5.2 - Euler s Method

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1 5. - Euler s Method Consider solving the initial-value problem for ordinary differential equation: (*) y t f t, y, a t b, y a. Let y t be the unique solution of the initial-value problem. In the previous section, two approximation methods, Picard s Method and the method using Taylor expansion at t a, are introduced to compute sequences of approximations y k t to the solution y t for a t b. This section, we study the first numerical method, Euler s Method, that computes the set y k N k 0 where y k y t k and a t 0 t... t N t N b. Assume that the initial-value problem is well-posed. Let h b N a,wherenis a positive integer. Define t k t k h or t k a kh.. Euler Method: The Taylor expansion of y t i at t t i is: y t i y t i h y t i y t i! t i t i y t i! t i t i y t i hf t i, y t i h y t i where t i is in the interval t i, t i. When h is small, y t i y t i hf t i, y t i.lety 0. Compute y i y i hf t i, y i, i,,...,n. The equation y i y i hf t i, y i is called the difference equation associated with Euler s Method. Example Use Euler s Method to approximate the solution of the initial-value problem: y y t, 0 t, y with h 0.. Use the obtained approximation y i and linear interpolations to approximate the values of y 0.55, y.3, and y.95. N h 0 0. i 0 i 0. 0.i, i 0,...,0. Compute by hand: i t i y i 0 t 0 0 y t 0. y t 0.4 y t y Use MatLab program: eulfun.m y-t^ ; a 0; b ; y0 0.5; h 0.; [tv,yv,n] eulfun(fun,a,b,y0,h)

2 i t i y i i t i y i Graph y i : plot(tv,yv) axis([006]) title( dy/ y-t^, 0 t, y(0) / ) 6 dy/=y-t +, 0<=t<=, y(0)=/ Just for a comparison, we use h 0.4, 0., 0., 0.05 to compare y i and compare them with the true solution y t. The general solution of dy y t is: y t t t e t C. The solution for the initial-value problem: y 0 C 0.5, C 0.5, so y t 0.5e t. Use the MatLab program eulex.m

3 5.5 dy/=y-t +, y(0)=/, t in [0,] o- Euler method with h=0.4 -*- Euler method with h=0. -x- Euler method with h= Euler method with h=0.05 t - solution: y(t)=(t+) -0.5e We use the Newton Forward Difference Formula for a linear interpolation using data: t i, y i, t i,y i P t y i y i y i t i t t t i. i Approximate y 0.55 using t, y, t 3,y 3 : L t y y 3 y t t y 0.55 L Approximate y.3 using t 6, y 6, t 7,y 7 : L 6 t y 6 y 7 y t t y.3 L Approximate y.95 using t 9, y 9, t 0,y 0 : L 9 t y 9 y 0 y t t y.95 L Example Use Euler s Method to approximate the solution of the initial-value problem: dy tsin ty, 0 t, y 0 0 with h 0., h 0.05 and h 0.0. By hand: h 0. : 3

4 i t i y i 0 t 0 0 y 0 0 t 0. y sin t 0. y sin t y sin h 0.05 : i t i y i 0 t 0 0 y 0 0 t 0.05 y sin t 0. y sin t y sin Euler Method: dy/=+t*sin(t*y), t in [0,pi], y(0)=0 h=0. h=0.05 h= Example Use Euler s Method to approximate the solution of the initial-value problem: dy y t, 0 t, y 0 with h 0. and h Compare the numerical solutions with the approximation y 3 t by Picard s Method. 4

5 0 Euler Method: dy/=y +t, t in [0,], y(0)= h=0. h=0.05 y 3 (t) by Picard's Method Error Bound for Euler Method: Theorem Suppose that f is continuous and satisfies a Lipschitz condition with constant L on D t, y, a t b, y, and there exists a constant M such that y t M, for all t a, b. Let y t be the unique solution to the initial-value problem y t f t, y, a t b, y a, and y 0, y,..., y N be the approximation generated by Euler s Method for some positive number N. Then for each i 0,,,...,N y t i y i hm L el t i a. Proof: 5

6 y t i y i y t i hf t i, y t i h y t i y i hf t i, y i y t i y i h f t i, y t i f t i, y i h y t i y t i y i h f t i, y t i f t i, y i h y t i y t i y i hl y t i y i h M hl y t i y i h M hl hl y t i y i h M h M hl y t i y i h M hl hl i y t 0 y 0 h M hl i hl i... hl 0 h M hl i hl h M hl i hl hm L hl h t i a, t i a ih, i t i a h hm L hl hl L t i a hm L el t i a hm hl i L y t i y i hm L el t i a How do we find or estimate M (without computing y t explicitly)? If f is differentiable in t and y, then y t dy d y t f t,y t t f t,y t f t,y t t f t,y t f t,y t f t,y t dy f t,y t t f t,y t f t,y t Example a. Find an error bound when Euler s Method is used to approximate the solution of the initial-value problem: y y t, 0 t, y with h 0., h 0.. b. Determine as large as possible a value of h so that the approximation error is less than 0.. Find the Lipschitz constant L and an upper bound of y M : f t,y y t f, L. y t f t,y t t f t,y t f t,y t t y t We cannot get an upper bound for y since there is no upper bound for y in D. In this case, 6

7 y t t 0.5e t, y t t 0.5e t, y t 0.5e t y t graphically, let M. Or, we may find the upper bound of y t graphically: y M y t i y i hm L el t i a h e t i 0 h e x a. y t 0.5e t h 0. y t i y i h 0. y t i y i i 0.38 i i 0.95 i i i i i i i b. Findh such that Let h y t i y i h e 0. h 0. e Exercises:. For each initial-value problem, a. compute by hand, y 0, y and y using Euler s method with h 0.5; and b. find the approximation error bound for each approximation y i for i,. () y 4 t y t4, t 3, y () y sin y et,0 t, y 0 0 cos y (3) y y, t 4, y 0 t 7

8 . For each initial-value problem, a. compute by hand, y 0, y and y using Euler s method with h 0.5; b. compute the true error y t i y i for each y i obtained in a.; c. find an upper bound for the approximation error for each approximation y i obtained in a. and compareitwiththetrueerrorobtainedinb; d. determine if f t,y satisfies a Lipschitz condition on given D and if so, give the Lipschitz constant; e. determine if the initial-value problem has a unique solution over the given D; and f. use the MatLab function eulfun.m to compute y i and then plot both sequences y i and y t i (y t is given) for h 0.5 and h 0.. Turn in your graphs and program. () y et y,0 t, y 0 y t e t, D t,y ; 0 t, y () y t t tan y, 0 t, y 0 4 y t sin y t, D t,y ; 0 t,0 y 4 (3) y t y, 0 t 4, y 0 y t e t t, D t,y ; 0 t 4, y 3. For each initial-value problem, a. find as small as possible an upper bound for the true error y t i y i ; b. use the MatLab function eulfun.m to compute y i with h 0.5 and h 0.5; and c. compare the true errors with the approximation errors obtained in a. for h 0.5 and h 0.5. () y 4t y y,0 t, y 0, y t t () y y e ty,0 t 0.9, y 0 0, y t e t tan e t 4. Extra points: () Consider the population model y ry y y k y. The first term on the right side is known as the logistic growth term and the second one represents harvesting/predation of the species by some other species. The parameters r and k are called the natural growth rate of the population and the environmental carrying capacity, respectively Let r 0.4, k 0 and initial population.44. Use Euler method to determine the eventual population level reached from the initial population 8