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## ORDINARY DIFFERENTIAL EQUATIONS

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**ORDINARY DIFFERENTIAL EQUATIONS**ENGR 351 Numerical Methods for Engineers Southern Illinois University Carbondale College of Engineering Dr. L.R. Chevalier Dr. B.A. DeVantier**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.**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)**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.**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.**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**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)**Ordinary Differential Equations**• A second order differential equation would have the form: } does not necessarily have to include all of these variables**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:**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).**Ordinary Differential Equations**• An ordinary differential equation of order n requires that n conditions be specified. • Boundary conditions • Initial conditions**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**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.**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.**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**Scope of Lectures on ODE**• Boundary value problems • Case studies**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.**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**Review of Analytical Solution**At this point lets consider initial conditions. y(0)=1 and y(0)=2**What we see are different**values of C for the two different initial conditions. The resulting equations are:**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**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**Example**For the initial condition y(1)=1, determine y for h = 0.1 analytically and using Euler’s method given:**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**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**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.**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**y**Use this “average” slope to predict yi+1 xixi+1 {**y**y xixi+1 x xi xi+1**y**x xi xi+1**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**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**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.**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**, Incremental Functioncan be interpreted as a representative**slope over the interval**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**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)**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**By setting these two equations equal to each other and**recalling: we derive three equations to evaluate the four unknown constants**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?**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.**Consider the following:**Case 1: a2 = 1/2 Case 2: a2 = 1 These two methods have been previously studied. What are they?**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.**Case 2: a2 = 1**This is the Improved Polygon Method.**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**Example**Evaluate the following ODE using Heun’s Methods**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