Contempoay Engineeing Sciences, Vol., 08, no. 95, 4743-4749 HIKAI Ltd, www.m-hikai.com https://doi.og/0.988/ces.08.8950 A Suvey of Azimuthal Angle Eigenvalues of the Laplace Equation Luz aía ojas Duque eseach in Statistics Epidemiology GIEE) Fundación Univesitaia del Áea Andina, Peeia, Colombia Peeia, Colombia Pedo Pablo Cádenas Alzate Depatment of athematics GEDNOL Univesidad Tecnológica de Peeia Peeia, Colombia José Geado Cadona Too Depatment of athematics GIEE Univesidad Tecnológica de Peeia Peeia, Colombia Copyight c 08 Luz aía ojas Duque et al. This aticle is distibuted unde the Ceative Commons Attibution License, which pemits unesticted use, distibution, epoduction in any medium, povided the oiginal wok is popely cited. Abstact In this wok we show that the eigenvalues of the Laplacian with homogeneous Neumann B.C. cannot be positive we solve the Diichlet BVP fo the Laplace equation u = 0 in the egion between two concentic sphees = = ). Finally, we show the simplifications in the solution if Diichlet bounday data ae independent of the Azimuthal angle. Keywods: Laplace equation, Azimuthal angle, spheical hamonics
4744 Luz aía ojas Duque et al. Intoduction Let us denote by ω = ω, ω,..., ω n ) the unit nomal vecto outwad) to the bounday. Note that ω is a vecto field that is defined at each point of the bounday. That u is an eigenfunction of the Laplacian with the homogeneous Neumann condition means that thee is a numbe λ called the eigenvalue coesponding to u, such that [] u = λu, ) in ω u=0 on, whee ω u is the nomal deivative ω u = ω u x ω n u xn We equie u to be not identically zeo, since the zeo function u = 0 satisfies Eq.) fo any λ. ecall Geen s fist identity ω u = ω ω u ω u, which can be poven by applying the divegence theoem to the vecto field ω u. Now, multiplying by u integating ove we get u u = λ u. Theefoe, by Geen s identity we obtain u ω u u = λ which implies that λ u = u, ) then due to the homogeneous Neumann condition, the bounday integal tem is null. Now, si λ 0, then u must be nonzeo in ode to be an eigenfunction []. Diichlet case Fo the Diichlet case, by a simila easoning it is concluded that λ 0. Now, if thee is an eigenfunction u with λ = 0, then in u = 0 on. u = 0, u,
Laplace equation 4745 Hee, the identically-zeo function u satisfies this equation it is the only solution). Finally, thee is no nontivial function u = that satisfies u = 0, concluding that all eigenvalues ae nonzeo, i.e., λ < 0 [3].. Diichlet B.V.P fo the Laplace equation Hee, employing spheical coodinates, the equation we need to solve is [4] { u = 0, {, θ, φ) : < < } u, θ, φ) = fθ, φ), u, θ, φ) = gθ, φ). The 3-dimensional Laplacian in spheical coodinates is u = u ) u θ sin θ) θ sin θ u φφ sin θ, o u = u u u θθ cot θ u θ sin θ u φφ. Now, substituting u, θ, φ) = )Nθ, φ) into u = 0 we obtain If, we multiply by N N N N θθ cot θ N θ sin θ N φφ we get = N θθ N N θ cot θ N φφ N N sin θ = λ Hee, the equation fo the spheical hamonics, we have the solution fo n = 0,,... m = ±,..., ±n 3) with λ = n )n. Nθ, φ) = Y n,m θ, φ), whee Theefoe, the spheical hamonics ae explicitly given by Y n,m θ, φ) = sin m θ)p n m ) cos θ)ϕ m φ), ϕ m φ) = { cosmφ), m 0 sinmφ), m < 0. Hee, P n is the Legende polynomial of degee n [6]. Taking into account that λ = n )n, the equation fo is = nn ).
