Differential Equations

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• Differential Equations
• Polynomial Approximations
• Ordinary Differential Equations
• Differential and Algebraic Equation Systems
• Partial Differential Equations

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Finite Difference Approximations - I Evenly
spaced points
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Finite Difference Approximations - II We have
What is
? Taylor series
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Finite Difference Approximations III Central
difference approximation Other approximations
can be derived with any number of forward and
backward steps and of any desired order (some
combinations are impossible, however).
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A Single Differential Equation
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Initial Value Problem (IVP)
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Boundary Value Problem (BVP)
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Integration Methods - I
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Integration Methods II Taylor series
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• Integration Methods III
• is related to the slope of the
  solution curve.
• Very many different methods
• Euler
• Runge-Kutta
• and much more

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Simple Euler Method

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Less-simple Euler Methods
and more
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Semi-Implicit Euler Method
Implicit Euler


D
Taylor series for
(
)
f
,
x
h
y
Combine, drop terms in and rearrange
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Runge-Kutta Methods - I
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c, a, and w are numerical coefficients chosen to
satisfy certain conditions. is the number of
terms. 4 term methods are very popular.
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Runge-Kutta Methods - II 1. Make a first
(tentative) step with the Euler method.
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Runge-Kutta Methods - III 2. Evaluate slope at
intermediate point.
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Runge-Kutta Methods - IV 3. Use the adjusted
slope and make a second (also tentative)
step from the initial point.
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Runge-Kutta Methods - V 4. Evaluate function at
additional points and use this information
to further adjust the slope to be used at the
start
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Runge-Kutta Methods - VI 5. Evaluate function
at as may other points as required and make
further adjustments to the slope to be used at
the start.
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Runge-Kutta Methods - VII 6. Combine all the
estimates to make the actual step.
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Differential Equation System

in matrix form
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Eulers Method for Systems

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Runge-Kutta Method for Systems
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Is there more? Yes, a great deal, but first
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• Stability
• Stability of the integration method
• Stability of the ODE being integrated

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Exercise Lamberts Problem Use an explicit
Runge-Kutta method to integrate these equations
from the initial condition to t 100. Plot
the variables u and v as a function of
t. Investigate the effect of step size, h, on the
solution.

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• Stiff Differential Equations
• t1 gtgtgt t2
• Difficult to solve with explicit Euler or RK
• Use implicit or semi-implicit methods
• Gears method (historically important)

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Semi-Implicit Runge-Kutta Method I Michelsens
method

A
(
)
(
)
D
f
(
)
y
x
3




,
,
,
,
R
1.037609496
R
1
b
a
.4358665215
1
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2
4



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b
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.8349304835
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2
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Extrapolation Methods - I
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Extrapolation Methods - II
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• BESIRK
• Third order SIRK method of Michelsen
• BS-like extrapolation
• Prime number step sequence

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• Extrapolation Methods III
• Bulirsch-Stoer
• Any integration method
• Step pattern

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• Differential Algebraic Systems
• ODE AE DAE
• Example
• Engineering examples
• Residue curves
• Process dynamics
• PDE solved by Method of Lines (MOL)

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Index The index of a system of DAEs is the
number of times we need to differentiate the
algebraic equations to obtain a system with only
differential equations. Index 0 very easy
(ODE) Index 1 straightforward Index 2 more
difficult Index 3 VERY difficult Not all ODE
integration methods can solve DAE problems!
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An Index 3 Problem
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An Index 3 problem
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Index Determination Exercise 1
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Index Determination Exercise 2
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Index Determination Exercise 3
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Index Determination Exercise 4
where the ks are constants.
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Index Determination Exercise 5
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• Solving Differential Algebraic Systems
• DAE systems infinitely stiff (even index 1)
• Cant use Euler or ordinary RK methods
• Implict or semi-implicit methods

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• Partial Differential Equations
• More than one independent variable.
• Linear if A, B, and C do not depend on x or y.

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• Classification of PDEs
• Case 1 D lt 0 Elliptic equation
• e.g. Laplace eqn.
• Case 2 D 0 Parabolic equation e.g.
  Diffusion eqn.
• Case 3 D gt 0 Hyperbolic equation e.g. Wave
  eqn. e.g. Convective transport eqn.

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Boundary Value and Initial Value Problems
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Finite Difference Methods
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• Method of Lines
• Approximate derivatives in all but one
  direction.
• Leaves ODE (or DAE) system to be solved.

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Example Heat Transfer in a Slab - I
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Example Heat Transfer in a Slab II Apply
finite difference approximations to x-direction
derivatives. This equation holds at all
interior points, but not at the Boundaries where
(assuming 11 points altogether).
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Example Heat Transfer in a Slab III Collect
all of the equations together (many more
similar expressions omitted) This is a
differential algebraic equation (DAE) system, not
a purely ODE system. We can integrate this
system using a DAE solver.
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Example Heat Transfer in a Slab IV The
initial condition
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Example Heat Transfer in a Slab V
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Example Heat Transfer in a Slab VI
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• Method of Lines - Summary
• Easy to use
• Can handle nonlinear terms in the PDE
• Can handle derivative and mixed form boundary
  conditions
• Right hand side is banded
• Any discretization method can be used
• Number of lines can be very large
• System of equations often is stiff
• Need good ODE or DAE solver

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