User:Bartlete2/EGM6322Team2Hwk1

=Homework #1 PEAII Spring 2014 Team 2=

Statement of Problem #1
(a) Verify that polynomial (1) is a Laguerre polynomial except for a minus sign.

(b) Find the expressions of Pn(u) for various values of N and compare to the Laguerre polynomials.

(c) Plot the associated Laguerre polynomials for n=3 and α=0,-1,-2,-3,and -4.

Equation (1-1):


P_{N}(u) = u+\sum_{n=2}^{N} \dfrac{(-1)^{n-1}(N-1)(N-2)...(N-n+1)}{n!(n-1)!} u^{n} $$

Solution #1
(a) The polynomial is a Laguerre polynomial if is is a finite polynomial solution to the generalized Laguerre differential equation, equation 2.

Equation (1-2):


uP''(u)+(\alpha+1-u)P'(u)+NP(u)=0 $$

The first and second derivatives of the polynomial are given in equations 3 and 4.

Equation (1-3):


P'_{N}(u) = 1+\sum_{n=2}^{N} \dfrac{n(-1)^{n-1}(N-1)(N-2)...(N-n+1)}{n!(n-1)!} u^{n-1} $$

Equation (1-4):


P''_{N}(u) = \sum_{n=2}^{N} \dfrac{n(n-1)(-1)^{n-1}(N-1)(N-2)...(N-n+1)}{n!(n-1)!} u^{n-2} $$

After substituting the polynomial and its first and second derivatives into the Laguerre differential equation, changing the indices so that the independent variable u in the summation is raised to the same power, n, and collecting terms within the common range from n=2,3,...,N-1, the following equation was derived.

Equation (1-5):


(\alpha+1)+(\alpha+1)u-N(\alpha+1)u-(\alpha+1)\sum_{n=2}^{N-1} \dfrac{(-1)^{n-1}(N-1)(N-2)...(N-n)}{(n!)^{2}} u^{n} $$

Letting \alpha equal -1, the equation does reduce to zero. Since the term n=N and all the subsequent terms contain the factor N-N=0,the polynomial is finite. Therefore, the polynomial in equation 1, is a finite series solution to the Laguerre differential equation.

However, it could still be a multiple of the Laguerre polynomial. Using the Rodriguez function, Equation 1-6, for a second order polynomial with α=-1, the Laguerre polynomial in Equation 1-7 was generated.

Equation (1-6):


L_{n}^{\alpha}(u)=\dfrac{u^{-\alpha}exp(u)}{n!}\dfrac{d^{n}}{du^{n}}(exp(-u)u^{n+\alpha}) $$

Equation (1-7):


L_{2}^{-1}(u)=\dfrac{1}{2}(u^{2}-2u) $$

After comparing Equation 1-7 to the polynomial solution for N=2 in the polynomial comparison table below, it was determined that P(u)=-L-1(u).

Table: Polynomial Comparison
(c) Using the Rodriguez function, Equation 1-6, the associated Laguerre polynomials of the third order for α=0,-1,-2,-3, and -4 are shown in the graph.

Statement of Problem #2
(a) Use Equation 2-1 to determine the behavior of S(u) as u approaches ui.

Equation 2-1:


\sum_{j=1,j\neq{i}}^{N-1} \dfrac{(-1)^{n-1}(N-1)(N-2)...(N-n)}{(n!)^{2}} u^{n} $$

(b) Also using Equation 2-1, prove equation 2-2 as u approaches ui.

Equation 2-2:


\dfrac{P''(u)}{P'(u)}\to1 $$

Solution of Problem #2
(a) P(u) is defined in equation 2-3 and the derivative is displayed in Equation 2-4.

Equation 2-3:


P(U)=\prod_{j=1}^{N}(u-u_{j}) $$

Equation 2-4:


P'(U)=\sum_{i=1}^{N}\prod_{j=1,j\neq{i}}^{N}(u-u_{j}) $$

Equation 2-5 is the definition of S(u).

Equation 2-5:


S(U)=\dfrac{P'(u)}{P(u)} $$

Substituting equations 2-3 and 2-4 into equation 5 yields a summation for N terms. Substituting Equation 1 for the N-1 terms (excluding j=i) and evaluating the behavior of the function as u approaches ui results in equation 2-6.

Equation 2-6:


S(U)\sim{\dfrac{1}{u-u_{i}}} $$

(b) Equation 2-7 is the derivative of S as u approaches ui.

Equation 2-7:


S'(U)\sim{\dfrac{-1}{u-u_{i}}^{2}} $$

Using the chain rule the following equation can be defined using Equation 2-5, the definition of S.

