User:Egm6341.s11.team4.KTH/HW4

=Problem 4.3=

Given
To prove Simple Simpson Error Theorem. Let, where,

Objectives
Show that

Solution
From given equation, the first derivative of e(t) is below.

Differentiate once more from eq(3.1) we get eq(3.2).

Substitute 0 into the eq(3.2), we get eq(3.3).

From the given equation about G(t). Calculate the first derivatives.

Calculate the first derivatives from eq(3.4).

Substitute 0 into the eq(3.5).

Given
From the equation of SSET.

Objectives
Find the relationship between $$ \zeta $$ and $$ \xi $$.

Solution
In the proof of SSET, we used transformation of variable.

Let

Substitute 1,-1 into the eq(2.2), we get eq(5.3) and eq(5.4).

From eq(5.3) and eq(5.4), we can get eq(5.5) and eq(5.6).

From addition of eq(5.5) and eq(5.6), we get eq(5.7)

From subtraction of eq(5.5) and eq(5.6), we get eq(5.8)

Substitute eq(5.7) and eq(5.8) into eq(5.2), we get eq(5.9).