ODE / IV Problems - Cornell University

ODE / IV ProblemsCS3220 - Summer 2008

Jonathan Kaldor

Differential Equations

• So far, we have looked at problems involving one or more variables, with either linear or nonlinear relationships between them

• A Differential Equation is simply an equation (or system of equations) that involves both variables and their derivatives


• For instance, a differential equation might be:dx/dt = sin(t x)

• This says that the change of the variable x as the variable t changes is according to the equation sin(t x).

• In this case, we have two variables. We call t the independent variable (it is the variable we can change) and x the dependent variable


• As you might imagine, including derivatives of variables as well as the variables themselves complicates things further

• The solution methods we’ve looked at so far are not applicable in this case

• Special case: when the derivative of x depends only on t (i.e. dx/dt = f(t)), in which case we can use quadrature)

Applications

• Differential Equations appear everywhere

• Wave propagation

• Heat dissipation

• Mechanics

Applications

• One of the most familiar rules of mechanics is in fact a differential equation: Newton’s Second Law of Motion:F = ma (Force = Mass x Acceleration)

• If the position of an object is denoted by x, then its velocity (the change in position over time) is dx/dt. The acceleration is then the change in velocity over time, or d2x/dt2 (the second derivative of x w.r.t t)

Applications

• The second law of motion is then:F = m d2x / dt2

• Oftentimes, the force on an object depends only on the current position, velocity, and time, giving us:F(x, dx/dt, t) = m d2x / dt2

• This is a differential equation relating the acceleration of an object to a function of its current position, velocity, and time

Applications

• Some examples of force functions:

• Planetary motion (force is due to gravity, which is dependent on positions alone)

• Springs (again, force is due only to position of object / compression of spring)

• Air resistance (complex function, but the faster something is moving the more resistance it experiences - dependent on velocity and position [higher up == less air])

Classifications of Differential Equations

• There are two main classifications of DE problems: Ordinary Differential Equations (ODEs) and Partial Differential Equations (PDEs)

• An ODE has only one independent variable, and has derivatives with respect to that independent variable


• So, the example we discussed is an ODE: there is a single independent variable (t) and one or more dependent variables (x) with derivatives taken with respect to the independent variable only

• A PDE consists of several independent variables, with derivatives taken with respect to each other (so you may have x, y, dx/dy, dy/dx, etc...)

Classification of Differential Equations

• Examples of ODEs:dx/dt = sin(t x)d2x/dt2 = - x / ‖x‖3

• Examples of PDEs:du/dt = d2u/dx2 (heat dissipation)d2u/dt2 = d2u/dx2 (wave propagation)


• We will focus solely on ODEs in this class

• I will use x’ or x to denote the derivative of a variable with respect to the independent variable (i.e. dx/dt)


• We can also talk about systems of differential equations, much like how we had systems of linear/nonlinear equations.

• So, for instance, our Newtonian Mechanics ODE can be expressed in the 2D plane asfx(x, y, dx/dt, dy/dt) = m d2x/dt2

fy(x, y, dx/dt, dy/dt) = m d2y/dt2


• As usual, it is much easier to represent this using vector notation. We can represent systems of ODEs as

f(x, x, ... , t) = 0

where x is now an n vector, and f represents a set of n equations. This is known as the implicit form of an ODE

Explicit Form

• We would also like to convert our problem into explicit form: on the right hand side is the highest derivative used, and the left hand side is a function of the variables and all lower derivatives

• In general, for an order-n ODE we then have f(x, x’, ... x(n-1),t) = x(n)

• For some ODEs, this is not possible (need root finding strategies instead)


• Given an ODE, the order of the problem is the highest derivative involved in the problem.

