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# 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 [email protected] 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
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 = 0 , L (2) u = 0 on z = 0 (3) u = 1 on z = L (4) Problem “B”: Heat conduction in a circular cylinder 2 u = 0 for 0 < r < a, 0 < z < L (5) with the assumption that u = u ( r, z, only) (that is, no θ dependence), and with the boundary conditions: u = 0 on r = a (6) u = 0 on z = 0 (7) u = 1 on 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 = 0 on r = a, π/ 2 φ π (10) u = 1 on 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 ∂r + 1 r 2 2 u ∂θ 2 + 2 u ∂z 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 ρ 2 ∂u ∂ρ + 1 sin φ ∂φ ∂u ∂φ sin φ + 1 sin 2 φ 2 u ∂θ 2 (16) where x = ρ sin φ cos θ, y = ρ sin φ sin θ, z = ρ cos θ, that is, ρ 2 = x 2 + y 2 + z 2 (17) and where 0 θ < 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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PDE14 - Lecture Notes on PDE's Separation of Variables and...

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