ࡱ>  T#` Mbjbj5G5G W-W-g;< < < < xxx4i4i4i8li<k<2l x"xxy $AhL!xxxx< < xy'222x&< xxy2x22 vxyl 0쎜4iv:f0Rtzx%2|eɢ< ԰^%%%xxxxN_$ _< < < < < <  CHAPTER VII Numerical Solution of Ordinary Differential Equations (ODE) I. Definition An equation that consists of derivatives is called a differential equation. Differential equations have applications in all areas of science and engineering. Mathematical formulation of most of the physical and engineering problems lead to differential equations. So, it is important for engineers and scientists to know how to set up differential equations and solve them. Differential equations are of two types ordinary differential equation (ODE) partial differential equations (PDE). An ordinary differential equation is that in which all the derivatives are with respect to a single independent variable. Examples of ordinary differential equation include  EMBED Equation.3 , EMBED Equation.3 ,  EMBED Equation.3  EMBED Equation.3  EMBED Equation.3 , EMBED Equation.3  Note: In this first part, we will see how to solve ODE of the form  EMBED Equation.3  In another section, we will discuss how to solve higher order ordinary differential equations or coupled (simultaneous) differential equations. But first, How to write a first order differential equation in the above form? Example 1  EMBED Equation.3  is rewritten as  EMBED Equation.3  In this case  EMBED Equation.3  Example 2  EMBED Equation.3  is rewritten as  EMBED Equation.3  In this case  EMBED Equation.3  II. Eulers Method   HYPERLINK "http://scienceworld.wolfram.com/biography/photo-credits.html" \l "Euler"  We will use Eulers method to solve an ODE under the form:  EMBED Equation.3  At  EMBED Equation.3 , we are given the value of  EMBED Equation.3  Let us call  EMBED Equation.3  as  EMBED Equation.3 . Now since we know the slope of  EMBED Equation.3 with respect to  EMBED Equation.3 , that is,  EMBED Equation.3 , then at  EMBED Equation.3 , the slope is  EMBED Equation.3 . Both  EMBED Equation.3 and  EMBED Equation.3  are known from the initial condition EMBED Equation.3 .  SHAPE \* MERGEFORMAT  Figure 1. Graphical interpretation of the first step of Eulers method. So the slope at x=x0 as shown in the figure above Slope  EMBED Equation.3   EMBED Equation.3  Thus  EMBED Equation.3  If we consider  EMBED Equation.3 as a step size EMBED Equation.3 , we get  EMBED Equation.3 . We are able now to use the value of  EMBED Equation.3  (an approximate value of  EMBED Equation.3 at EMBED Equation.3 ) to calculate EMBED Equation.3 , which is the predicted value at  EMBED Equation.3 ,  EMBED Equation.3   EMBED Equation.3  Based on the above equations, if we now know the value of  EMBED Equation.3  at  EMBED Equation.3 , then  EMBED Equation.3  This formula is known as the Eulers method and is illustrated graphically in Figure 2. In some books, it is also called the Euler-Cauchy method.  SHAPE \* MERGEFORMAT  Figure 2. General graphical interpretation of Eulers method.  - Additional resources to analyze the above example  Figure 3. Comparing exact and Eulers method. The problem was solved again using a smaller step size. The results are given below in Table 1. For smaller steps Step size,  EMBED Equation.3  EMBED Equation.3  EMBED Equation.3  EMBED Equation.3 480 240 120 60 30 -987.81 110.32 546.77 614.97 632.77 1635.4 537.26 100.80 32.607 14.806 252.54 82.964 15.566 5.0352 2.2864   Figure 4. Comparison of Eulers method with exact solution for different step sizes.  Figure 5. Effect of step size in Eulers method. The exact solution of the ordinary differential equation is given by the solution of a non-linear equation as:  EMBED Equation.3  The solution to this nonlinear equation is  EMBED Equation.3  It can be seen that Eulers method has large errors. This can be illustrated using Taylor series.  EMBED Equation.3   EMBED Equation.3  As you can see the first two terms of the Taylor series  EMBED Equation.3  are the Eulers method. The true error in the approximation is given by  EMBED Equation.3  The true error hence is approximately proportional to the square of the step size, that is, as the step size is halved, the true error gets approximately quartered. However from Table 1, we see that as the step size gets halved, the true error only gets approximately halved. This is because the true error being proportioned to the square of the step size is the local truncation error, that is, error from one point to the next. The global truncation error is however proportional only to the step size as the error keeps propagating from one point to another. II. Runge-Kutta 2nd order Eulers method was derived from Taylor