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Review Initial Value Problems Euler’s Method Summary
THE EULER METHOD
P.V. Johnson
School of Mathematics
Semester 1 2008
Review Initial Value Problems Euler’s Method Summary
OUTLINE
1 REVIEW
2 INITIAL VALUE PROBLEMS
The ProblemPosing a Problem
3 EULER’S METHOD
MethodErrors
4 SUMMARY
Review Initial Value Problems Euler’s Method Summary
OUTLINE
1 REVIEW
2 INITIAL VALUE PROBLEMS
The ProblemPosing a Problem
3 EULER’S METHOD
MethodErrors
4 SUMMARY
Review Initial Value Problems Euler’s Method Summary
OUTLINE
1 REVIEW
2 INITIAL VALUE PROBLEMS
The ProblemPosing a Problem
3 EULER’S METHOD
MethodErrors
4 SUMMARY
Review Initial Value Problems Euler’s Method Summary
OUTLINE
1 REVIEW
2 INITIAL VALUE PROBLEMS
The ProblemPosing a Problem
3 EULER’S METHOD
MethodErrors
4 SUMMARY
Review Initial Value Problems Euler’s Method Summary
ERRORS
Be careful when interpreting results from a computer
Errors can arise in all sorts of places
Using Taylor Series to approximate a differential
Truncation errors
Review Initial Value Problems Euler’s Method Summary
ERRORS
Be careful when interpreting results from a computer
Errors can arise in all sorts of places
Using Taylor Series to approximate a differential
Truncation errors
Review Initial Value Problems Euler’s Method Summary
EULER’S METHOD
Initial value problems for ODEs
Solving ODEs numerically
Euler’s Method
Truncation errors again
Review Initial Value Problems Euler’s Method Summary
OUTLINE
1 REVIEW
2 INITIAL VALUE PROBLEMS
The ProblemPosing a Problem
3 EULER’S METHOD
MethodErrors
4 SUMMARY
Review Initial Value Problems Euler’s Method Summary
THE INITIAL VALUE PROBLEM
Here we will look at the solution of ordinary differentialequations of the type, say
dydx
= f (x , y), a ≤ x ≤ b
subject to an initial condition
y(a) = α
Review Initial Value Problems Euler’s Method Summary
THE INITIAL VALUE PROBLEM
Here we will look at the solution of ordinary differentialequations of the type, say
dydx
= f (x , y), a ≤ x ≤ b
subject to an initial condition
y(a) = α
Review Initial Value Problems Euler’s Method Summary
EXAMPLES
CONSIDER THE PROBLEM:
dydx
= y(
1 −y4
)
, x ≥ 0
with the initial condition
y(0) = 1
What can we say about the solution?
How can we go about solving it?
Review Initial Value Problems Euler’s Method Summary
EXAMPLES
CONSIDER THE PROBLEM:
dydx
= y(
1 −y4
)
, x ≥ 0
with the initial condition
y(0) = 1
What can we say about the solution?
How can we go about solving it?
Review Initial Value Problems Euler’s Method Summary
EXTENSION TO HIGHER ORDER ODES
We normally would like to solve higher order ODEs
We can do this by rearranging them as a system of firstorder ODEs
EXAMPLE
y ′′ − 2xyy ′ + y2 = 1, y(1) = 1, y ′(1) = 2.
The equivalent first order system is:
(y1(x), y2(x))T = (y(x), y ′(x))T ,f1(x , y1, y2) = y2(x),
f2(x , y1, y2) = 1 + 2xy1(x)y2(x) − y21 (x),
and initial condition
(y1(1), y2(1))T = (1, 2)T .
Review Initial Value Problems Euler’s Method Summary
EXTENSION TO HIGHER ORDER ODES
We normally would like to solve higher order ODEs
We can do this by rearranging them as a system of firstorder ODEs
EXAMPLE
y ′′ − 2xyy ′ + y2 = 1, y(1) = 1, y ′(1) = 2.
The equivalent first order system is:
(y1(x), y2(x))T = (y(x), y ′(x))T ,f1(x , y1, y2) = y2(x),
f2(x , y1, y2) = 1 + 2xy1(x)y2(x) − y21 (x),
and initial condition
(y1(1), y2(1))T = (1, 2)T .
