User:Egm6322.s12.team2.steele.m2/Mtg38

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

[[media:Pea1.f11.mtg38.djvu|Page 38-0]]
Plan: $$\Rightarrow$$Sheffler polynomial sequence (PEA2) $$\Rightarrow$$Heat conduction on a sphere Transient heat equation Steady state, homogeneous, constant Conductivity: Laplace equation Spherical coordinates

[[media:Pea1.f11.mtg38.djvu|Page 38-1]]
=Sheffler Polynomial Sequence= Generalize [[media:Pea1.f11.mtg37.djvu|Eqn(1) p.37-1]] to the case in which the factor r in the exponent is a function (non-constant):

Similar to [[media:Pea1.f11.mtg37.djvu|Eqn(1) p.37-2]], rewritten in the following form.

With the variable t used in the series to distinguish with the variable x, expand the function $$r(\cdot)$$ into series

Using (2)-(3) in (1) leads to

[[media:Pea1.f11.mtg38.djvu|Page 38-2]]
The coefficients $$\{s_0(x), s_1(x), s_2(x), \ldots\}$$ in Eqn(4) p.38-1, which are polynomials, collectively form what is called the Sheffler sequence having $$e^{x \cdot r(t)} \varphi(t)$$ as generating function. Sheffler sequence plays an important role in orthogonal polynomials with important applications in many areas: Probability, physics, biology, mathematical economy. Roman 1984, The umbral calculus, p.2. Koekoek et al. 2010, Hypergeometric orthogonal polynomials …, p.265. More in PEA2 S12…

[[media:Pea1.f11.mtg38.djvu|Page 38-3]]
=Heat Conduction on a Sphere= Transient heat equation

See also p.19-8

For a homogeneous isotropic material:

[[media:Pea1.f11.mtg38.djvu|Page 38-4]]
Consider the following particular case: Homogeneous, isotropic, constant conductivity

Laplace operator acting on u(x)

M space dimension (1, 2, or 3)

[[media:Pea1.f11.mtg38.djvu|Page 38-5]]
Consider an infinitesimal segment $$\mathbf ds$$ (vector) in 3-D space: See graph on this page with vectors in bold below from the equation.

[[media:Pea1.f11.mtg38.djvu|Page 38-6]]
Kronecker delta:

General curvilinear coordinates: $$(\xi_1, \xi_2, \xi_3)$$ See p4-4 Spherical coordinates Astronomy convention

Math/physics conv.:

See image on this graph for spherical coordinates.

[[media:Pea1.f11.mtg38.djvu|Page 38-7]]
$$\theta \in [-\frac{\pi}{2}, + \frac{\pi}{2}]$$ Latitude (astronomy, geography) (1) angle measured from the equator $$\bar \theta \in [0, \pi]$$	 angle measured from the North Pole (2)