Mark Boady, Pavel Grinfeld, Jeremy Johnson

Approximation of Fourier Series to Determine Laplace-Dirichlet Eigenvalues Mark Boady, Pavel Grinfeld, Jeremy Johnson

MOTIVATION EVALUATION RESULTS In 2004, Grinfeld and Strang [3] posed the following boundary problem. The CMS provides expressions for the derivatives of λ but evaluation is Evaluating more terms increases accuracy but also the time and number of What is the series in 1/N for the simple Laplace eigenvalues λN on a regular dependent on the surface. terms that need to be simplified. polygon with N sides under the Dirichlet boundary condition? The first four Z terms in the series were computed by hand in 2012 [4]. Our goal is to automate i λ1 = − C∇iu∇ udS. (7) the process and derive more terms in the series. S Z The Calculus of Moving Surfaces (CMS), an extension of tensor calculus 2 α i δC i i ∂u λ2 = (C Bα ∇ u∇iu − ∇ u∇iu − 2C∇ ∇iu to deforming manifold, provides an analytic method to describe the change S δτ ∂t caused by the deformation of a surface. The Laplace eigenvalues on the unit 2 m i − 2C N ∇iu∇ ∇mu)dS (8) circle under the Dirichlet boundary condition are known. We have applied the CMS to analyze the change in the eigenvalues between the unit circle and N- The surface deformation is encapsulated in the surface velocity, C, and the sided regular polygon[2]. It remains to evaluate these expressions and form a ∂u partial derivatives of u, ∂t . The terms only need to be evaluated at t = 0. Taylor series for the eigenvalues on the N-sided polygon. This problem is interesting in applied mathematics[5], physics[1] and pat- − cos(θ) + cos(π/N) C|t=0 = (9) We believe that all values will converge to integer multiple of the ζ function tern recognition[6]. It also presents a number of analytical challenges associ- cos(θ) iζ(j)ρk in the form N j for integers i,j, and k. ated with the singularities present in the problem. This problem has important To work with this expression we represent it as a Fourier series, implications for the analysis of numerical methods in which a smooth bound- P∞ ikNθ c0(k)e where ary is replaced by an approximated shape. k=∞ ( π2 − 3N 2 k = 0 c0(k) = (−1)k (−1)kπ2 5(−1)k (10) SURFACE DEFORMATION N 2k2 + 3N 4k4 − N 4k4 + ··· otherwise The Laplace-Dirichlet Eigenvalue problem is described by the following Next, we evaluate λ1 in terms of the Fourier coefficients. set of equations for a boundary S embedded in space Ω. 2 2 2 2π ρ 4u = −λu (1) λ1 = −2c0(0)ρ = (11) 3N 2 u|S = 0 (2) For λ2, the result includes convolutions of coefficients. 1 Z Our current Taylor series up to N 5 is u2dΩ = 1 (3) Ω √ 2 2 δu1 π2ρ2 ζ (3) ρ2 14 π4ρ2 λ2 = 2conv(c0, c0)(0)ρ − 2c1(0)ρ + 4 πconv c0, |r=1 (0) (12) λ(t = 1) = ρ2 + 2/3 + 4 + The unknowns, u and λ, can be found when S is the unit circle and Ω is de- δr N 2 N 3 45 N 4 scribed in polar coordinates. 2 2 ∂u 1 ρ −45360 ζ (5) ρ + 453600 ζ (5) Where u1(k), is the k-th coefficient of the Fourier series for ∂t . The ex- + (17) J (ρr) 22680 N 5 u(r, θ) = √ 0 (4) pressions for ui are derived from the CMS and will also include convolutions πJ1(ρ) of the ci coefficients. 2 CHALLENGES λ = ρ (5) ( c (0)(ρrJ (ρr)−J (ρr)) 0 √1 0 k = 0 πJ1(ρ) Although the results are simple, the expression swells rapidly during evalu- u1(k) = c (k)ρJ (ρr) (13) In this expression, Jm is the m-th Bessel Function and ρ is the n-th root of J0. 0 kN√ ation. The highest we can currently determine λ4 is with convolutions of range J (ρ) π otherwise The surface will start at t = 0 with the unit circle and deform into an N-sided kN m = −32 ··· 32. For this sum, we need to evaluate 22,494 terms which took regular polygon at time t = 1. The area will be constant for the deformation. By approximating these convolutions, we can determine the solution to λ2. over an hour. Our goal is to optimize the calculations using methods such as

2 4 2 parallel processing to solve λ4 for range m = −256 ··· 256 in minutes. 8ζ(3)ρ 32π ρ −5 λ2 = + + O(N ) (14) N 3 45N 4 REFERENCES [1] P. Amore. Spectroscopy of drums and quantum billiards: perturbative and non-perturbative CONVOLUTIONS results. J. Math. Phys., 51:052105, 2010. Convolutions are approximated using a custom Maple library. [2] M. Boady, P. Grinfeld, and J. Johnson. A Term Rewriting System for the Calculus of Moving Surfaces. International Symposium on Symbolic and Algebraic Computation (ISSAC), 2013. m X [3] P. Grinfeld and G. Strang. Laplace eigenvalues on polygons. Computers and Mathematics conv(a, b)(i) = a(k)b(i − k) (15) with Applications 48:1121–1133, 2004. k=−m [4] P. Grinfeld and G. Strang. Laplace eigenvalues on regular polygons: A series in 1/N. Journal We will use a Taylor series to describe λ(t = 1) in terms of the i-th deriva- of Mathematical Analysis and Applications 385(1):135–149, 2012. Deeper convolutions require more terms approximate. [5] P. Guidotti and J. V. Lambers. Eigenvalue characterization and computation for the laplacian tives of λi(t = 0) = λi. on general 2-d domains. Numerical Functional Analysis and Optimization. 29:507-531, m m ! 1 1 1 X X 2008. λ(t = 1) = λ + λ + λ + λ + λ + ··· (6) conv(c, conv(a, b))(i) = c(j) a(k)b((i − j) − k) (16) 1 2! 2 3! 3 4! 4 [6] M. Khabou, L. Hermi, and M. Rhouma. Shape recognition using eigenvalues of the dirichlet j=−m k=−m laplacian. Pattern Recognition, 40(1):141-153, 2007.