Equation 2-8:


\dfrac{P''(U)}{P'(u)}=S(u)+\dfrac{S'(u)}{S(u)} $$

Substituting equation 2-6 and 2-7 into equation 2-8, multiplying the last term by (u-ui)/(u-ui), and evaluating the behavior of the system as u approaches ui yields P'&apos;(u)/P'(u) approaches 1 as u approaches ui.

Equation 2-8:


\dfrac{P''(U)}{P'(u)}\sim{1+\dfrac{1}{u-u_{i}}-\dfrac{1}{u-u_{i}}} $$

Statement of Problem #3
Show that Equation 3-1 holds where P(u) was defined in equation 1-1 and the derivatives were presented in equations 1-2 and 1-3.

Equation (3-1):


P''(u)-P'(u)=cu^{-1}P(u) $$

Solution of Problem #3
Substituting the first and second derivatives of P(u) into equation 3-1, Equation 3-2 is determined.

Equation (3-2):


\sum_{n=2}^{N}\dfrac{n(n-1)(-1)^{n-1}(N-1)(N-2)...(N-n+1)}{n!(n-1)!}u^{n-2}-1-\sum_{n=2}^{N}\dfrac{n(-1)^(n-1)(N-1)(N-2)...(N-n+1)}{n!(n-1)!}n^{u-1} $$

Changing the index so that the independent variable, u, is always raised to the n-1 power and then factoring the n(n-1) out of the factorial in the first summation and then combining the summations within the common range from n=2,3,...,N, leads to Equation 3-3.

Equation (3-3):


1-N-1-\sum_{n=2}^{N}(N-n+n)\dfrac{(-1)^{n-1}(N-1)(N-2)...(N-n+1)}{n!(n-1)!}u^{n-1} $$

After factoring the constant N and u -1 from all of the terms, Equation 3-4 is defined.

Equation (3-4):


-Nu^{-1}(u+\sum_{n=2}^{N}\dfrac{(-1)^{n-1}(N-1)(N-2)...(N-n+1)}{n!(n-1)!}u^{n}) $$

Finally, the definition of P(u) was substituted back into Equation 3-4 to prove Equation 3-5.

Equation (3-5):


-Nu^{-1}P(u)=cu^{-1}P(u) $$

Statement of Problem #4
(a) Use the Frobenius Method to find the power series solution for the differential equation 4-1 for a fixed N.

Equation 4-1:


uP''(u)-uP'(u)+NP(u)=0 $$

(b) Identify a finite series solution and an infinite series solution, if they exist. (c) Is it possible to find a closed form solutioin for the infininte series solution?

Solution of Problem #4
(a) The Method of Frobenius applies to power series solutions about regular singular points. In equations 4-2 and 4-3, the point u0=0 is proved to be a regular singular point because the limits exist.

Equation 4-2:


lim_{u\to{0}}u(-1)=0 $$

Equation 4-3:


lim_{u\to{0}}u^{2}(\dfrac{N}{u})=0 $$

In the Method of Frobenius, the power series solution where uo=0 (Equation 4-4) and its first and second derivatives (Equation 4-5 and 4-6, respectively) are substituted into the differential equation (Equation 4-1).

Equation 4-4:


P(u)=\sum_{n=0}^{\infty}c_{n}(u)^{n+r} $$

Equation 4-5:


P'(u)=\sum_{n=0}^{\infty}(n+r)c_{n}(u)^{n+r-1} $$

Equation 4-6:


P''(u)=\sum_{n=0}^{\infty}(n+r)(n+r-1)c_{n}(u)^{n+r-2} $$

Equation 4-7:


\sum_{n=0}^{\infty}(n+r)(n+r-1)c_{n}(u)^{n+r-1}-\sum_{n=0}^{\infty}(n+r)c_{n}(u)^{n+r}+N\sum_{n=0}^{\infty}c_{n}(u)^{n+r}=0 $$

After changing the index of the first term so that the independent variable, u, is always raised to the power (n+r) and combining the summation within the common range from n=0,1,...,infinity, Equation 4-7 becomes Equation 4-8.

Equation 4-8:


c_{0}(r)(r-1)u^{r-1}+\sum_{n=0}^{\infty}((n+r+1)(n+r)c_{n+1}+(N-r-n)c_{n})(u)^{n+r}=0 $$

Examination of the first term of Equation 4-8, with c0 not equal to zero by definition and u as the independent variable, leads to the indicial equation (Equation 4-9), with the two roots, 0 and 1.