u = 0, in {,, ') : < < }, u,, ') = f, '), u,, ') = g, '). ecall that the 3-dimensional Laplacian in spheical coodinates is u'' u ) u sin ) sin sin Luz aı a cot Duque et al. ojas = u u u u u''. sin 7 Let us look fo a solution that can be witten as u,, ') = ), '). Substituting this y 00 into the equation u = 0, we get u= 4746 cot 00 0 '' = 0, sin multiplying by y 0 y eaanging, we have y '' 00 0 cot = =. SOLUTIONS TO SELECTED POBLES FO ASSIGNENTS 3, 4 sin A vaiation of Poblem fom Assignment 4 We 4ecognise the equation given by the second equality sign as the equation fo the spheical hamonics, so the fo Laplace each = 0,,y..u.,= m = 0, ±,..., ±n, we have the solution y value y fo y 0 nequation Statement. Solve the Diichlet bounday poblem 0 infoy each the egion between two concentic sphees of adii. What simplifications do we get = Yn,mangle, '), in the solution pocess if the Diichlet bounday data ae independent of, the') azimuthal longitude) '? with ecall that the spheical hamonics ae explicitly given by Solution. Employing spheical coodinates, the equation we need to solve is u = 0, in {,, ') : < Figue B.: Plots of the eal-valued spheical hamonic basis functions. Geen indicates positive values m m ) values., ') = f, '), ed<indicates }, negative n,m u, u,, ') = g, ').m Y, ') = sin ) Pn cos ) '), whee FigueLaplacian : eal-valued spheical hamonics basis functions. ecall that the 3-dimensional in spheical coodinates is = nn ). cosm ) fo m 0, m ') = sinm ) fo m < 0, only the fequencies of the function up to some theshold, obtaining an n th ode b-limited Pn is the Legende polynomial of degee n. u'' function u )appoximation u sin ) whee f: u= f of the oiginal sin sin cot n l = u u u u f ~!) = X ux ''.y m ~!) f lm. B.) sin l =0 m= l l Let us look fo a solution that can be witten as u,, ') = ), '). Substituting this into the equation u = 0, we getlow-fequency functions can be well appoximated using only a few bs, as the numbe of coefficients inceases, cot highe fequency signals can be appoximated moe accuately. 00 0 '' = 0, It is often convenient toefomulate sin the indexing scheme to use a single paamete multiplying by i = l l ) m. have With this convension it is easy to see that an n th ode appoximation can be eaanging, we econstucted coefficients, 0 '' using n cot = =. sin nx!) = equation y i ~!) f i. fo the spheical We ecognise the equation given by the second equality sign asf ~the i =0 hamonics, so fo each n = 0,,..., fo each m = 0, ±,..., ±n, we have the solution 00 B.), ') = Yn,m, '), a) with ). n =,,=3, 4nn fom top to bottom, b) n = 3, 4, 5, 6 fom top to bottom, m =,..., n m =given 0,... by, n fom left to ight fom left to ight ecall that the spheical hamonics ae explicitly n,m cosm ) fo m 0, Yn,m, ') = sin m ) Pn m ) cos ) m '), whee m ') = sinm )Figue fo m <. 0, Spheical hamonics Ynm Figue : eal-valued spheical hamonics Y 0,..., n. fo n =,, 3, 4 m = whee Pn is the Legende polynomial of degee n. a) n =,, 3, 4 fom top to bottom, m = 0,..., n fom left to ight b) n = 3, 4, 5, 6 fom top to bottom, m =,..., n fom left to ight Figue 3: eal-valued spheical hamonics Yn,m fo n = 3, 4, 5, 6 m = Figue. Spheical hamonics Ynm,..., n. Hee, taking ) = α gives αα ) α = nn ),
Laplace equation 4747 which has the solutions α = n α = n ). We conclude that fo each n N 0, fo each m Z, we have two independent solutions u, θ, φ) = n Y n,m θ, φ) u, θ, φ) = n) Y n,m θ, φ), so the geneal solution is given by a linea combination of all u, θ, φ) = n n=0 m= n A n,m n B ) n,m Y n n,m θ, φ) The coefficients A n,m B n,m ae to be found the bounday conditions. Then, we have, n fθ, φ) = fn,my n,m θ, φ) n=0 m= n n gθ, φ) = gn,my n,m θ, φ), n=0 m= n Finally A n,m B n,m A n,m n B n,m n = g n,m A n,m = n g n,m f m,n n B n,m = n fn,m n gm,n. n Now, when the bounday data ae independent of φ. The spheical hamonics Y n,m that do not depend on φ ae the ones with m = 0, i.e, the zonal hamonics hence the solution is simply u, θ, φ) = Y n,0 θ, φ) = Z n θ) = P n cos θ), n=0 A n n B ) n P n n cos θ).
