User:Egm6322.s14.team2/report1

=Report 1 PEA 2 Spring 2014 Team 2=

Given Information for R1.1
The following polynomial is given as a solution to an associated Laguerre differential equation:

The associated Laguerre differential equation is given by:

$$P_N$$ is a solution to the associated Laguerre differential equation where $$\alpha=-1$$.

Assignment for R1.1
(a)   Verify that the polynomial in ($$) is a Laguerre polynomial except for a minus sign.

(b)   Find the expressions of $$P_N(u)$$ for various values of N, and compare them to a table of Laguerre polynomials.

(c)  Plot the associated Laguerre polynomials for $$ N=3$$, and $$\alpha=0,-1,-2,-3,-4$$.

Solution to R1.1

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On our honor, we did this assignment on our own, without looking at the solutions in previous semesters or other online solutions.
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Part a: Comparing the Laguerre Polynomial
The falling factorial term in ($$) can be rewritten with a Pochhammer symbol.

Where,

The Pochhammer symbol can be related to a binomial coefficient by the equation:

Plugging ($$) into ($$) results in:

The binomial coefficient has the following closed form for integer arguments:

{{NumBlk|:|$$\binom{N}{n}=\left\{\begin{matrix} \frac{N!}{n!\,(N-n)!} & 0\leq n \leq N\\ 0& \text {otherwise} \end{matrix}\right.$$ |$$|}}

Substituting the closed form in ($$) into ($$) gives:

The associated Laguerre polynomial has the closed form:

For the the case where $$\alpha=-1$$, ($$) has the form:

Using the expression ($$) for the binomial term in ($$) yields:

{{NumBlk|:|$$\binom{N-1}{N-n}=\left\{\begin{matrix}

\frac{(N-1)!}{(N-n)!\,(n-1)!} & 0\leq N-n \leq N-1\\

0& \text{otherwise}

\end{matrix}\right.$$ |$$|}}

Plugging ($$) into ($$) gives:

It should be noted that the summation index in ($$) changes from $$n=0$$ to $$n=1$$. This is because when $$n=0$$ the binomial coefficient is zero and cannot be represented by the factorial expression. This term must be brought outside of the sum, but since the term is zero, all that results is an index change.

The first term of ($$) is brought outside the summation.

Comparing ($$) to ($$), it is seen that the following relation holds:

Part c: Graph of the Generalized Laguerre Polynomial for Several Values of α
When $$N=3$$, the generalized Laguerre polynomial can be found by the following equation :



When $$ \alpha = 0$$ the generalized Laguerre polynomial is identically equal to the Laguerre polynomial which has 3 real distinct roots. For $$ \alpha = -1$$, the polynomial also has 3 real distinct roots, but they are closer to the origin, with one root located at the origin. The polynomial when $$ \alpha = -2$$ has three real roots with a double root at the origin. The polynomial when $$ \alpha = -3$$ has 3 real repeated roots at the origin, and the polynomial when $$ \alpha = -4$$ has a single real root and 2 complex roots.

Given Information for R1.2
($$) represents the non-dimensional equilibrium position for particle i.

Assignment for R1.2
Problem is stated in lecture notes 66a

(a) Use ($$) to determine the behavior of $$ S(u) $$ as $$ u \to u_i $$.

(b) Also using ($$), prove ($$) as $$ u \to u_i $$.

Solution to R1.2

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On our honor, we did this assignment on our own, without looking at the solutions in previous semesters or other online solutions.
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Part a: Characterize the Behavior of S(u)
($$) is the definition of S(u), a summation of N terms.

Substituting ($$)for the N-1 terms (excluding j=i) in ($$) and evaluating the behavior of the function as u approaches ui results in ($$).

Part b: Characterize the Behavior of P(u)
P(u) is defined in equation ($$) and the derivative is displayed in ($$).

Substituting ($$) and ($$) into ($$), ($$) was established.

Using the chain rule the following equation can be defined using ($$).

($$) is the derivative of S as $$ u \to u_i $$.

Substituting ($$) and ($$) into ($$), multiplying the last term by (u-ui)/(u-ui), and evaluating the behavior of the system as $$ u \to u_i $$ yields P''(u)/P'(u) approaches 1 as $$ u \to u_i $$.

Given Information for R1.3
($$) was given.

Assignment for R1.3
(a) Show that ($$) holds. (b) Show that c=-N. (c) Deduce that P(u) satisfies the associated Laguerre differential equation.

Solution to R1.3

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On our honor, we did this assignment on our own, without looking at the solutions in previous semesters or other online solutions.
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Part a: Establish the Relationship Between P(u) and Its Derivatives
The definition of PN(u) is given in ($$).

Through a binomial expansion ($$) can be written as power series with N terms beginning with n=1 since u1 is fixed at zero. The first and second derivatives of the power series representation of PN are presented in ($$) and ($$).

Substituting ($$) through ($$) into ($$) in such a way that the independent variable u is always raised to the power of n and combining the summations:

Which leads to the recurrence relationship:

After evaluating the first few constants in ($$), the closed form expression for the cn:

Note: cN is equal to 1 and c1 is the constant:

Factoring a (-1) out of the (n-1) factors in the product and then substituting the coefficients back into ($$):

Part b: Determining the Order of the Laguerre Polynomial
Substituting ($$) into ($$) where n=N, P′′(u)=0, simplifies ($$) to ($$).

Therefore, c=-N.

Part c: Satisfying the Laguerre Polynomial
Also, comparing ($$) and ($$) to the associated Laguerre differential equation and the associated Laguerre polynomials, it is clear that P(u) satisfies the Laguerre differential equation.

Given Information for R1.4
The following finite power series,

Is given as a solution to the associated Laguerre differential equation where $$\alpha=-1$$.

Assignment for R1.4
Verify that the solution given by ($$) has the following coefficients:

and,

Solution for R1.4

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On our honor, we did this assignment on our own, without looking at the solutions in previous semesters or other online solutions.
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Taking the derivatives of ($$) gives the following:

Note: The summation index for ($$) must start at 2; we are assuming a polynomial solution. As such, derivatives must also remain polynomials (powers of $$u$$ must be non-negative). The n=1 term of the second derivative is zero because of the factor n-1; therefore, the summation is evaluated from n=2.

Plugging ($$), ($$) and ($$) into ($$) gives:

A change of index is needed to equate the power of $$u$$ in the first sum.

Since the summation indices are all identical, ($$) can be written under a single summation.

In order for ($$) to hold true, the term inside the square brackets must sum to zero giving the recurrence relation:

With a shift of index ($$) can be written as:

Table 4.1 shows how the coefficients can be expressed as a function of a1. Based on ($$), and Laguerre polynomials in general, the lowest order term has a coefficient of 1. Any arbitrary constant can be multiplied by the Laguerre polynomials to obtain a solution to the Laguerre differential equation.

Using the results of Table 4.1 and setting a1 equal to 1, the solution to ($$) can be written as the following series:

=Contributing Team Members=
 * Cameron Stewart Solved problems 1and 4
 * Elizabeth Bartlett Solved problem 2 and 3
 * Kevin Frost Reviewed and formatted
 * Christopher Neal Reviewed and formatted

For this report all members solved all of the problems individually. The transcription of the problems into the Wikiversity report was handled by only two members. Subsequent reports will have contributions from each team member.

=References=