Equation 4-9:


r(r-1)+0r+0=0 $$

The summation in Equation 8 leads to the recurrence formula, Equation 10, where the constants (cn) are expressed as a function of the lower order constant (cn-1).

Equation 4-10:


c_{n}=\dfrac{c_{n-1}(n+r-N-1)}{(n+r)(n+r-1)} $$

Now using the first exponent of the differential equation, r=1, and rewriting the expression for the coefficients greater than one in terms of C0, leads to Equation 4-11, where (1-N)n is the Pochhammer symbol.

Equation 4-11:


c_{n}=\dfrac{c_{0}(1-N)_{n}}{(n+1)!(n)!} $$

(b) Substituting equation 4-11 into the power series expression (Equation 4-4) where r=1 gives Equation 4-12. Since each term of the summation beyond n=N contains a factor N-N=0 in the Pochhammer function, Equation 12 is a closed form finite series solution to the differential equation.



P(u)=c_{0}u+\sum_{n=1}^{N}\dfrac{c_{0}(1-N)_{n}}{(n+1)!(n)!}(u)^{n+1} $$

Given Information
The following equations is given

and

Problem Statement
1. Find the values of the integral of the function (7-1) by other means (exact or approximate), then evaluate these integrals with the Gauss-Laguerre quadrature for comparison.

2. Find and appropriate transformation to show the equality of (7-2) and (7-3). Find the value of $$ J_0(1/2) $$ in some table and use the Gauss-Laguerre quadrature to find an approximate value of the integral to compare.

Solution
First the limits of integration were changed to fit the form of the gauss-laguerre quadrature by making the substitution u=x-2 and the identity e-ueu was substituted into the equation:

According to the gauss-laguerre quadrature the integral can be approximated by the summation

where ui are the n roots of the nth order Laguerre polynomial and wi is

Using the definition of a line integral of f along c,

where

Using Cauchy's Integral Theorem

Using symmetry,

where

=Report 1 PEA 2 Spring 2014 Team 2=

Given Information
The following equation is given for PN and the Laguerre polynomials:

Problem Statement
Compare Equation 3-1 to the various finite power series expressions for the associated Laguerre polynomials in Equations 3-2 through 3-4.

Solution

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First, Equation 3-1 can be expressed in a closed form in equation 3-5 by identifying the product (N-1)(N-2)...(N-n+1) as a difference of the factorials (N-1)! and (N-n)!.

With α=-1, PN(u) in Equation 3-5 equals negative LN(-1) as shown in Equation 3-6 by factoring out a negative one and expressing the summation within the common range n=(2,∞],

where (n)N-n in Equation 3-2, the rising factorial denoted by the pochhammer symbol which is the product between all consecutive integers between n and n+N-n-1 was replaced by the difference of factorials in Equation 3-7.

Again with α=-1, Equation 3-3 can be expressed in the same form where the binomial coefficient was replaced by its closed form solution in Equation 3-8.

Finally, Equation 3-4 can be expressed similarly using the same equation for the binomial coefficient with N choose n and the product relationship in Equation 3-9.

Given Information
The following equation is given:

xi is written in spherical coordinates:

Problem Statement
Show that the infinitesimal length ds in equation 2-1 can be written in spherical coordinates, Equation 2-2.

Solution

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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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The derivatives of xi are given in equations 4-5 through 4-7.

Equation 4-8 was developed by substituting the squared derivatives in the summation of equation 4-1 and combining like terms.

Using the Pythagorean Trigonometric Identity, Equation 4-8 reduces to Equation 4-9.

Given Information
The following equation is given for P2(x), the Legendre Polynomial of degree n=2:

Problem Statement
Show that Equation 5-2 is a second solution to the Legendre differential..

Solution
The Legendre differential equation of degree n=2 is given in equation 5-3.

Using reduction of order, some function, v, times P2(x) and the first two derivatives of the product are substituted into the Legendre differential equation in equation 5-7. Furthermore, since P2(x) is a solution to the Legendre equation the coefficient of the first order term, v, reduces to zero. Since the lowest order term cancels out, the second order differential equation is reduced to a first order differential equation where w=v' and w'=v'&apos; in Equation 5-8 which can be solved through separation of variables.

Through partial fraction decomposition, the integral is

Through integration

Using properties of the natural logarithm and then inverting it using the exponential function, equation 5-9 becomes

Substituting v' back into Equation 5-10 and again using partial fraction decomposition

Completing the integration

Substituting v multiplied by the arbitrary constant 8 back into Equation 5-4 then identifying the new solution as Q2