2 Ordinary Differential Equations: Initial Value Problems

Transcription

1 Ordinar Differential Equations: Initial Value Problems Read sections 9., (9. for information), 9.3, 9.3., 9.3. (up to p. 396), Review questions 9.3, 9.4, 9.8, 9.9, Two Examples.. Foxes and Rabbits Assume that we have a forest where foxes and rabbits are living. The forest contains infinite resources of feed for the rabbits. So rabbits could live as the would if the would not meet foxes. As soon as a rabbit meets a fox the former will be eaten b the fox. On the other hand, rabbits are the onl feed for foxes. If a fox does not meet a rabbit sufficientl often, it will die. Let t be the number of rabbits at a certain time t. Similarl, let t be the number of foxes. Without the presence of foxes the number of rabbits will increase proportional to the number of individuals. The probabilit of a meeting between foxes and rabbits is proportional to the product of both populations. This gives rise to the following differential equation for the number of rabbits: d dt α rabbits reproduction β rabbits death Similarl, the differential equation for the number of foxes reads d dt γ foxes death δ foxes survival Such a sstem is called a Lotka-Volterra equation. This sstem was introduced in 95. In order to describe the two populations completel, we need to know the number of individuals at a certain point in time, sa t. This additional information is called initial values:.. Van der Pol Equation In the 9-ies van der Pol investigated a nonlinear amplifier circuit. The mathematical model he proposed is a differential equation of the form d dt ε The parameter ε describes some electrical properties of the amplifier circuit. Again, one needs initial values in order to single out a unique solution. Because the differential equation is of second order, two initial conditions are needed: α d dt β

2 . Generalities About Ordinar Differential Equations Differential equation are often obtained if one describes a continuous process. Most often the state of a sstem is described b a function of time t: t. A differential equation is a mathematical model which relates the values of a function and its derivatives and which must hold at ever point in time t. The notion ordinar indicates that there are onl derivatives with respect to one independent variable involved. The other independent variables are treated as parameters. The order of a differential equation is the order of the highest derivative which is present in the equation. The Lotka-Volterra equation is an example of a sstem of first order while the van der Pol equation is a second order equation. Assume that we are given a (scalar) differential equation of k-th order k f t k We define k new dependent variables u t t u t t u k. t k t The original k-th order differential equation is then equivalent to the following first-order sstem with k equations: u u. uk uk u u 3. u k f t u u k Hence, it is sufficient to consider onl sstems of first-order ordinar differential equations In more detail, this notation means t f t t t t d i f i t n t t i n dt i t i i n Note that this transformation is rather common. Almost all available codes in program libraries are designed to solve sstems of first-order ordinar differential equations. For instance, this is the approach taken in MATLAB.

3 .3 One-Step Methods In order to simplif the notation we will assume for the moment that we want to solve one scalar differential equation, onl, that means n. It is almost obvious to see how the considerations and methods can be generalized to sstems of equations. The differential equation at hand is the mathematical model of a process which the scientist or engineer is interested in. Depending on the application he might be interested in asking different questions: The engineer might be interested in knowing how the solution curve looks like in detail. The mathematical question would be: How does the solution behave on a given finite interval t T? Another engineer might be interested in a more qualitative picture, for example if the intended working point is reached if the sstem is switched on. The mathematical question would be: How does the solution behave on an infinite interval t? Of course, one cannot compute over infinite time intervals, but one wants to obtain a solution over a large interval such that the solution behavior in a vicinit of infinit becomes clear. Usuall, the mathematical model contains a lot of parameters, most of them with uncertainties. The engineer might wonder how sensitive the solution is to changes in those parameters. The mathematical question would be: What is the conditioning of the differential equation? Some states of the sstem ma be of special interest. In our introductor example the question might be when all animals are dead. The mathematical question would be: Does there exist a point in time where the solution has special well-defined properties? All these questions require special algorithms and specialized programs. The basic question, however, is if the ordinar differential equation can be solved reliabl and efficientl. There are onl ver few problems where an analtic (or, closed form) solution can be provided. Therefore, we are almost alwas required to use numerical methods. Most of the numerical methods for solving initial value problems for ordinar differential equations are based on a discretization method which is called the finite difference method. In contrast to an analtical solution, we do no longer attempt to provide an approximation of the solution for all points t t T. Instead we tr to compute onl a finite number (sufficientl man, of course) of approximations k t k to the true values t k for selected points t t t t k T such that t t t t k T Such a finite sequence of points is called a grid. The general idea consists of the following process: At t t, the initial value t is known. Then we tr to compute a value t b using the function f, the previous value, and the grid points t t. Our problem is now reduced to finding a solution to the problem t f t t t 3

