PDE14 - Lecture Notes on PDE's: Separation of Variables and...

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Lecture Notes on PDE’s: Separation of Variables and Orthogonality Richard H. Rand Dept. Theoretical & Applied Mechanics Cornell University Ithaca NY 14853 rhr2@cornell.edu http://audiophile.tam.cornell.edu/randdocs/ version 14 Copyright 2008 by Richard H. Rand 1
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R.Rand Lecture Notes on PDE’s 2 Contents 1 Three Problems 3 2 The Laplacian 2 in three coordinate systems 4 3 Solution to Problem “A” by Separation of Variables 5 4 Solving Problem “B” by Separation of Variables 7 5 Euler’s Differential Equation 8 6 Power Series Solutions 9 7 The Method of Frobenius 11 8 Ordinary Points and Singular Points 13 9 Solving Problem “B” by Separation of Variables, continued 17 10 Orthogonality 21 11 Sturm-Liouville Theory 24 12 Solving Problem “B” by Separation of Variables, concluded 26 13 Solving Problem “C” by Separation of Variables 27
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R.Rand Lecture Notes on PDE’s 3 1 Three Problems We will use the following three problems in steady state heat conduction to motivate our study of a variety of math methods: Problem “A”: Heat conduction in a cube 2 u = 0 for 0 <x<L, 0 <y<L, 0 <z<L (1) with the assumption that u = u ( x,z, only) (that is, no y dependence)), and with the boundary conditions: u =0 on x ,L (2) u z = 0 (3) u =1 z = L (4) Problem “B”: Heat conduction in a circular cylinder 2 u = 0 for 0 <r<a, 0 (5) with the assumption that u = u ( r, z, only) (that is, no θ dependence), and with the boundary conditions: u r = a (6) u z = 0 (7) u z = L (8) Problem “C”: Heat conduction in a sphere 2 u = 0 for 0 <ρ<a (9) with the assumption that u = u ( ρ,φ, only) (that is, no θ dependence), and with the boundary conditions: u r = a, π/ 2 φ π (10) u r = a, 0 φ<π/ 2 (11) Here φ is the colatitude and θ is the longitude.
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R.Rand Lecture Notes on PDE’s 4 2 The Laplacian 2 in three coordinate systems Rectangular coordinates 2 u = 2 u ∂x 2 + 2 u ∂y 2 + 2 u ∂z 2 (12) Circular cylindrical coordinates 2 u = 2 u ∂r 2 + 1 r ∂u + 1 r 2 2 u ∂θ 2 + 2 u 2 (13) where x = r cos θ, y = r sin θ, that is, r 2 = x 2 + y 2 (14) and where 0 θ< 2 π (15) Spherical coordinates 2 u = 1 ρ 2 ± ∂ρ ² ρ 2 ³ + 1 sin φ ∂φ ² sin φ ³ + 1 sin 2 φ 2 u 2 ´ (16) where x = ρ sin φ cos θ, y = ρ sin φ sin θ, z = ρ cos that is, ρ 2 = x 2 + y 2 + z 2 (17) and where 0 2 π, 0 φ π (18)
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R.Rand Lecture Notes on PDE’s 5 3 Solution to Problem “A” by Separation of Variables In this section we solve Problem “A” by separation of variables. This is intended as a review of work that you have studied in a previous course. We seek a solution to the PDE (1) (see eq.(12)) in the form u ( x,z )= X ( x ) Z ( z ) (19) Substitution of (19) into (12) gives: X ±± Z + XZ ±± = 0 (20) where primes represent differentiation with respect to the argument, that is, X ± means dX/dx whereas Z ± means dZ/dz . Separating variables, we obtain Z ±± Z = - X ±± X = λ (21) where the two expressions have been set equal to the constant λ because they are functions of the independent variables x and z , and the only way these can be equal is if they are both constants.
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This note was uploaded on 03/31/2008 for the course T&AM 310 taught by Professor Phoenix during the Spring '07 term at Cornell University (Engineering School).

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PDE14 - Lecture Notes on PDE's: Separation of Variables and...

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