Numerical Solution of Ordinary Differential Equation - PowerPoint PPT Presentation

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Numerical Solution of Ordinary Differential Equation

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Numerical Solution of Ordinary Differential Equation A first order initial value problem of ODE may be written in the form Example: Numerical methods for ordinary ... – PowerPoint PPT presentation

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Title: Numerical Solution of Ordinary Differential Equation


1
Numerical Solution of Ordinary Differential
Equation
  • A first order initial value problem of ODE may be
    written in the form
  • Example
  • Numerical methods for ordinary differential
    equations calculate solution on the
    points, where h is the steps size

2
Numerical Methods for ODE
  • Euler Methods
  • Forward Euler Methods
  • Backward Euler Method
  • Modified Euler Method
  • Runge-Kutta Methods
  • Second Order
  • Third Order
  • Fourth Order

3
Forward Euler Method
  • Consider the forward difference approximation for
    first derivative
  • Rewriting the above equation we have
  • So, is recursively calculated as

4
  • Example solve
  • Solution
  • etc

5
Graph the solution
6
Backward Euler Method
  • Consider the backward difference approximation
    for first derivative
  • Rewriting the above equation we have
  • So, is recursively calculated as

7
  • Example solve
  • Solution
  • Solving the problem using backward Euler method
    for yields
  • So, we have

8
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9
Graph the solution
10
Modified Euler Method
  • Modified Euler method is derived by applying the
    trapezoidal rule to integrating So, we
    have
  • If f is linear in y, we can solved for
    similar as backward euler method
  • If f is nonlinear in y, we necessary to used the
    method for solving nonlinear equations i.e.
    successive substitution method (fixed point)

11
  • Example solve
  • Solution
  • f is linear in y. So, solving the problem using
    modified Euler method for yields

12
Graph the solution
13
Second Order Runge-Kutta Method
  • The second order Runge-Kutta (RK-2) method is
    derived by applying the trapezoidal rule to
    integrating
  • over the interval . So, we have
  • We estimate by the forward euler
    method.

14
  • So, we have
  • Or in a more standard form as

15
Third Order Runge-Kutta Method
  • The third order Runge-Kutta (RK-3) method is
    derived by applying the Simpsons 1/3 rule to
    integrating
  • over the interval . So, we have
  • We estimate by the forward euler
    method.

16
  • The estimate may be obtained by forward
    difference method, central difference method for
    h/2, or linear combination both forward and
    central difference method. One of RK-3 scheme is
    written as

17
Fourth Order Runge-Kutta Method
  • The fourth order Runge-Kutta (RK-4) method is
    derived by applying the Simpsons 1/3 or
    Simpsons 3/8 rule to integrating
    over the interval . The formula of RK-4
    based on the Simpsons 1/3 is written as

18
  • The fourth order Runge-Kutta (RK-4) method is
    derived based on Simpsons 3/8 rule is written as
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