ordinary differential equations n.
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  1. ORDINARY DIFFERENTIAL EQUATIONS ENGR 351 Numerical Methods for Engineers Southern Illinois University Carbondale College of Engineering Dr. L.R. Chevalier Dr. B.A. DeVantier

  2. Ordinary Differential Equations…where to use them The dissolution (solubilization) of a contaminant into groundwater is governed by the equation: where kl is a lumped mass transfer coefficient and Cs is the maximum solubility of the contaminant into the water (a constant). Given C(0)=2 mg/L, Cs = 500 mg/L and kl= 0.1 day-1, estimate C(0.5) and C(1.0) using a numerical method for ODE’s.

  3. Ordinary Differential Equations…where to use them A mass balance for a chemical in a completely mixed reactor can be written as: where V is the volume (10 m3), c is concentration (g/m3), F is the feed rate (200 g/min), Q is the flow rate (1 m3/min), and k is reaction rate (0.1 m3/g/min). If c(0)=0, solve the ODE for c(0.5) and c(1.0)

  4. Ordinary Differential Equations…where to use them Before coming to an exam Friday afternoon, Mr. Bringer forgot to place 24 cans of a refreshing beverage in the refrigerator. His guest are arriving in 5 minutes. So, of course he puts the beverage in the refrigerator immediately. The cans are initially at 75, and the refrigerator is at a constant temperature of 40.

  5. Ordinary Differential Equations…where to use them The rate of cooling is proportional to the difference in the temperature between the beverage and the surrounding air, as expressed by the following equation with k = 0.1/min. Use a numerical method to determine the temperature of the beverage after 5 minutes and 10 minutes.

  6. Ordinary Differential Equations • A differential equation defines a relationship between an unknown function and one or more of its derivatives • Physical problems using differential equations • electrical circuits • heat transfer • motion

  7. Ordinary Differential Equations • The derivatives are of the dependent variable with respect to the independent variable • First order differential equation with y as the dependent variable and x as the independent variable would be: • dy/dx = f(x,y)

  8. Ordinary Differential Equations • A second order differential equation would have the form: } does not necessarily have to include all of these variables

  9. Ordinary Differential Equations • An ordinary differential equation is one with a single independent variable. • Thus, the previous two equations are ordinary differential equations • The following is not:

  10. Ordinary Differential Equations • The analytical solution of ordinary differential equation as well as partial differential equations is called the “closed form solution” • This solution requires that the constants of integration be evaluated using prescribed values of the independent variable(s).

  11. Ordinary Differential Equations • An ordinary differential equation of order n requires that n conditions be specified. • Boundary conditions • Initial conditions

  12. Ordinary Differential Equations • An ordinary differential equation of order n requires that n conditions be specified. • Boundary conditions • Initial conditions consider this beam where the deflection is zero at the boundaries x= 0 and x = L These are boundary conditions

  13. consider this beam where the deflection is zero at the boundaries x= 0 and x = L These are boundary conditions P a yo In some cases, the specific behavior of a system(s) is known at a particular time. Consider how the deflection of a beam at x = a is shown at time t =0 to be equal to yo. Being interested in the response for t > 0, this is called the initial condition.

  14. Ordinary Differential Equations • At best, only a few differential equations can be solved analytically in a closed form. • Solutions of most practical engineering problems involving differential equations require the use of numerical methods.

  15. Scope of Lectures on ODE • One Step Methods • Euler’s Method • Heun’s Method • Improved Polygon • Runge Kutta • Systems of ODE • Adaptive step size control

  16. Scope of Lectures on ODE • Boundary value problems • Case studies

  17. Specific Study Objectives • Understand the visual representation of Euler’s, Heun’s and the improved polygon methods. • Understand the difference between local and global truncation errors • Know the general form of the Runge-Kutta methods. • Understand the derivation of the second-order RK method and how it relates to the Taylor series expansion.

