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1、1,Ordinary Differential Equations,Equations which are composed of an unknown function and its derivatives are called differential equations. Differential equations play a fundamental role in engineering because many physical phenomena are best formulated mathematically in terms of their rate of chan
2、ge.,v- dependent variable t- independent variable,by Lale Yurttas, Texas A&M University,Part 7,2,When a function involves one dependent variable, the equation is called an ordinary differential equation (or ODE). A partial differential equation (or PDE) involves two or more independent variables. Di
3、fferential equations are also classified as to their order. A first order equation includes a first derivative as its highest derivative. A second order equation includes a second derivative. Higher order equations can be reduced to a system of first order equations, by redefining a variable.,by Lal
4、e Yurttas, Texas A&M University,Part 7,3,ODEs and Engineering Practice,Figure PT7.1,by Lale Yurttas, Texas A&M University,Chapter 25,4,Figure PT7.2,by Lale Yurttas, Texas A&M University,Chapter 25,5,Runga-Kutta Methods Chapter 25,This chapter is devoted to solving ordinary differential equations of
5、the form Eulers Method,by Lale Yurttas, Texas A&M University,Chapter 25,6,Figure 25.2,by Lale Yurttas, Texas A&M University,Chapter 25,7,The first derivative provides a direct estimate of the slope at xi where f(xi,yi) is the differential equation evaluated at xi and yi. This estimate can be substit
6、uted into the equation: A new value of y is predicted using the slope to extrapolate linearly over the step size h.,by Lale Yurttas, Texas A&M University,Chapter 25,8,Not good,by Lale Yurttas, Texas A&M University,Chapter 25,9,Error Analysis for Eulers Method/ Numerical solutions of ODEs involves tw
7、o types of error: Truncation error Local truncation error Propagated truncation error The sum of the two is the total or global truncation error Round-off errors,by Lale Yurttas, Texas A&M University,Chapter 25,10,The Taylor series provides a means of quantifying the error in Eulers method. However;
8、 The Taylor series provides only an estimate of the local truncation error-that is, the error created during a single step of the method. In actual problems, the functions are more complicated than simple polynomials. Consequently, the derivatives needed to evaluate the Taylor series expansion would
9、 not always be easy to obtain. In conclusion, the error can be reduced by reducing the step size If the solution to the differential equation is linear, the method will provide error free predictions as for a straight line the 2nd derivative would be zero.,by Lale Yurttas, Texas A&M University,Chapt
10、er 25,11,Figure 25.4,by Lale Yurttas, Texas A&M University,Chapter 25,12,Improvements of Eulers method,A fundamental source of error in Eulers method is that the derivative at the beginning of the interval is assumed to apply across the entire interval. Two simple modifications are available to circ
11、umvent this shortcoming: Heuns Method The Midpoint (or Improved Polygon) Method,by Lale Yurttas, Texas A&M University,Chapter 25,13,Heuns Method/ One method to improve the estimate of the slope involves the determination of two derivatives for the interval: At the initial point At the end point The
12、two derivatives are then averaged to obtain an improved estimate of the slope for the entire interval.,by Lale Yurttas, Texas A&M University,Chapter 25,14,Figure 25.9,by Lale Yurttas, Texas A&M University,Chapter 25,15,The Midpoint (or Improved Polygon) Method/ Uses Eulers method t predict a value o
13、f y at the midpoint of the interval:,by Lale Yurttas, Texas A&M University,Chapter 25,16,Figure 25.12,17,Runge-Kutta Methods (RK),Runge-Kutta methods achieve the accuracy of a Taylor series approach without requiring the calculation of higher derivatives.,Increment function,ps and qs are constants,b
14、y Lale Yurttas, Texas A&M University,Chapter 25,18,ks are recurrence functions. Because each k is a functional evaluation, this recurrence makes RK methods efficient for computer calculations. Various types of RK methods can be devised by employing different number of terms in the increment function
15、 as specified by n. First order RK method with n=1 is in fact Eulers method. Once n is chosen, values of as, ps, and qs are evaluated by setting general equation equal to terms in a Taylor series expansion.,by Lale Yurttas, Texas A&M University,Chapter 25,19,Values of a1, a2, p1, and q11 are evaluat
16、ed by setting the second order equation to Taylor series expansion to the second order term. Three equations to evaluate four unknowns constants are derived.,by Lale Yurttas, Texas A&M University,Chapter 25,20,We replace k1 and k2 in to get or Compare with and obtain (3 equations-4 unknowns),by Lale Yurttas, Texas A&M University,Chapter 25,21,Because we can choose an infinite number of values for a2, there are an infinite number of second-order RK
