Roberto s Notes on Integral Calculus Chapter 3: Basics of differential equations Section 6. Euler s method. for approximate solutions of IVP s

Transcription

1 Roberto s Notes on Integral Calculus Chapter 3: Basics of differential equations Section 6 Euler s method for approximate solutions of IVP s What ou need to know alread: What an initial value problem is. How to construct the linearization of a function at a point. What ou can learn here: How to obtain an approximate solution to an initial value problem (IVP), without solving its ODE. We have seen onl a few basic methods for solving ODE s and man more are available. However, not ever ODE can be solved exactl and when the cannot, it ma be nice to have a method that provides an approximate solution. Such methods do exist and the rel on intense computational work and are therefore best suited for computer programming. Here we shall focus not on the computational part, but on the principles and ideas behind one such method. As the title of the section suggests, the method I am referring to is Euler s method, named after the great mathematician Leonhard Euler. It can be applied to an first order initial value problem in whose ODE the first derivative can be isolated, that is, an ODE that can be written in the form: ' F( x, ) In this situation, the ODE allows us to compute the slope of the solution that contains an given point (x, ). This value and the point itself allow us to construct a linear approximation of the function near that point, thus leading us to a new point nearb. Continuing the process allows us to construct not a formula, but a table of values whose corresponding graph approximate the solution we are seeking. Here is the formal description of the process. Euler s strateg for approximating the solution of a first order IVP If a first order initial value problem can be written as: ' F( x, ), x 0 0 where x0, 0 is a given point, an approximation to its solution f () x can be obtained as follows:. Determine a step size h corresponding to a small change of x. This should be chosen small enough to allow a good approximation, but large enough to make the computations feasible in a reasonable number of steps.. Use the linear approximation formula Integral Calculus Chapter 3: Basics of differential equations Section 6: Euler s method for approximate solutions Page

2 ( ) ( ) f x h f x f x h and the starting point x0, 0 to obtain an approximation to the coordinate of the point at x x h: 0 f ( x ) F x, h 0 0 0, This generates the new point x. 3. Repeat the same process b using, x to obtain an estimate of the next point, x., 4. Keep repeating the same process, b using the iterative formula ( ), f x f x F x h n n n n to obtain as man points as needed All we know about this phantom function is a point on it and a formula for its slope at an point, F x, the given point x b using the slope, 0, 0. So, we start b constructing the tangent line at F x. 0 0 On this tangent line we can identif a point nearb, at x x0 h, for a small value of h. We call this new point x., Wait a minute! What exactl are we accomplishing with all these formulae and how do I know that the work? Excellent point! Here is a visual representation of the process. We are tring to construct an unknown function curve. f x, represented here with a dashed Notice, however, that this second point is not on the solution we are seeking, but on a different one near it. This means that for the next step we will use the tangent line to a nearb solution, therefore adding inaccurac of the approximation. Still, we now have a new point and we can repeat the procedure to obtain a third point and more. Integral Calculus Chapter 3: Basics of differential equations Section 6: Euler s method for approximate solutions Page

3 So, b using this procedure we build each step on the approximation of the previous one. This implies that the approximations usuall tend to become worse as we move awa from the original point. B using small steps, that is, small values of h, we hope not to wander too far from to the solution we want. Notice also that this method is iterative, that is, it repeats each step on the basis of the previous one. Example: 3 x, We are here in a suitable setting to use Euler s method to approximate the value of, sa, (). To do that, we first pick a value for h that is not too large or small. If we tr h 0., we get the following iteration formula: / 3 / x 0. n n n n n n B appling the formula repeatedl, we obtain the following table of values: xn n / n Therefore () But we can also look at a plot of the points so obtained, in the hope of extracting more information about the solution of the whole IVP: x n n What kind of additional information? Technical fact If the plot obtained from Euler s method shows a fairl linear pattern, it means that the solution is also linear and hence the approximation, which uses tangent lines, is fairl good. If the plot obtained from Euler s method shows an upward concavit, it means that the tangent lines used for the approximation are below the curve and hence we have an underestimate of the true values. If the plot obtained from Euler s method shows a downward concavit, it means that the tangent lines used for the approximation are above the curve and hence we have an overestimate of the true values. If the plot shows marked changes in concavit, it means that the tangent lines used in the approximations diverge quickl from the real solutions and hence the estimate is likel not good. 0-3 For instance, in our previous example we can see that the estimated solution seems concave down. This tells us that the approximation we obtained is likel an overestimate and the true value of () is likel lower than Moreover, the curve seems to be more concave at the beginning and flatter later. This ma indicate an unsatisfactor qualit of the approximation. Likel? Seems? Ma? Not too sure, are we? No, we are not! Remember that this is a method to approximate a solution and, like all such methods, there is the big question of how trustworth our conclusions are. There is a whole area of mathematics devoted to that, but it is beond our current goals. What we can tr for now, however, is to compare the approximation to the exact solution when we can compute the latter. Here is an example. Integral Calculus Chapter 3: Basics of differential equations Section 6: Euler s method for approximate solutions Page 3

