Solving ODEsUC Berkeley

Fall 2004, E77http://jagger.me.berkeley.edu/~pack/e77

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Numerical Soln of Differential Equations

Recall the population problem you worked out in HW–Independent variable: time.–Dependent variables: eggs, larvae, adults–Governing rule for their “one-step” evolution

An initial condition (values for e0, v0 and a0) and a for loop leads to the solution (over time).

Here, the variables evolve discretely in time. This is called a discrete-time equation. What if the variables evolve continuously in time?

Differential Equations

Applying the “conservation laws” that make up a scientific field often leads to differential equations. Examples

–“rate of change of linear momentum of a mass particle” equals the forces applied to the particle

–“rate of change of internal energy of a system” equals the net flow of energy into the system

–“rate-of-change of concentration of a molecule” equals a sum of terms proportional to the products of concentrations of other molecules

Rate-of-change means derivative. We’ll use to denote derivative of y with respect to its argument.

If the argument is time, convention is to use to denote derivative.

Ordinary Differential Equations

Initially, use the following notation–Independent variable, x–Dependent variable, y

• y is a function of x, y(x) (notate the derivative as y´(x))

–Known function, f

Given information–Initial condition

• The value of y(0) is known, say y0

–The “governing equation,” which gives y´ in terms of y

Goal: find “solution”, namely a function y which satisfies–The initial condition, and–The governing equation for all x

Fact: For most right-hand-sides, there exists a unique soln

The rate-of-change of y at x depends only on the value of x, and the value of y(x)

Example: Ordinary Differential Equations

Consider a simple example

Exact Solution is easy:

Consider a less-simple example

Exact solution is (integrating factor)

Ordinary Differential Equations

An “initial value” problem–the initial value of y as known/given–the goal is to compute y for “later” (more positive) values of x.

A “final value” problem–the final value of y as known/given–the goal is to compute y for “earlier” (less positive) values of x.

A “boundary value” problem–the initial value of some components of y is known/given–the final value of some components of y is known/given–the goal is to compute y for the intermediate values of x.

Initial value and final value problems are basically the same. Boundary value problems are different.

Ordinary Differential Equations

Initially, use the following notation–Independent variable, x

–Many dependent variables, y1, y2, …, yn

–Known function, f

Given information–Initial condition

• The value of y(0) is known, say y1,0, y2,0, …, yn,0

–The governing equation, which gives y´ in terms of y

The rate-of-change of y at x depends only on the value of x, and all of the values of y(x)

Coupled, 1st-order Ordinary DiffEq

Use the following terminology–Coupled, since there are n dependent variables, and their

values influence the rate-of-change of each other–1st order, since only 1 derivative appears–Ordinary, since there is only one independent variable

Write concisely

Satellite orbiting Earth

The motion of a point-mass, m, orbiting a stationary massive object M is governed by

Accelerations in radial and tangential directions

Forces in radial and tangential directions

Converting to first-order form

Starting with

Define

Write each in terms of all of the

Two, 2nd-order coupled ODEs

Beam Theory (CE 130)

Consider a beam.

Under the influence of a distributed load...

The relationship between the load, w(x), and the transverse deflection v(x) is

Boundary-value problem!

w(x), distributed load, in Newtons/meter, function of location

it deflects!

Beam Theory: convert to first-order form

Governing equation is

With boundary conditions

Define

reformulated

Euler Approximation to solution

Write the system of equations

Here, f maps a scalar (x), and an n-by-1 vector (y) into a n-by-1 vector.

Consider an IVP. The value of y at x0 is given as y0. What is the solution for x>x0?

Euler Approximation to solution

At x0, the governing equation implies

Use forward finite difference approximation to the left-hand side (ie., the derivative)

Equate these, giving the approximate value for y(x0+h)

known

Euler Approximation to solution

The Euler approximation to the solution is obtained by repeating this process using

–the forward finite difference approximation to the derivative, and

– the approximate solution as the value of y at x0+h.

The forward finite difference says

But we don’t know these.

Approximate value is available via the previous step,

Use this approximate value, and continue in this fashion.

Euler Approximation to solution

Let yE denote the Euler solution. Specifically, denotes the approximate solution at the kth “step”.

Using a stepsize of h, the Euler approximate solution is evolved (as a discrete equation)

Euler psuedo-code

function [x,y] = eulersolve(fh,xspan,y0,h)

n = length(y0); % assume column vector

x = xspan(1):h:xspan(2);

M = length(x);

y = zeros(n,M);

y(:,1) = y0;

xk = xspan(1);

for k=1:M-1

yk = y(:,k);

y(:,k+1) = yk + h*feval(fh,xk,yk);

xk = xk + h;

end

y = y.’; % matlab convention(see ode23/45)

Adaptive StepSizes

In some algorithms, called adaptive, the stepsize is–adjusted automatically, –based on the magnitude of f.

In this case, the stepsize is written hk and the iteration is

More details, as usual, in Math 128A

Higher Order Methods (Runge-Kutta)

In the Euler method, the approximate solution at the next step is based on evaluating the function f at one point,

In Runge-Kutta algorithms, the approximate solution at the next step is based on evaluating the function f at several points.

More details, as usual, in Math 128A

ode23, ode45: Runge-Kutta

Syntax is

[x,y] = ode45(fh,xspan,y0)

The input arguments are–fh is a function handle to the function describing the governing

equation. Called repeatedly, by ode45, with two arguments,

feval(fh,xvalue,yvalue). Should return a column vector representing y´

–xspan, 1-by-2 row vector, [X0 Xfinal]–y0, initial condition, n-by-1 column vector

The output arguments are–x, M-by-1 vector of x-values where approx soln is computed

–y, M-by-n “solution”.

–Approximate solution at x(k)is y(k).’

Example: Ordinary Differential Equations

Simple example

First, write

function yp = cclin(x,y)

yp = -2*y;

Complex example

function yp = ncclin(x,y)

yp = (-0.75+sin(x))*y;

Then

>> [xS,yS] = ode45(@cclin,[0 5],1.5);

>> [xC,yC] = ode45(@ncclin,[0 10],1.5);