x1 A, x2
  B, x3 C, x4 D, and beta the product
  vector.
• Then, click the pink arrow button to solve
  it.

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Matrix Example (5)
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Using Polymath to Solve ODE

• In many physical systems, we have differential
  equations to describe the rate of change of the
  system. In order to find the status of this
  system at any time, we need to integrate those
  differential equations.
• E.g. batch reactor
• A ? B
• Assume first order kinetics
• We know the rate of consumption of species A is
• We can solve the problem by integrate the above
  differential equation by hand (analytical
  solution)

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Numerical Integration (1)

• For those differential equations with complicated
  functions that cannot be integrate analytically,
  or
• When we use computer to solve those problems.
• We will need numerical integration to solve those
  ODE.

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Numerical Integration (2)

• What we know
• Initial concentration A0
• Kinetic parameter k
• Rate of reaction at any given A.
• What we do not know
• Concentration At at any given time t.

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Numerical Integration (3)

• Then we can guess the new concentration A by
  using the know reaction rate (slope)

At A0 - slope ? t
? t
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Numerical Integration (4)

• Repeat the above calculation and well have
  approximate solution of the ODE

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Numerical Integration (5)

• Smaller steps ? more accurate result
• However, more computation time (resource) is
  required.
• Many numerical integration methods are developed
  in order to improve accuracy.
• We need to balance in between accuracy and
  computation resource.

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Stiff Systems

• System of ODE where the dependent variables
  change on various time (independent variable)
  scales which differ by many orders of magnitude
  are called Stiff systems.
• We need to choice the appropriate method in
  Polymath to handle this problem.

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Polymath Demonstration

• Independent variable
• Equation input (a) differential equation, (b)
  explicit equation.
• Parameter input as explicit equation
• Numerical integration method
• Typo error
• Limitation on number of equations
• Decision statement / step function

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ODE Example - Batch Reactor (1)

• Irreversible reaction
• A ? B
• First order kinetic rA -kA
• In Polymath click the Ordinary Differential
  Equations Problem button

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ODE Example - Batch Reactor (2)

• Click the Add New Differential Equation button to
  input the differential equation
• Then type in the equation that describes the rate
  of change of A per time. We also need to
  provide the initial value of A0 (let say, 10M)

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ODE Example - Batch Reactor (3)

• Also need to input the differential equation for
  dB/dt.
• Then we need to define the rate of reaction rA
  by input an explicit equation. Click the Add New
  Explicit Equation button
• Type in the rate equation
• Add one more explicit equation to define the
  value of k.

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ODE Example - Batch Reactor (4)

• Finally, define the initial and find time (t0 and
  tfinal). Click the Define Initial and Final
  Values of the Independent Variable button
• Once we have the sufficient information, Polymath
  will show the Solve button. Click it to solve
  this model

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ODE Example - Batch Reactor (5)
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Step Function

• Typical step function
• We can create a step function by using decision
  making statement
• if (t lt t1) then (0) else (1)
• OR
• if (t gt t1) then (1) else (0)
• Above two statements are identical.

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ODE Example - CSTR (1)

• A ? B
• First order kinetic rA -kA
• Flow rate F, reactor volume V, feed stream
  reactant conc. Afeed.
• Increase reactant conc. in feed stream at certain
  time.
• t lt t1, Afeed Af1t t1, Afeed Af2.

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ODE Example - CSTR (2)

• Define the feed stream reactant concentration by
  using the step function
• Of course, we need to define the time for the
  step function.

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ODE Example - CSTR (3)
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ODE Example - CSTR (4)