Show that the two-step method
![{\displaystyle y_{n+1}=2y_{n-1}-y_{n}+h[{\frac {5}{2}}f(x_{n},y_{n})+{\frac {1}{2}}f(x_{n-1},y_{n-1})]\qquad (1)\!\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a9a1631e5fe622c0abef1e78022d20d049550f71)
is of order 2 but does not satisfy the root condition.
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Method
finds an approximation for
such that
.
Let
be the
th point of evaluation where
is the starting point and
is the step size.
Substituting into the second term on the right hand side of
and simplifying yields
Since
, we also take Taylor Expansion of
about
Substituting and simplifying yields,

Hence
shows that (1) is a method of order 2.
The Characteristic equation of (1) is

Giving the roots


clearly does not satisfy
Give an example to show that the method (1) need not converge when solving .
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Let
. Then
. We have the difference equation
which has general solution (use the roots)
If
, then
as
Hence,
. Therefore if
, then
.
Consider the boundary value problem

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Prove that has at most one solution
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Let
and
be solutions. Let
.
By subtracting the two equations and their conditions we have
Multiplying by test function
and integrating by parts from 0 to 1, we want to find
such that for all
Let
. Then, we have
Since
,
, and
are all
,
. Hence
.
Discretize the problem. Take a uniform partition of

Use the three point difference formula for and the simplest difference formula for the boundary condition at . Write the resulting system as a matrix vector equation where
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The three point difference formula for
is given by
Substituting into
with our difference formula we have in matrix formulation
We can eliminate the
variable by using the approximation
which implies
Using this relationship and the three difference formula, we have
Since
, we can eliminate the
variable by substituting into the n-1 equation.
Prove that the equation found in has a unique solution
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Since the matrix
is diagonally dominant, the system
has unique solution.
Transform the problem into an equivalent problem with homogeneous boundary conditions.
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Let
,a solution of the boundary value problem, be represented as the sum of solutions to two different boundary value problems i.e.
where
Suppose
. Then
and
which implies
and hence
Substituting into
, we then have
which implies
Since
, we have
Therefore an equivalent boundary value problem with hemogenous boundary conditions is given by
Using the problem's notation, we want to find
such that for all
, we have
The above comes from integrating by parts and applying the boundary conditions.
To show that
is unique, we show that the hypothesis of the Lax-Milgram Theorem are met.
Consider the approximation of by piecewise linear finite elements. Define precisely the piecewise linear finite element subspace (use the partition (3)). Show that the finite element problem has a unique solution.
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Let be a nonlinear function with zero :
, .
Consider the iteration
, .
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Prove (4) is locally convergent.
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is a fixed point iteration. By the contraction mapping theorem, if
is a contraction in some neighborhood of
then the iteration converges at least linearly.
We have to show there exists
such that
.
By the mean value theorem we have that
, that is
for some
in our neighborhood of
.
In particular,
, implying that
is a contraction and that the iterative method converges at least linearly.
Show that the convergence is at least quadratic.
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where
satisfies
when
.
Then, we obtain
.
Write the Newton iteration and compare it with (4)
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The Newton iteration looks like this:
Where B is the inverse of the Jacobian of f.
That is,
in the Newton Iteration gives (4).