series as:  EMBED Equation.3  This can be considered to be Runge-Kutta 1st order method. The true error in the approximation is given by  EMBED Equation.3  Now let us consider a 2nd order method formula. This new formula would include one more term of the Taylor series as follows:  EMBED Equation.3  Let us now apply this to a simple example:  EMBED Equation.3   EMBED Equation.3  Now since y is a function of x,  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  The 2nd order formula would be  EMBED Equation.3   EMBED Equation.3  You could easily notice the difficulty of having to find  EMBED Equation.3  in the above method. What Runge and Kutta did was write the 2nd order method as  EMBED Equation.3  where  EMBED Equation.3   EMBED Equation.3  This form allows us to take advantage of the 2nd order method without having to calculate EMBED Equation.3 . But, how do we find the unknowns  EMBED Equation.3 ,  EMBED Equation.3 ,  EMBED Equation.3  and  EMBED Equation.3 ? Equating the above equations:  EMBED Equation.3  and  EMBED Equation.3  gives three equations.  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  Since we have 3 equations and 4 unknowns, we can assume the value of one of the unknowns. The other three will then be determined from the three equations. Generally the value of  EMBED Equation.3  is chosen to evaluate the other three constants. The three values generally used for  EMBED Equation.3  are EMBED Equation.3 ,  EMBED Equation.3  and EMBED Equation.3 , and are known as Heuns Method, Midpoint method and Ralstons method, respectively. II.1. Heuns method Here we choose EMBED Equation.3 , giving  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  resulting in  EMBED Equation.3  where  EMBED Equation.3   EMBED Equation.3  This method is graphically explained in Figure 6.  SHAPE \* MERGEFORMAT  Figure 6. Runge-Kutta 2nd order method (Heuns method). II.2. Midpoint method Here we choose EMBED Equation.3 , giving  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  resulting in  EMBED Equation.3  where  EMBED Equation.3   EMBED Equation.3  II.3. Ralstons method Here we choose  EMBED Equation.3 , giving  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  resulting in  EMBED Equation.3  where  EMBED Equation.3   EMBED Equation.3   - Additional resources to analyze the above example  Figure 7. Heuns method results for different step sizes. Using smaller step size would increases the accuracy of the result as given in Table 1 and Figure 3 below. Effect of step size for Heuns method Step size, h EMBED Equation.3  EMBED Equation.3  EMBED Equation.3 480 240 120 60 30-393.87 584.27 651.35 649.91 648.211041.4 63.304 -3.7762 -2.3406 -0.63219160.82 9.7756 0.58313 0.36145 0.097625 Now, let us compare Eulers method and Runge-Kutta 2nd order method results: Comparison of Euler and the Runge-Kutta methods Step size, h EMBED Equation.3 EulerHeunMidpointRalston480 240 120 60 30-987.84 110.32 546.77 614.97 632.77-393.87 584.27 651.35 649.91 648.211208.4 976.87 690.20 654.85 649.02449.78 690.01 667.71 652.25 648.61  Figure 8. Comparison of Euler and Runge Kutta methods with exact results over time. NOTE: How do these three methods compare with results obtained if we found  EMBED Equation.3  directly? We know that since we are including first three terms in the series, if the solution is a polynomial of order two or less (that is, quadratic, linear or constant), any of the three methods are exact. But for any other case the results will be different. Consider the following example  EMBED Equation.3  If we directly find the EMBED Equation.3 , the first three terms of Taylor series gives  EMBED Equation.3  where  EMBED Equation.3   EMBED Equation.3  For a step size of EMBED Equation.3 , using Heuns method, we find  EMBED Equation.3  The exact solution  EMBED Equation.3  gives  EMBED Equation.3   EMBED Equation.3  Then the absolute relative true error is  EMBED Equation.3   EMBED Equation.3  For the same problem, the results from the Euler and the three Runge-Kutta method are given below Comparison of Eulers and Runge-Kutta 2nd order methods y(0.6)ExactEulerDirect 2ndHeunMidpointRalstonValue0.962390.49551.09301.10121.09741.0994 EMBED Equation.3 %48.51413.57114.42314.02914.236 III. Runge-Kutta 4th order Runge-Kutta 4th order method is based on the following  EMBED Equation.3  where knowing the value of  EMBED Equation.3  at  EMBED Equation.3 , we can find the value of y=yi+1 at  EMBED Equation.3 , and  EMBED Equation.3  The above equation is equated to the first five terms of Taylor series  EMBED Equation.3  Knowing that  EMBED Equation.3  and  EMBED Equation.3   EMBED