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Numerical Solution of Ordinary Differential Equation

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 . Numerical Methods for ODE. Euler Methods

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Numerical Solution of Ordinary Differential Equation

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  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. Graph the solution

  9. 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)

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

  11. Graph the solution

  12. 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.

  13. So, we have Or in a more standard form as

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

  15. 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

  16. Fourth Order Runge-Kutta Method • The fourth order Runge-Kutta (RK-4) method is derived by applying the Simpson’s 1/3 or Simpson’s 3/8 rule to integratingover the interval . The formula of RK-4 based on the Simpson’s 1/3 is written as

  17. The fourth order Runge-Kutta (RK-4) method is derived based on Simpson’s 3/8 rule is written as

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