4 subject to the initial condition t Obviousl, this problem can be handled in the same wa as before thus providing an approximation t at t. This process can be continued until T is reached. More general, if we know k, then we can compute k. The difference between two consecutive grid points is called the step size h k, h k t k t k In practical computations, one uses varing (or adaptive) step sizes. For our purposes it is sufficient to consider onl constant step sizes h k h such that.3. Euler s Method We start b writing down the Talor expansion for the true solution: t k t kh where s k t k t k t k t k t k h t k h f t k t k h s k h s k. This gives rise to the following recursive method: given k k h f t k for k Example.. We consider the following simple initial value problem for t The exact solution of this problem is t e t. Let us choose a step size of.5. The -th approximation is the initial guess. The next approximation is obtained for t t h 5 5. According to Euler s method, this becomes h f 5 5 The exact solution is 5 e Let us take one additional step. The next grid point is t t h. Euler s approximation gives This gives the following little table: h f

5 t k k t k error It is relativel simple to implement Euler s method in a MATLAB function. The input arguments to such a function should be the data of the problem at hand: the function f, the interval t T, the initial value, and the step size h. The output should include the computed grid points t k as well as the approximations k. This gives rise to the following code: function [t,] = Euler(f,interval,,h) t = interval(); T = interval(); % and f are assumed to provide column vectors = ; t = t; k = ; tk = t; while tk+h < T+h/ k = k+h*feval(f,tk,k); = [,k]; tk = tk+h; t = [t,tk]; end This function can now be used for solving our little example problem. The result is given in Figure. clear clf f = inline(, t, ); [t,] = Euler(f,[,],,.5); % plot the result plot(t,,t,, o ) xlabel( t ) label( ) title( A simple example ) Euler s method is said to be a one-step method because the computation of the new value k at t k depends onl on information at the previous grid point t k. Such a method can be written down more generall as k k hφ t k k h 5

6 .5 A simple example t Figure : A simple example.3. Accurac The most important question is how accurate the numerical results is and if the accurac becomes better and better if the step size h becomes smaller and smaller. As usual, there are two tpes of error sources: discretization errors which appear because the differential equation is onl approximated; rounding errors. We will in the following consider onl the discretization error because, in the context of the solution of ordinar differential equations, the rounding error is usuall negligible. What we are mostl interested in is the global error e k k t k Unfortunatel, this error is not available explicitl. Even reliable and practicall useful estimates are hard to obtain. Opposed to that, it appears to be rather eas to describe the local error. The latter is the error which arises if one takes onl one step of the method with an errorfree k t k : This statement does not hold for other algorithms! l k k t k t k hφ t k t k h t k 6

7 Example.. For Euler s method, we can compute l k t k t k h s k h f t k t k t k h f t k t k t k h t k h s k Hence, we can write l k O h. A method has the order of accurac (or simpl the order) p if l k O h p This means that Euler s method has the order. Wh is the order of a method important? One can show that the order of magnitude of the global error for one-step methods can be estimated b that of the local error, that is, for a method of order p it holds e k O h p if the step size h is sufficientl small. Example.3. What are the consequences of the order of a method? The user of a method is usuall interested in obtaining a solution with a given accurac with minimal computation time. Therefore, it is probabl a good idea to use step sizes as large as possible. If a method provides a result with a certain accurac for a given step size h, a reduction of the step size b a factor a leads to a reduction of the error b a factor of a p if we have a p-th order method. This is illustrated in the following table where we have used the factor a which amounts to step size halving. if the order p is 3 4 the error is reduced b So the general rule is to alwas appl a higher order method if there are no other reasons for not doing it..3.3 Runge-Kutta methods There are a number of higher order one-step methods available. Most of them belong to the huge class of Runge-Kutta methods. As examples, we provide two methods of this tpe which were invented more than ears ago. 7

