User:Eml5526.s11.team2.brenner/hwk2

=Problem 2.4: =

Solution
=Problem 2.6:=

Solution
=Problem 2.8: =

Given
Equations provided below can be found in Eqn (1) & (2) mtg. 10-4

Find
Show that 2.8.1 and 2.8.2 are equivalent.

Ensure that $$\displaystyle {u^h}(x) $$ satisfies the essential and natural boundary conditions.

where 2.8.4 is the essential boundary condition and 2.8.5 is the natural boundary condition

(Method I)
Taking the derivative with respect to x of both sides of (2.8.3) provides the following

Following dividing (2.8.6) through by $$\displaystyle P({u^h}(x)) $$ leads to

The weighted function $$\displaystyle {w^h}(x) $$ per Eqn (1) mtg. 10-3 takes the following form

Substituing (2.8.8) into (2.8.6) provides the following

Selecting a linear polynomial basis $$ \displaystyle \{ 1,x,{x^2}\} $$ on which to expand (2.8.9) and making use of the fact that (2.8.1) & (2.8.2) are equal to zero

Recall the requirement for the weighting function to be zero for $$\displaystyle w^h(x=1) = 0 $$ implies that

Making use of the Bubnov-Galerkin Method and we can construct a trial solution $$\displaystyle u^h(x) $$ of the form

Taking into account (2.8.8) and the requirement that the weighting function satisfy $$\displaystyle w^h(x=1) = 0 $$ reduces (2.8.12) to the following

Thereby satisfying (2.8.4) and the requirement on the essential boundary condition.

(Method B)
Begin by demonstrating that (8.2.1) is equivalent to (8.2.2). Assume the following finite approximation for the weighting function $$\displaystyle {w^h}(x) $$ which can be found in   Eqn (1) mtg. 10-3

Substituting (2.8.6) into (2.8.1) leads to the following expression

Expanding the right hand side of the summation found in (2.8.7) leads to the following form

(2.8.8) can be represented as a sum of integrals on $$\displaystyle \Omega $$

Next unexpand the summation of integrals in (2.8.9) and group the coefficients of $$\displaystyle \sum\limits_{i = 1}^n $$

Enforcing the elimination of the trivial solution requires that $$\displaystyle c_i \ne 0 $$. This leads to the following which satisfies (2.8.2).

Recall the requirement for the weighting function to be zero, which is as follows

Making use of the Bubnov-Galerkin Method we can construct a trial solution $$\displaystyle u^h(x) $$ of the form

Taking into account (2.8.13) and the requirement that the weighting function satisfy $$\displaystyle w^h(x=1) = 0 $$ (2.8.13) reduces

Thereby satisfying (2.8.4) which is the requirement of the essential boundary condition.

Continue by demonstrating that (8.2.2) is equivalent to (8.2.1).

Multiple (8.2.2) by $$\displaystyle \sum\limits_{i = 1}^n $$ which provides the following

Reorganize (8.2.13) as follows

Per (2.8.6) substitute (2.8.16 ) into (2.8.15) which results in the following

Expand (2.8.17) to a difference of integrals $$\displaystyle \int_{\Omega} $$

Using (2.8.8) and (2.8.21) $$\displaystyle g(x) $$ can also be represented by

Substituting (2.8.19) into (2.8.18) provides the following

Assume the following finite approximation for the trial solution $$\displaystyle {w^h}(x) $$ which can be found in   Eqn (2) mtg. 10-3

Which modifies (2.8.19) into the following

substituting (2.8.6) into (2.8.22) satifies the proof.