User:Egm6341.s11.team2.zou/HW2

=Problem 2.3: "Finding Taylor series"=

''' This problem was solved without referring to S10 homework. '''

Refer to lecture slide from [[media:nm1.s11.mtg7.djvu|mtg-7]] for the problem statement and table shown below.

Given:
Function $$\displaystyle f(x)$$ is,

Find:
A)Expand $$\displaystyle {{e}^{x}}$$ in Taylor series up to order $$\displaystyle n$$ with remainder $$\displaystyle {{R}_{n+1}}[{{e}^{x}};x]$$.

B)Find Taylor series of $$\displaystyle f(x)$$ up to order $$\displaystyle n$$ with remainder $$\displaystyle {{R}_{n+1}}[f;x]$$.

Solution:
A)Expand $$\displaystyle {{e}^{x}}$$ in Taylor series up to order $$\displaystyle n$$ with remainder $$\displaystyle {{R}_{n+1}}[{{e}^{x}};x]$$. According the Taylor series theorem ,

Function $$\displaystyle g(x)={{e}^{x}}$$ can be expanded at $$\displaystyle {{x}_{0}}=0$$ as,

where the remaider $$\displaystyle {{R}_{n+1}}[{{e}^{x}};x]$$ is

where $$\displaystyle \xi \in [0,x]$$.

B)Find Taylor series of $$\displaystyle f(x)$$ up to order $$\displaystyle n$$ with remainder $$\displaystyle {{R}_{n+1}}[f;x]$$.

From Eq(3.2) we have,

Substituting Eq(3.4) into Eq(3.1) yield,

where the the Taylor series expansion is,

and the remainder is

from which we can conclude,

which satisfy Eq(1) in Mtg7.

=Problem 2.7: "Verification of orthogonality of Legendre polynominals"=

Refer to lecture slide from [[media:nm1.s11.mtg9.djvu|mtg-9]] for the problem statement and table shown below.

Background:
Legendre polynominals are given by :

Note that they are all defined on the domain $$\displaystyle (-1,+1)$$.

The first 6 Legendre polynominals are :

While the Gramian matrix is defined as:

Find:
A) Construct Gramian Matrix $$\displaystyle \Gamma $$ with respect to Legendre polynominals from order $$\displaystyle 0$$ to $$\displaystyle 5$$ and show that it‘s diagonal.

B) Compute the determinant of $$\displaystyle \Gamma$$.

C)Discuss the orthogonality of Legendre polynominals.

Solution:
A) Construct Gramian Matrix $$\displaystyle \Gamma $$ with respect to Legendre polynominals from order $$\displaystyle 0$$ to $$\displaystyle 5$$.

When the order of Legendere polynominals is less than or equal 2, we have,

As the order goes higher we may use WolframAlpha to help us construct the Gramian matrix(click on the link to check the results):

which means the Gramian matrix $$\displaystyle \Gamma $$ is,

which is apparently a diagonal matrix

B) Compute the determinant of $$\displaystyle \Gamma $$.

Since $$\displaystyle \Gamma $$ is a diagonal matrix, the determinant of it is

C)Discuss the orthogonality of Legendre polynominals.

From Eq(7.2) we can conclude that within order of 5 the Legendre polynominals are orthogonal to each other.

In fact Legendre polynominals have such orthogonality property :

where $$\displaystyle {{\delta }_{mn}}$$ is the Kronecker delta.

The proof of the above statement is given by A.K.Lal et al.

=Problem 2.14: "Expand the specific interpolation interval to arbitrary domains"=

Refer to lecture slide from [[media:nm1.s11.mtg11.djvu|mtg-11]] for the problem statement and table shown below.

Given:
Tables have been established for Lagrangian Interpolation method to acquire weights on different points for integrations like:

Find:
Expand the applicable domain from $$\displaystyle [-1,+1]$$ to an arbitrary one, i.e. make the tables established applicable to integrations like:

Solution:
Integration such as Eq $$\displaystyle \int_{a}^{b}{f(x)dx}$$ can be transformed to $$\displaystyle \int_{-1}^{1}{f(x)dx}$$ by transformation of variables $$\displaystyle x=x(t)$$such that:

where $$\displaystyle \bar{f}(t)=f\left( x(t) \right){x}'(t)$$. Note that this transformation requires that $$\displaystyle x=x(t)$$ is a continuously differentiable function and that:

For simplicity, let‘s assume $$\displaystyle x(t)$$ to be a linear function such that:

Combine Eqs(14.1) and Eq(14.2) we have,

And,

Then we can transform the integration $$\displaystyle I(f)=\int_{a}^{b}{f(x)dx}$$ into,

which meets the prerequisite of tables for Lagrangian Interpolation in Mtg11.

=References=