4748 Luz aía ojas Duque et al. Theefoe, so we look fo a solution u, θ) = )ϕθ) athe than u, θ, φ) = )Nθ, φ) we get = ϕ ϕ ϕ cot θ ϕ = λ. We ecognise the equation fo ϕ as the equation leading to the Legende equation x )y xy = λy, whee yx) = ϕθ) with x = cos θ, then we have a solution ϕθ) = P n cos θ) with λ = nn ) fo each n = 0,,... [7]. 3 Conclusion In this pape we demonstate that the eigen-values of the Laplacian on with homogeneous Neumann bounday conditions cannot be positive. Also, we have poven that the eigenvalues of the Laplacian on with homogeneous Diichlet bounday condition) ae stictly negative. Theefoe, fo the Diichlet case, the Laplace equation has a unique solution with the homogeneous Diichlet bounday condition. We have solve the Diichlet bounday value poblem fo the Laplace equation u = 0 in the egion between two concentic sphees of = =. Acknowledgements. We would like to thank the efeee fo his valuable suggestions that impoved the pesentation of this pape ou gatitude to the Depatment of athematics of the Univesidad Tecnológica de Peeia Colombia), the goup GEDNOL eseach in Statistics Epidemiology GIEE). efeences [] P. Cádenas Alzate, A suvey of the implementation of numeical schemes fo linea advection equation,advances in Pue athematics, 4 04), no. 8, 467-479. https://doi.og/0.436/apm.04.4805 [] O. Agawal, Genealized Eule-Lagange equations tansvesality conditions fo FVPs in tems of the Caputo deivative, Jounal of Vibation Contol., 3 007), 7-37. https://doi.og/0.77/07754630707747
Laplace equation 4749 [3] Dinh Nho Hao, B.T. Johansson, D. Lesnic, Pham inh Hien, A vaiational method appoximations of a Cauchy poblem fo elliptic equations, J. Algoithms Comput. Technol., 4 00), 89-9. https://doi.og/0.60/748-308.4..89 [4]. Fontelos, Fundamentos atemáticos de la Ingenieía, Libeía- Editoial Dykinson, Vol. 0, 007. [5] K. Atkinson, The Numeical Solution of Integal Equations of the Second Kind, Cambidge: Cambidge Univesity Pess, 997. https://doi.og/0.07/cbo9780566340 [6] C. uñoz, Pedo Pablo Cadenas Alzate, Anibal unoz Loaiza, An iteative method fo solving two special cases of nonlinea PDEs, Contempoay Egineeing Sciences, 0 07), no., 545-553. https://doi.og/0.988/ces.07.765 [7] P. Cádenas Alzate, Jose Geado Cadona, Luz aia ojas A special case on the stability accuacy fo the D heat equation using 3-Level θ-schemes, Applied athematics, 6 05), no. 3, 476-483. https://doi.og/0.436/am.05.63045 eceived: Octobe 7, 08; Published: Novembe, 08