Equation.3  Based on equating the above equations, one of the popular solutions used is  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  Solving the same example as above, but considering Runge-Kutta 4th order method, gives the following results:  Figure 9. Comparison of Runge-Kutta 4th order method with exact solution for different step sizes. The effect of step size on the value of the calculated temperature at t=480 seconds, can be seen on the following table and figure. Value of temperature at time, t=480s for different step sizes Step size,  EMBED Equation.3  EMBED Equation.3  EMBED Equation.3  EMBED Equation.3 480 240 120 60 30-90.278 594.91 646.16 647.54 647.57737.85 52.660 1.4122 0.033626 0.00086900113.94 8.1319 0.21807 0.0051926 0.00013419  Figure 10. Effect of step size in Runge-Kutta 4th order method. We can also compae the exact results with Eulers method (Runge-Kutta 1st order method), Heuns method (Runge-Kutta 2nd order method) and Runge-Kutta 4th order method.  Figure 11. Comparison of Runge-Kutta methods of 1st, 2nd, and 4th order. There are other versions of the fourth order method just like there are several versions of the second order methods. Formula developed by Kutta is  EMBED Equation.3  where  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  IV. Solving higher order ODE Until now, we have solved using Eulers and Runge-Kutta methods first order ordinary differential equations of the form  EMBED Equation.3  But now, the question is: what do we do to solve simultaneous or coupled differential equations or differential equations that are higher than first order? For example an nth order differential equations:  EMBED Equation.3  By assuming  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  =  EMBED Equation.3  The above equations represent n first order differential equations as follows  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  Example Rewrite the following differential equation as a set of first order differential equations:  EMBED Equation.3  The ordinary differential equation would be rewritten as follows. Assume  EMBED Equation.3  Then  EMBED Equation.3  Substituting this in the given second order ordinary differential equation gives  EMBED Equation.3   EMBED Equation.3 . The set of two simultaneous first order ordinary differential equations complete with the initial conditions then is  EMBED Equation.3   EMBED Equation.3 . Now one can apply any of numerical methods used for first order ordinary differential equations to solve the above first order differential equations.     Numerical Methods for Eng [ENGR 391] [Lyes KADEM 2007] PAGE  PAGE 99 Ordinary Differential Equations Example A ball at 1200K is allowed to cool down in air at an ambient temperature of 300K. Assuming heat is lost only due to radiation, the differential equation for the temperature of the ball is given by  EMBED Equation.3  Find the temperature at  EMBED Equation.3 seconds using Runge-Kutta 2nd order method (with Heuns method). Assume a step size of  EMBED Equation.3  seconds. Example A ball at 1200K is allowed to cool down in air at an ambient temperature of 300K. Assuming heat is lost only due to radiation, the differential equation for the temperature of the ball is given by  EMBED Equation.3  Find the temperature at  EMBED Equation.3 seconds using Eulers method. Assume a step size of  EMBED Equation.3  seconds. Swiss mathematician who was tutored by  HYPERLINK "http://scienceworld.wolfram.com/biography/BernoulliJohann.html" Johann Bernoulli. He worked at the Petersburg Academy and Berlin Academy of Science. He had a phenomenal memory, and once did a calculation in his head to settle an argument between students whose computations differed in the fiftieth decimal place. Euler lost sight in his right eye in 1735, and in his left eye in 1766. Nevertheless, aided by his phenomenal memory (and having practiced writing on a large slate when his sight was failing him), he continued to publish his results by dictating them. Euler was the most prolific mathematical writer of all times finding time (even with his 13 children) to publish over 800 papers in his lifetime. He won the Paris Academy Prize 12 times. When asked for an explanation why his memoirs flowed so easily in such huge quantities, Euler is reported to have replied that his pencil seemed to surpass him in intelligence.  HYPERLINK "http://scienceworld.wolfram.com/biography/Arago.html" Franois Arago said of him "He calculated just as men breathe, as eagles sustain themselves in the air" (Beckmann 1971, p.143; Boyer 1968, p.482).  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