4 Example: tan, 0 6 This initial value problem can be solved exactl, since it is separable, and it gives the function: x e sin whose graph is shown here. Let us see how Euler s method performs here. If we use h 0., the iterative formula becomes: F x, h 0. tan n n n n n n We now implement it for 3 steps: x0, 0 0,. tan x, 0., tan x 3 x, 0.6,.989, 0.4, tan If we plot these points against the real solution, we obtain this graph: Notice that the concavit of the solution is up and, correspondingl, we get underestimates. The qualit of the approximation is not bad, but will it sta this good for long, or will it deteriorate quickl as we consider more points? Fortunatel here the domain is limited, because of the inverse sine function, so the danger is not big, but in other cases it ma be. So, should we alwas use h=0.? No, an small number ma be reasonable and in practical applications it ma be wise to tr several values of h and observe how different the corresponding estimates are. When the are ver different, a smaller step size should be used. But remember that a smaller step size implies more steps to get to where we need to go or to get a sufficientl large section of the solution. It s all a question of balance Summar Euler s method allows us to approximate the solutions of an initial value problem b using tangent lines at several points, close to each other. Since each step moves the estimated approximation to a different solution, such approximations ma worsen with an increasing number of steps. The step size should be small enough to not exacerbate the cumulative approximations, but large enough to allow the needed estimate. Don t get confused in the midst of the notation an formulae! Common errors to avoid Integral Calculus Chapter 3: Basics of differential equations Section 6: Euler s method for approximate solutions Page 4

5 Learning questions for Section I 3-6 Review questions:. Describe how Euler s method works. Memor questions:. To which differential equations can we appl Euler s method?. What is the iterative formula for Euler s method? Computation questions: For each of the initial value problems presented in questions -0, and for the corresponding desired value: a) Decide on a suitable value of the step size. b) Construct the iterative formula of Euler s method. c) Appl the formula to obtain the required value of the function. d) Plot the point and observed the concavit to determine whether what ou obtained is an overestimate or an underestimate. e) If possible, solve the problem exactl and compare it to Euler s approximation.. x, () ; find (.) 6. x, ; find 5., 4 3; find 5 3. ' 3 x e,, find 0 4. ' x, 0 ; find x x 0, ; find 7. cos( x), (0) ; find (0.6) 8. x, 0 ; find ' cos x, x 3 ', (), find., find 0.5 Integral Calculus Chapter 3: Basics of differential equations Section 6: Euler s method for approximate solutions Page 5

6 Theor questions:. For what tpe of ODE s is Euler s method usable?. Wh is it necessar to use Euler s method for certain ODEs? 3. For what kinds of solutions is Euler s method guaranteed to provide an exact solution of the problem? 4. Does Euler s method provide a better approximation for smaller or larger values of h? 5. If the points obtained with Euler s method form a downward concavit, what can we sa about the approximation it provides? 6. If the points obtained with Euler s method form an upward concavit, what can we sa about the approximation it provides? 7. What is one disadvantage of using a small step size in Euler s method? 8. Besides the iterative formula itself, what other approximation occurs in Euler s method? 9. If Euler s method provides a good approximation to a given ODE, what can be said about the second derivative of the solution it approximates? 0. If Euler s method provides an underestimate, what can be said about the second derivative of the solution it approximates?. If Euler s method provides an overestimate, what can be said about the second derivative of the solution it approximates?. When will Euler s method provide a ver poor estimate? Application questions:. In order to complete a certain design, ou must find an approximate solution for the initial value problem cos( x ); (0) 0 for x between 0 and 0. Set up the formula for Euler s approximation and use it to compute the first three steps of this approximation, b using a suitable value of h.. A battle between an Evil Empire arm of E soldiers and a Rebel Alliance group of R fighters is fought according to the Lanchester ODE de 5R dr E. If at the start of the battle E 000 and R 300, what will be the size of the Empire s arm when the Rebels have lost 30 fighters, as estimated b Euler s method with a new estimate for each 0 Rebels lost? What questions do ou have for our instructor? Integral Calculus Chapter 3: Basics of differential equations Section 6: Euler s method for approximate solutions Page 6