User:Egm6322.s12.team2.steele.m2/Mtg37

=EGM6321 - Principles of Engineering Analysis 1, Fall 2011=

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Plan: $$\Rightarrow$$Polynomial series as trial solution: Frobenius method Motivation Frobenius’ general polynomial series 1st homogeneous solution by Frobenius 2nd homogeneous solution by variation of parameters Example $$\Rightarrow$$Non-homogeneous Legendre equation (L2-ODE-VC) $$\Rightarrow$$Gravitational effects: GRACE satellites

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=Polynomial Series as Trial Solution: Frobenius Method= Motivation: All trial solutions used so far, including those with singularities, i.e., [[media:Pea1.f11.mtg36.djvu|Eqns(3)-(4) p.36-3]], (1) p.36-4, (2)-(3) p.36-10 [[media:Pea1.f11.mtg36.djvu|Eqns(2)-(3) p.36-10]], can be put in the following general form:

A property of the above trial solution is the common factor $$e^{rx}$$. But is (1) the most general trial solution ? Unlikely. What could be then the most general trial solution? Exponential function as a polynomial series:

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The function $$\varphi(x)$$ in (1) p.37-1 [[media:Pea1.f11.mtg37.djvu|Eqn(1) p.37-1]],can also be expanded into a polynomial series: Multiplying (1) and (4) p.37-1, as shown in (1) p.37-1, leads to another polynomial series. Frobenius considered the following general polynomial series that accommodates singularities in the trial solution that accommodates singularities in the trial solution, and that includes (3)-(4) p.36-3 [[media:Pea1.f11.mtg36.djvu|Eqns(3)-(4) p.36-3]], [[media:Pea1.f11.mtg36.djvu|Eqn(1) p.36-4]], [[media:Pea1.f11.mtg36.djvu|Eqns(2)-(3) p.36-10]],, as particular cases:

$$c, d_0, d_1, ...$$ unknown coefficients to be determined. Obtain 1st homogeneous solution in terms of one constant $$d_0$$ (to be determined using boundary condition or initial condition).

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The 2nd homogeneous solution for homogeneous L2-ODE-VC, or the complete solution in the case non-homogeneous L2-ODE-VC, can be obtained by variation of parameters. Example: The Legendre polynomials (finite series) can be obtained by the Frobenius method as the 1st homogeneous solutions of the Legendre equation [[media:Pea1.f11.mtg7.djvu|Eqn(1) p.7-1]]. $$\{P_n(x), \ n=0, 1, 2, \ldots \}$$ Legendre polynomials The corresponding 2nd homogeneous solutions of the Legendre equation can be obtained from the 1st homogeneous solutions by variation of parameters. $$\{Q_n(x), \ n=0, 1, 2, \ldots \}$$  Legendre functions Together, the pairs of homogeneous solutions are called $$\{P_n(x), Q_n(x)], \ n = 0, 1, 2, \ldots \}$$ Legendre functions

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Specifically: p.35-2 Given: $$u_1(x) = x = P_1(x)$$ [[media:Pea1.f11.mtg7.djvu|Eqn(3) p.7-1]] Found: $$u_2(x) = \frac{x}{2} \log \frac{1+x}{1-x}-1 = Q_1(x)$$						(4) p.7-1 Using variation of parameters (King 2003 p.33). R*6.8: Non-homogeneous L2-ODE-VC

For each of the above L2-ODE-VC, do

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a)	Find the 1st homogeneous solution by trial solution (see King 2003 p.28 Pb.1.1 ab) b)	Find the complete solution by variation of parameters with the following excitation:

(end Example) R6.9: Non-homogeneous Legendre equation (L2-ODE-VC)

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Homogeneous Legendre equation: [[media:Pea1.f11.mtg6.djvu|Eqn(2) p.6-5]], (1) p.7-1 [[media:Pea1.f11.mtg7.djvu|Eqn(1) p.7-1]] Given the 1st homogeneous solution being $$u_1(x) = P_1(x) = x$$							(3) p.7-1 Find the solution y(x) by variation of parameters. Note: There is a misprint in King 2003 p.34 Example: The L2-ODE-VC should read as in [[media:Pea1.f11.mtg32.djvu|Eqns(4)-(5) p.32-5]],; thus

From the Abel formula in King 2003 p.10(1.7), we have:

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Recall the irregularities of Uranus’ orbit, p.34-7. =Gravitational Effects:  GRACE Satellites= GRACE = Gravity Recovery and Climate Experiment See image. Artist rendering of GRACE twin satellites http://en.wikipedia.org/wiki/File:GRACE_artist_concept.jpg Animation: “Big Picture”  (Hulu video) Time markers on video: 34 min – 40 min http://www.hulu.com/watch/276628/big-picture

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“… the GRACE … satellites, launched in 2002, have used tiny variations in Earth’s gravity to infer changes in the masses of the Antarctic and Greenland ice sheets. Several years of GRACE results and other data now show conclusively that both ice sheets are losing mass and contributing to global sea-level rise. The physical processes … include surface melt, glacier flow, and snowfall.” Terminologies for scientists and the public. Somerville & Hassol 2011, Communicating the science of climate change, Physics Today, Oct.