• For instance, Newton’s Second Law of Motion is an order-two ODE (because it involves the second derivative of position)F = m d2x / dt2

Converting to Order 1

• The methods we will look at assume that we have a first order ODE. This appears to be a rather limiting requirement

• Fortunately, any order-2 or higher method can be converted to an equivalent order-1 ODE


• Suppose we have an order-2 ODE in one variable (so we have some d2x/dt2 term). Create a new variable y, and add the new equation y = dx/dtIt then follows that dy/dt = d2x/dt2, so we havent changed the problem at all. However, if we replace all occurrences of d2x/dt2 with dy/dt, then we have reduced the order to 1


• Example: F = ma

Classifications of ODEs

• One final classification: in order to solve ODEs, we need to integrate (or approximate the integral).

• Much like indefinite integrals, we end up with a family of possible solutions - in order to end up with a single unique solution, we also need to specify additional constraints


• Two types of additional constraints are initial values and boundary values

• Initial values specify initial values for each of the dependent variables. Boundary values specify the behavior of the problem on the boundaries (more common in PDEs)

• We will use initial value conditions


• For instance, in our F=ma ODE, there are a family of possible solutions. In order to narrow it down to one, we need to specify an initial position and initial velocity

Initial Value Problem

• Given all of this, we can now express our ODE in standard form:u = f(u,t)u0 = [initial conditions]

• Note: this gives us a starting value, and a function that gives us the derivative of the function at u,t

Integrating ODEs

• For simplicity, lets assume our ODE is a single equation in one variable; i.e. we havex’ = f(x,t)x0 = [initial conditions]

• We would like to find the function x(t) that represents the value of x at each time t. Obviously, we have x(0) = x0

Integrating ODEs

t

x

x0

f(x0, t0)

Integrating ODEs

• We have the function f(x, t) which tells us how x changes as t changes (the slope of x relative to t). Given x(0) = x0, we’d like to find x1 = x(h), where h > 0 is the timestep

• Easiest approach: use the derivative at x(0) and assume the function is linear. Thus, x(h) = x0 + hf(x0, t0)

Integrating ODEs

t

x

x0

f(x0, t0)

x1

h

Integrating ODEs

t

x

x0

f(x0, t0)

x1

h

f(x1, t1)

h

x2

Integrating ODEs

• This is known as Forward Euler:

for k = 1, 2, ... xk = xk-1 + h f(xk-1, tk-1) tk = tk-1 + hend

Stability and Accuracy

• This is obviously an approximation to the actual ODE, controlled by the choice of stepsize h

• Leads to two questions

• Stability of integration

• Accuracy of result

Stability

• We treat the derivative as effectively constant for the length of the stepsize

• What happens if the derivative changes dramatically over the length of the stepsize?

Stability

• An example: lets take the Newtonian Mechanics example and include a spring force

• For a single particle, a spring force looks like F = -(||x - c|| - rest) (x - c)/||x - c|| where x is the particle position, c is the center of the spring, and rest is the rest length

Stability

• If our stepsize is too large, the force can change dramatically between xi and xi+1, but we aren’t aware of it -- we only “see” the new force at xi+1

• If we step too far, we can end up with an even larger force pushing back towards the center, which if we step too far again...

Stability

• (Example using Springies)

Stability

• Formally, we say that there are restrictions on the choice of stepsize h in order to ensure stability of the integration

• Stability of a stepsize is problem dependent AND method-dependent (we will look at other methods later)

Stability

• We can sometimes find analytical bounds for the stepsize that ensure stability

• Analyze the integration method on a toy example

Accuracy

• Even when the integration is stable, there is some amount of error due to the stepsize h

• Ideally, we would like to have an error estimate to allow us to intelligently choose the stepsize

Accuracy

• There are two ways of looking at the accuracy: how much error is introduced when going from xi to xi+1 (local error), and how much error is introduced when going from x0 to xn (global error)

• Local error: assume that xi is exact (zero error), and see how much error is introduced when moving to xi+1

Accuracy

• Global error: how much error is there comparing xn and x(tn), where x(t) is the true solution started at the initial condition x0 = x(0)

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ODE / IV Problems - Cornell University