8 Heun s Method. This method can be written rather compact in the form k h f t k k f t k k h f t k k It is convenient to write down the method in a more sstematic wa: k k k f t k k f t k h k h k k Heun s method has second order. Here we give a sample implementation of Heun s method in MATLAB. Do not use it for real application problems! hk function [t,] = Heun(f,interval,,h) t = interval(); T = interval(); % and f are assumed to provide column vectors = ; t = t; k = ; tk = t; while tk+h < T+h/ k = feval(f,tk,k); k = feval(f,tk+h,k+h*k); k = k+.5*h*(k+k); = [,k]; tk = tk+h; t = [t,tk]; end Example.4. Let us demonstrate the application of our code for solving the Lotka-Volterra equation. We use the following parameters: α β γ δ T 9 The initial values are 3 and 5. Let us choose for example h (which is rather large!). Then the following codes provides us with nice results (tr it out b ourself). 8

9 clear clf [t,] = Heun(@flotka,[,9],[3;5],.); plot(t,(,:),t,(,:), -- ) xlabel( time ) label( number of individuals ) title( Lotka-Volterra equation ) % % The function: Should appear in a separate file!!! function p = flotka(t,) p = [*()-.*()*();... -()+.*()*()]; Classical Runge-Kutta Method. This is probabl the best-known Runge-Kutta method. It should be noted, however, that this method is superseded b much more efficient methods. The classical method can be formulated as follows: This method is of order 4. k f t k k k f h t k h k k k 3 f h t k h k k k 4 f t k h k hk 3 h k k k 3 k 4 6 k Runge-Kutta Methods in MATLAB. MATLAB provides two (explicit) Runge-Kutta methods. ode45 is a fifth-order method proposed b Dormand and Prince. The other solver is ode3 having order three using a method b Bogacki and Shampine. Both methods use adaptive step size. According to our considerations about the consequences of different orders on the computational work it is proposed alwas to tr the Dormand-Prince method first..3.4 Automatic Step Size Control While examining the local error of Euler s method we found out that beside its dependence on h it depends essentiall on the second derivative of the solution. If one requires a certain bound on the local error, sa l k tol then the step size can be chosen larger where the second derivative is small, and smaller where the second derivative is larger. Such an idea can be realized. The resulting algorithm is called adaptive. It must be noted, however, that the control variable is the local error and not the global 9

10 error. This automatic step size control can onl work reliabl if the problem at hand is wellconditioned. A discussion of that can be found in the course book..4 Some Numerical Examples Here we show some applications. We will use the van der Pol equation as a running example. This can be a ver hard problem depending on the parameter ε. In order to simplif things we will choose ε. This preserves nice features while the problem is not too hard. There are different was of presenting the solution of a problem to the user. The worst one is obviousl to provide a table containing all results. The useful information ma be too much hidden in the numbers. Therefore, a graphical representation is usuall the method of choice. Before presenting a graph one should think ver carefull how the intended informations can be shown as informativel (and suggestive?) as possible. In the context of ordinar differential equations one ver often chooses among the following two possibilities: Plot trajectories. This means that some or all solution components i t are plotted as a function of the independent variable t. Plot a phase portrait. Here one plots two solution components i t j t in a i j - coordinate sstem. The independent variable t appears merel as a parameter of the curve. Our method of choice is MATLAB s ode45 solver. Figure contains a trajector plot and a phase portrait. The initial values are and Periodic solution 4 Periodic solution t Figure : Solution to the van der Pol equation The graphs in Figure 3 contain some representations which demonstrate different possibilities of data enhancement. We will use phase portraits. The first plot gives the raw data as the are provided b ode45. The second plot uses piecewise linear interpolation which is the standard algorithm of MATLAB s plot command. The final plot uses advanced interpolation techniques

11 which are built-in into ode45. Note that the latter interpolation provides a highl accurate interpolation even if the data points are rather sparsel distributed!. 4 Phase portrait: Raw data 4 Phase portrait: Linear interpolation Phase portrait: ode45 interpolation Figure 3: Interpolation of discrete data In the last plot (Figure 4), we compared the approximate solution obtained b Euler s method with a step size of and, respectivel, with the solution obtained b ode45.

12 5 Phase portrait: Euler method 4 3 ode45 Euler. Euler Figure 4: Euler solutions of the van der Pol equation