Review Initial Value Problems Euler’s Method Summary
EXTENSION TO HIGHER ORDER ODES
We normally would like to solve higher order ODEs
We can do this by rearranging them as a system of firstorder ODEs
EXAMPLE
y ′′ − 2xyy ′ + y2 = 1, y(1) = 1, y ′(1) = 2.
The equivalent first order system is:
(y1(x), y2(x))T = (y(x), y ′(x))T ,f1(x , y1, y2) = y2(x),
f2(x , y1, y2) = 1 + 2xy1(x)y2(x) − y21 (x),
and initial condition
(y1(1), y2(1))T = (1, 2)T .
Review Initial Value Problems Euler’s Method Summary
GENERAL SYSTEM OF FIRST ORDER ODES
The general system can be written
dYdx
= F (x , Y ), a ≤ x ≤ b,
where
Y = (y1(x), y2(x), ..., yn(x))T ,F = (f1(x , Y ), f2(x , Y ), ..., fn(x , Y ))T ,
with initial data
Y (a) = α,α = (α1, α2, ..., αn)
T .
Review Initial Value Problems Euler’s Method Summary
OUTLINE
1 REVIEW
2 INITIAL VALUE PROBLEMS
The ProblemPosing a Problem
3 EULER’S METHOD
MethodErrors
4 SUMMARY
Review Initial Value Problems Euler’s Method Summary
WELL-POSED PROBLEMS
DEFINITION:
An initial value problem is said to be well-posed if:
A unique solution, y(x), to the problem exists;
The associated perturbed problem has a unique solutionand is sufficiently close to the original solution
How do we go about showing this for a general problem?
What do we mean by a perturbed problem?
With a computer, we are always solving the perturbedproblem
First, we need to know about the Lipschitz condition
Review Initial Value Problems Euler’s Method Summary
WELL-POSED PROBLEMS
DEFINITION:
An initial value problem is said to be well-posed if:
A unique solution, y(x), to the problem exists;
The associated perturbed problem has a unique solutionand is sufficiently close to the original solution
How do we go about showing this for a general problem?
What do we mean by a perturbed problem?
With a computer, we are always solving the perturbedproblem
First, we need to know about the Lipschitz condition
Review Initial Value Problems Euler’s Method Summary
WELL-POSED PROBLEMS
DEFINITION:
An initial value problem is said to be well-posed if:
A unique solution, y(x), to the problem exists;
The associated perturbed problem has a unique solutionand is sufficiently close to the original solution
How do we go about showing this for a general problem?
What do we mean by a perturbed problem?
With a computer, we are always solving the perturbedproblem
First, we need to know about the Lipschitz condition
Review Initial Value Problems Euler’s Method Summary
THE LIPSCHITZ CONDITION
DEFINITION:
A function is said to satisfy the Lipschitz condition in thevariable y on a set D ⊂ R2 if a constant L > 0 exists such that:
|f (x , y1) − f (x , y2)| ≤ L|y1 − y2|
whenever (x , y1), (x , y2) ∈ D. The constant L is called theLipschitz constant for f .
Review Initial Value Problems Euler’s Method Summary
PERTURBED PROBLEM
THE PERTURBED PROBLEM
Let δ(x) be a pertubation to the problem so that δ(x) < ǫ for anyǫ > 0, then
dzdx
= f (x , z) + δ(x), a ≤ x ≤ b, z(a) = α + ǫ0
and the solution z(x) exists with
|z(x) − y(x)| < k(ǫ)ǫ for all a ≤ x ≤ b,
for some constant k(ǫ).
Review Initial Value Problems Euler’s Method Summary
THE THEORY
THEOREM
Suppose D = {(x , y)|a ≤ x ≤ b and −∞ < y < ∞}. If f iscontinuous and satisfies a Lipschitz condition in the variable yon the set D, then the initial-value problem
dydx
= f (x , y), a ≤ x ≤ b, y(a) = α
is well-posed.