  18. Specific Study Objectives • Realize that there are an infinite number of possible versions for second- and higher-order RK methods • Know how to apply any of the RK methods to systems of equations • Understand the difference between initial value and boundary value problems

  19. Review of Analytical Solution At this point lets consider initial conditions. y(0)=1 and y(0)=2

  20. What we see are different values of C for the two different initial conditions. The resulting equations are:

  21. One Step Methods • Focus is on solving ODE in the form h y yi+1 slope = f yi x This is the same as saying: new value = old value + slope x step size

  22. Euler’s Method • The first derivative provides a direct estimate of the slope at xi • The equation is applied iteratively, or one step at a time, over small distance in order to reduce the error • Hence this is often referred to as Euler’s One-Step Method

  23. Example For the initial condition y(1)=1, determine y for h = 0.1 analytically and using Euler’s method given:

  24. Error Analysis of Euler’s Method • Truncation error - caused by the nature of the techniques employed to approximate values of y • local truncation error (from Taylor Series) • propagated truncation error • sum of the two = global truncation error • Round off error - caused by the limited number of significant digits that can be retained by a computer or calculator

  25. ....end of example

  26. Higher Order Taylor Series Methods • This is simple enough to implement with polynomials • Not so trivial with more complicated ODE • In particular, ODE that are functions of both dependent and independent variables require chain-rule differentiation • Alternative one-step methods are needed

  27. Modification of Euler’s Methods • A fundamental error in Euler’s method is that the derivative at the beginning of the interval is assumed to apply across the entire interval • Two simple modifications will be demonstrated • These modification actually belong to a larger class of solution techniques called Runge-Kutta which we will explore later.

  28. Heun’s Method • Determine the derivative for the interval • the initial point • end point • Use the average to obtain an improved estimate of the slope for the entire interval

  29. y Use this “average” slope to predict yi+1 xixi+1 {

  30. y y xixi+1 x xi xi+1

  31. y x xi xi+1

  32. Improved Polygon Method • Another modification of Euler’s Method • Uses Euler’s to predict a value of y at the midpoint of the interval • This predicted value is used to estimate the slope at the midpoint

  33. Improved Polygon Method • We then assume that this slope represents a valid approximation of the average slope for the entire interval • Use this slope to extrapolate linearly from xi to xi+1 using Euler’s algorithm

  34. Runge-Kutta Methods Both Heun’s and the Improved Polygon Method have been introduced graphically. However, the algorithms used are not as straight forward as they can be. Let’s review the Runge-Kutta Methods. Choices in values of variable will give us these methods and more. It is recommend that you use this algorithm on your homework and/or programming assignments.

  35. Runge-Kutta Methods • RK methods achieve the accuracy of a Taylor series approach without requiring the calculation of a higher derivative • Many variations exist but all can be cast in the generalized form: { f is called the incremental function

  36. , Incremental Functioncan be interpreted as a representative slope over the interval

  37. NOTE: k’s are recurrence relationships, that is k1 appears in the equation for k2 which appears in the equation for k3 This recurrence makes RK methods efficient for computer calculations

  38. Second Order RK Methods

  39. Second Order RK Methods • We have to determine values for the constants a1, a2, p1 and q11 • To do this consider the Taylor series in terms of yi+1 and f(xi,yi)

  40. Now, f’(xi , yi ) must be determined by the chain rule for differentiation The basic strategy underlying Runge-Kutta methods is to use algebraic manipulations to solve for values of a1, a2, p1 and q11

  41. By setting these two equations equal to each other and recalling: we derive three equations to evaluate the four unknown constants

  42. Because we have three equations with four unknowns, we must assume a value of one of the unknowns. Suppose we specify a value for a2. What would the equations be?

  43. Because we can choose an infinite number of values for a2 there are an infinite number of second order RK methods. Every solution would yield exactly the same result if the solution to the ODE were quadratic, linear or a constant. Lets review three of the most commonly used and preferred versions.

  44. Consider the following: Case 1: a2 = 1/2 Case 2: a2 = 1 These two methods have been previously studied. What are they?

  45. Case 1: a2 = 1/2 This is Heun’s Method with a single corrector. Note that k1 is the slope at the beginning of the interval and k2 is the slope at the end of the interval.

  46. Case 2: a2 = 1 This is the Improved Polygon Method.

  47. Ralston’s Method Ralston (1962) and Ralston and Rabinowitiz (1978) determined that choosing a2 = 2/3 provides a minimum bound on the truncation error for the second order RK algorithms. This results in a1 = 1/3 and p1 = q11 = 3/4

  48. Example Evaluate the following ODE using Heun’s Methods

  49. Third Order Runge-Kutta Methods • Derivation is similar to the one for the second-order • Results in six equations and eight unknowns. • One common version results in the following Note the third term NOTE: if the derivative is a function of x only, this reduces to Simpson’s 1/3 Rule