Numerical Solutions of ODE

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Numerical Solutions of ODE

• Dr. Asaf Varol
• avarol_at_mix.wvu.edu

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What is ODE and PDE

• A differential equation is an equation which
  involves derivatives of one or more dependent
  variables. If there is only one independent
  variable involved in the equation(s), then the
  derivatives are referred to as ordinary
  derivatives. If, however, there is more than one
  independent variable in the equation, then
  partial derivatives (PDE) with respect to each of
  the independent variables are used.

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Linear first-order ODEs
dy/dx x y y x y
du/dx u 2 u u 2
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Non-Linear first-order ODEs
dy/dx x cos(y) y x cos(y)
du/dt u2 2 u u2 2
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Linear, second-order ODEs
d2y/dx2 dy/dx xy
y -2y 0.1y
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Non-Linear, second-order ODEs
d2y/dx2 dy/dx xy-y
y -2y 0.1(y)2
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Homogeneous ODEs

• Homogeneous ODE is an equation which contains the
  dependent variable or its derivatives in every
  term.

d2y/dx2 dy/dx xy-y
y -2y 0.1(y)2
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Partial Differential Equation
First order, linear PDE where for a given function u u( x, t) x and t are the independent variables ? is the independent variable.
Second-order linear PDE Here, x and y are the independent variables.
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Eulers Method
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MATLAB (Euler)
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Plot (Euler)
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Example

• EULER METHODS
• Solving a simple ODE with Eulers Method
• Consider the differential equation y f( x, y )
  on a x b. Let
• y x y 0 x 1 a 0, b 1,
  y(0) 2.
• First, we find the approximate solution for h0.5
  (n 2), a very large step size.
• The approximation at x1 0.5 is
• y1y0 h (x0 y0) 2.0 0.5 (0.0 2.0) 3.0
• Next, we find the approximate solution, we use n
  20 intervals, so that h 0.05.

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Solution with MATLAB (Euler)
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Plot (Euler)
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Modified Euler Method
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Higher Order Taylor Methods

• One way to obtain a better solution technique is
  to use more terms in the Taylor series for y in
  order to obtain higher order truncation error.
  For example, a second-order Taylor method uses
• y(xh)y(x)hy(x)(h2/2)y(x)O(h3)
• O(h3) is the local truncation error

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Solving a Simple ODE with Taylors Method

• Consider the differential equation
• yx y 0 x 1 with a initial condition
  y(0) 2.
• To apply the second order Taylor method to the
  equation, we find
• yd/dx( x y) 1 y 1 x y
• This gives the approximation formula
• y(x h)y(x)hy(x)(h2/2)y(x)

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Contd

• yi1yih(xiyi)(h2/2)(1xiyi)
• For n2 (h0.5), we find
• y1y0h(x0y0)(h2/2)(1x0y0)
• 20.5(02)((.5)2/2)(102)3.375
• y2y1h(x1y1)(h2/2)(1x1y1)
  3.3750.5(0.53.375)((0.5)2/2)(10.53.375)5.92
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MATLAB Program f. Taylor
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Plot (Taylor)
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RUNGE-KUTTA METHODS

• Runge-Kutta methods are the most popular methods
  used in engineering applications because of their
  simplicity and accuracy. One of the simplest
  Runge-Kutta methods is based on approximating the
  value of y at xi h/2 by taking one-half of the
  change in y that is given by Eulers method and
  adding that on to current value yi. This method
  is known as the midpoint method.

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Midpoint Method

• k1hf(xi,yi) Change in y given by Eulers method.
• k2hf(xi0.5h,yi0.5k1) Change in y using slope
  estimate at midpoint

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Solving a Simple ODE with Midpoint Method

• Consider the differential equation
• yx y 0 x 1 with a initial condition
  (a0.0, b0.0), y(0) 2.
• First, we find the approximate solution for h0.5
  (n2), a very large step size.
• k1hf(x0,y0)0.5(0.02.0)1.0
• k2hf(x00.5h,y00.5k1)0.5(0.00.50.52.00.51.
  0)1.375
• Y1y0k22.01.3753.375
• Next, we find the approximate solution y2 at
  point x20.02h1.0

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Contd

• k1hf(x1,y1)0.5(x1,y1)0.5(0.53.375)1.9375
• k2hf(x10.5h,y10.5k1)0.5(0.50.50.53.3750.5
  1.9375)2.547
• y2y1k23.3752.54695.922

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MATLAB Prog. f. Midpoint
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Plot (Midpoint)
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References

• Celik, Ismail, B., Introductory Numerical
  Methods for Engineering Applications, Ararat
  Books Publishing, LCC., Morgantown, 2001
• Fausett, Laurene, V. Numerical Methods,
  Algorithms and Applications, Prentice Hall, 2003
  by Pearson Education, Inc., Upper Saddle River,
  NJ 07458
• Rao, Singiresu, S., Applied Numerical Methods
  for Engineers and Scientists, 2002 Prentice Hall,
  Upper Saddle River, NJ 07458
• Mathews, John, H. Fink, Kurtis, D., Numerical
  Methods Using MATLAB Fourth Edition, 2004
  Prentice Hall, Upper Saddle River, NJ 07458
• Varol, A., Sayisal Analiz (Numerical Analysis),
  in Turkish, Course notes, Firat University, 2001