Computational Methods CMSC/AMSC/MAPL 460. Ordinary differential equations

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1 Computational Methods CMSC/AMSC/MAPL 460 Ordinar differential equations Ramani Duraiswami, Dept. of Computer Science Several slides adapted from Prof. ERIC SANDT, TAMU

2 ODE: Previous class Applications and eamples Standard form Eamples of converting equations to standard form Volterra equation Euler Method (an eplicit method) Bacward Euler Method (an implicit/nonlinear method) Predictor corrector methods

3 Toda Runge Kutta methods Matlab function RK45 Solve Volterra equation Multistep methods: Adams Bashforth Implicit methods: Adams Moulton

4 Runge-Kutta Methods General class of methods that use evaluations at intermediate points to achieve high order Derivation of the nd order RK method Loo for a formula of the tpe a b n n (, ) hf n n (, ) hf α h β n n Specification of α, β, a, and b provides the formula

5 Runge-Kutta Methods The initial conditions are: d d 0 f (, ) ( ) 0 To derive method we use the Talor series epansion including nd order terms d, d, ( ) ( ) h d! d ( ) h ( ) n n n n n n

6 Runge-Kutta Methods Epand the derivatives: d d d d d d [ f (, ) ] f f f f f The Talor series epansion becomes n n hf h f ff ( ) Have epressed second derivative in terms of st derivatives of f

7 Runge-Kutta Methods Compare with formula we want for Runge-Kutta n n ahf bhf α n h, β n hf The definition of the function ( ) Epand the net step ( ) f αh, βhf f αhf βhf f n n [ ] ( α β ) ahf bh f hf hf f n n n a b hf bαh f bβh f f

8 Runge-Kutta Methods Compare with the Talor series [ ] a b hf bαh f bβ h f f n n [ a b] αb βb 4 Unnowns

9 Runge-Kutta Methods The Talor series coefficients (3 equations/4 unnowns) [ a b], αb, βb If ou select a as 3 3 a, b, α, β 3 3 If ou select a as a b, α β Note: These coefficient would result in a modified Euler or Midpoint Method

10 Runge-Kutta Method ( nd Order) Eample Consider Eact Solution d d The initial condition is: The step size is: h 0. ( 0 ) Use the coefficients a b, α β

11 Runge-Kutta Method ( nd Order) Eample The values are (, ) hf i i (, ) hf h i i [ ] i i

12 Runge-Kutta Method ( nd Order) Eample The values are similar to that of the Modified Euler also a second order method Estimate Solution Eact Error n n 'n h'n *n *' n h(*'n )

13 Runge-Kutta Method ( nd Order) Eample [b] The values are a 3, b, α, β (, ) hf i i ( α, β ) hf h i i a b i i

14 Runge-Kutta Method ( nd Order) Eample [b] The values are Estimate Solution Eact Error n n 'n h'n *n *' n h(*'n ) Eact

15 Runge-Kutta Methods Fourth order Runge-Kutta method n /6( n 3 4 ) hf(,) h(f(h/, / ) 3 h(f(h/, / ) 4 h(f(h, 3 ))

16 4th -order Runge-Kutta Method f f 4 f 3 f f f 3 6 ( f f f f ) 4 i i h/ i h

17 Volterra eample Write a function in standard form function f rabfo(t,) % Computes for the Volterra model. % () is the number of rabbits at time t. % () is the number of foes at time t. global alpha % interaction constant t % a print statement, just so we can see how fast % the progress is, and what stepsize is being used f(,) *() - alpha*()*(); f(,) -() alpha*()*();

18 Stud its solution for various values of encounter % Run the rabbit-fo model for various values of % the encounter parameter alpha, plotting each % solution. global alpha for i:-:0, alpha 0^(-i) [t,] ode45('rabfo',[0:.:], [0,0]); plot(t,(:,),'r',t,(:,),'b'); legend('rabbits','foes') title(sprintf('alpha %f',alpha)); pause end

19 Runge-Kutta Method (4 th Order) Eample Consider Eact Solution d d The initial condition is: The step size is: h 0. e ( 0 )

20 The 4 th Order Runge-Kutta The eample of a single step: 3 4 h h f h f h (, ) [ ] ( ) ( f 0.f 0, 0. 0 ) ( 0.05,.05) ( 0.05,. / ) [ f ( h, )] 0.f ( 0.,.04988) 6 h, h, 0. [ ] n n f 0. f

21 Runge-Kutta Method (4 th Order) Eample The values for the 4 th order Runge-Kutta method f(,) f f 3 3 f 4 4 Change Eact

22 The 4 th Order Runge-Kutta The step sizes are: [ ] [ ] h ( ) ( ) ( ) ( ) ( ) h h h The net step would be:

23 One Step Method The one-step techniques These methods allow us to var the step size. Use onl one initial value. After each step is completed the past step is forgotten: We do not use this information.

24 Eplicit and One-Step Methods Up until this point we have dealt with: Euler Method Modified Euler/Midpoint Runge-Kutta Methods These methods are called eplicit methods, because the use onl the information from previous steps. Moreover these are one-step methods

Computational Methods CMSC/AMSC/MAPL 460. Ordinary differential equations

Computational Methods CMSC/AMSC/MAPL 460 Ordinar differential equations Ramani Duraiswami, Dept. of Computer Science Several slides adapted from Prof. ERIC SANDT, TAMU ODE: Previous class Standard form