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Unformatted text preview: 12.11 Solution of PDEs by Laplace Transforms We may transform a PDE involving both space and time into a family of PDEs involving spatial variables only. This is accomplished by the Laplace transform, which for a function f : (0 , ∞ ) → R is defined by f # ( s ) ≡ L [ f ]( s ) := ∞ integraldisplay f ( t ) e − st dt, s ∈ C . (391) if the integral exists (it does for a large class of functions, for Re( s ) large enough). L is a linear transform and has an inverse, L − 1 . Using integration by parts, we find that the Laplace transform of the nth derivative of F is given by L [ f ( n ) ]( s ) = s L [ f ( n − 1) ]( s ) − f ( n − 1) (0) . (392) By induction we conclude that L [ f ( n ) ]( s ) = s n L [ f ]( s ) − s n − 1 f (0) − ··· − sf ( n − 2) (0) − f ( n − 1) (0) . (393) Remark: Because of property ( 393 ), the Laplace transform is particularly suited to initial value problems, which model physical systems that are at rest for all times t < 0 and suddenly change at t = 0. If, on the other hand, the system is timeinvariant, it is better to use the Fourier transform instead. We may use ( 393 ) to eliminate the time derivatives in a PDE. Consider the following initial value problem for the wave equation: U tt = c 2 Δ U, in Ω × (0 , ∞ ) , c > , (394) U = 0 , U t = 0 , in Ω × { } . (395) This problem is not well posed, but we will take care of the boundary condi tions later. The Laplace transform u ( x ; s ) := L [ U ( x , · )]( s ) of the unknown function U ( x ,t ) satisfies the Helmholtz equation Δ u + k 2 u = 0 , in Ω , k := is c ∈ C . (396) Any boundary condition imposed on U on ∂ Ω × (0 , ∞ ) also needs to be Laplace transformed, which yields a new boundary condition for u on ∂ Ω. Thus we end up with a boundary value problem in Ω. For simple shapes of Ω, we can solve this problem analytically, as we have seen for a rectangle in 2D (Section 12.8 squiggleright double Fourier series), a disk in 2D (Section 12.9 squiggleright FourierBessel series), as well as in the case of a spherical boundary in 3D (Section 12.10 squiggleright spherical harmonics). 69 12.12 Application: hydrogenlike atomic orbitals A hydrogenlike atom is one with a single electron (charge − e , where e ≃ 1 . 60 · 10 − 19 C denotes the elementary charge) and a nucleus of charge Ze , with the atomic number Z ∈ N . The quantum state of the electron is described by the wave function Ψ : R 3 × (0 , ∞ ) → C : the function  Ψ  2 is the probability density for the position of the electron at time t > 0. The time evolution of the wave function Ψ is described by the Schr¨odinger equation i planckover2pi1 ∂ Ψ ∂t = − planckover2pi1 2 2 m e ΔΨ + V Ψ , (397) where i 2 = − 1 and where planckover2pi1 ≃ 1 . 05 · 10 − 34 Js denotes the reduced Planck constant, m e ≃ 9 . 11 · 10 − 31...
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This note was uploaded on 11/17/2011 for the course MATH 529 taught by Professor Staff during the Spring '08 term at UNC.
 Spring '08
 Staff
 Math

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