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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 U, in (0 , ) , c > , (394) U = 0 , U t = 0 , in { } . (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 u + k 2 u = 0 , in , k := is c C . (396) Any boundary condition imposed on U on (0 , ) also needs to be Laplace transformed, which yields a new boundary condition for u on . Thus we end up with a boundary value problem in . For simple shapes of , 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 : R 3 (0 , ) C : the function   2 is the probability density for the position of the electron at time t > 0. The time evolution of the wave function is described by the Schrodinger equation i planckover2pi1 t = planckover2pi1 2 2 m e + V , (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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 Spring '08
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 Math

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