Review Initial Value Problems Euler’s Method Summary
OUTLINE
1 REVIEW
2 INITIAL VALUE PROBLEMS
The ProblemPosing a Problem
3 EULER’S METHOD
MethodErrors
4 SUMMARY
Review Initial Value Problems Euler’s Method Summary
SOLVING AN INITIAL VALUE PROBLEM
The simplest method to solve an ODE is the Euler method
In order to solve, we must discretise the problem – make acontinuous (infinite) problem discrete (finite)
Divide up the interval [a, b] into n equally spaced intervals
bah
x x x xx0 1 2 i n... ...
Review Initial Value Problems Euler’s Method Summary
SOLVING AN INITIAL VALUE PROBLEM
Suppose y(x) is the unique solution to the ODE, and istwice differentiable
Apply a Taylor series approximation around xi then wehave
y(xi+1) = y(xi) + y ′(xi)h +12
y ′′(ξ)
where xi ≤ ξ ≤ xi+1 .
Review Initial Value Problems Euler’s Method Summary
SOLVING AN INITIAL VALUE PROBLEM
Suppose y(x) is the unique solution to the ODE, and istwice differentiable
Apply a Taylor series approximation around xi then wehave
y(xi+1) = y(xi) + f (xi , y(xi))h +12
y ′′(ξ)
where xi ≤ ξ ≤ xi+1 .
Since y is a solution to the ODE, we can replace y ′ by thefunction f (x , y)
Review Initial Value Problems Euler’s Method Summary
THE EULER METHOD
Assume wi is our approximation to y at xi , then
w0 = α
Then to find all subsequent values of w ,set the remainderto zero in the previous equation to obtain:
wi+1 = wi + hf (xi , wi), for each i = 0, 1, . . . , n − 1
Review Initial Value Problems Euler’s Method Summary
OUTLINE
1 REVIEW
2 INITIAL VALUE PROBLEMS
The ProblemPosing a Problem
3 EULER’S METHOD
MethodErrors
4 SUMMARY
Review Initial Value Problems Euler’s Method Summary
TRUNCATION ERRORS
We would like to be able to compare the errors for differentmethods
We can use the local truncation error – difference betweenequation and the approximation
For the Euler method we have:
τi+1(h) =yi+1 − (yi + hf (xi , yi))
h
=yi+1 − yi
h− f (xi , yi)
Review Initial Value Problems Euler’s Method Summary
TRUNCATION ERRORS
We would like to be able to compare the errors for differentmethods
We can use the local truncation error – difference betweenequation and the approximation
For the Euler method we have:
τi+1(h) =yi+1 − (yi + hf (xi , yi))
h
=yi+1 − yi
h− f (xi , yi)
Review Initial Value Problems Euler’s Method Summary
TRUNCATION ERRORS
We can calculate the truncation error as
τi+1(h) =h2
y ′′(ξi),
and if y ′′ is bounded by the constant M on the interval[a, b] then
|τi+1(h)| ≤h2
M.
Hence the truncation error is O(h).
A method with truncation error O(hp) is called an order pmethod
Review Initial Value Problems Euler’s Method Summary
TRUNCATION ERRORS
We can calculate the truncation error as
τi+1(h) =h2
y ′′(ξi),
and if y ′′ is bounded by the constant M on the interval[a, b] then
|τi+1(h)| ≤h2
M.
Hence the truncation error is O(h).
A method with truncation error O(hp) is called an order pmethod
Review Initial Value Problems Euler’s Method Summary
TRUNCATION ERRORS
We can calculate the truncation error as
τi+1(h) =h2
y ′′(ξi),
and if y ′′ is bounded by the constant M on the interval[a, b] then
|τi+1(h)| ≤h2
M.
Hence the truncation error is O(h).
A method with truncation error O(hp) is called an order pmethod
Review Initial Value Problems Euler’s Method Summary
SUMMARY
Need Lipschitz condition to say a problem is well PosedProblems
Euler’s Method is a first order method
wi+1 = wi + hf (xi , wi), for each i = 0, 1, . . . , n − 1
Truncation errors give a quick estimate of error bounds onthe problem
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