DOCSLIB.ORG

IX.4 the Hankel Transform November 24, 2020 803

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
IX.4 the Hankel Transform November 24, 2020 803 Chapter IX The Integral Transform Methods IX.4 The Hankel Transform November 24, 2020 803 IX.4 The Hankel Transform IX.4.1 The Hankel Transform – Definition and Operational Properties 804 IX.4.2 Instantaneous Heat Source 807 IX.4.3 Heat Equation with Instantaneous Line Source 808 IX.4.4 Solution of PDE with Application of Two Integral Transforms 816 (Fourier and Hankel Transforms) – Diffusion of Light in Semi-Infinite Space IX.4.5 Finite Hankel Transform IX.4.5.1 Circular Domain (FHT-1) 820 IX.4.5.2 Development of Velocity Profile in a Pipe under a Pressure Gradient IX.4.5.3 Annular Domain (FHT-2) 827 IX.4.5.4 Viscous Fluid Flow between Two Cylinders IX.4.7.8 Some additional notes on Hankel Transform 842 804 Chapter IX The Integral Transform Methods IX.4 The Hankel Transform November 24, 2020 IX.4.1 HANKEL TRANSFORM The physical domain where we are supposed to apply the Hankel transform is the infinite 2-dimensional plane with polar coordinates (r,θ ). The Hankel transform which we consider in this section will be applied for transformation of the differential operator 1u∂∂ν 2 Lu ≡ ru− (1) ∂∂ 2 Hermann Hankel rr r r (1839-1873) where ν ∈ is a parameter and domain is 0r≤ <∞. The particular case of this operator with ν = 0 is the radial term of Laplacian in cylindrical coordinates. We assume that the function ur( ) is bounded for all r and that the function u and its derivative have radial decaying at infinity: ∂u <∞ = = ur( ) , ru r→∞ 0 , r 0 (2) ∂r r→∞ The bounded solution of the singular Sturm-Liouville problem (eigenvalue problem): Lu= µ u has non-trivial solution when the parameter is non-positive, let us rename it as µλ= − 2 . That yields the equation Jν (λ r) is a Bessel function 2 1u∂∂ν 2 of the 1st kind of order ν . r−=− uuλ (3) rr∂∂ r r 2 Jν (λ r) is the bounded solution which is the Bessel equation of order ν , the bounded solutions st of the Bessel Equation of which are the Bessel functions of the 1 kind 2 22 2 r y′′++ ry ′ (λν r −) y = 0 Jrν (λ ) for r∈∞[ 0, ) The self-adjoint form of the operator (1) is 1u∂∂ν 2 Lu ≡ ru− (4) rr∂∂ r r Jr0 ( ) Therefore, the weight function is pr( ) = r, which is used for Jr( ) 1 definition of the weighted inner product. Jr2 ( ) Formally, the Hankel transform can be constructed from the two- dimensional Fourier transform with transition to polar coordinates and application of the integral representation of the Bessel functions [Debnah]. Then the Hankel transform and the inverse Hankel transform are defined as ∞ Hankel Transform of order ν Hν { fr( )} = fνν(λλ) = ∫ f( r) J( r) rdr (5) 0 ∞ −1 Inverse Hankel transform of order ν Hfνν{ (λ )} = fr( ) = ∫ fνν(λ) J( λ r) λλ d (6) 0 Chapter IX The Integral Transform Methods IX.4 The Hankel Transform November 24, 2020 805 Apply the Hankel transform (5) of order ν to δµ(r − ) , µ > 0 ∞ Hrν {δµ( − )} =∫δµλ(r − ) Jν ( r) rdr 0 = Jν (λµ) µ Then δµ(r − ) the inverse Hankel transform (6) of Jν (λµ) µ −1 δµ(r − ) = HJνν{ (λµ) µ} = ∞ = ∫ Jνν(λµ) µ J( λ rd) λ λ 0 ∞ = µ∫ Jνν( λ rJ) ( λµ) λ d λ 0 from which follows the orthogonality of Bessel functions: ∞ δµ(r − ) λµ λ λ λ = µ > Orthogonality of Bessel functions ∫ Jνν( ) J( rd) , 0 (7) 0 µ Then we can formulate the Hankel Integral Theorem: ∞∞ = µ λµ µ µ λ λ λ The Hankel Integral Theorem fr( ) ∫∫ f( ) Jνν( ) d J( r) d (8) λµ=00 = Proof using the orthogonality relationship (7): ∞∞ µ λµ µ µ λ λ λ ∫∫f( ) Jνν( ) dJ( rd) λµ=00 = ∞∞ = µ λµ λ λ λ µ µ ∫∫f( ) Jνν( ) J( rd) d µλ=00= ∞ λµ( − r) = ∫ f (µ ) µ dµ p0= µ = fr( ) ■ The Hankel integral theorem equation (8) can be used for definition of the Hankel transform pair (5,6). However, it should be noted, that equation (8) in this notes is derived by application of equations (5,6) assuming that they are true. We can state there is no circular reasoning (derivation of the result starting with assumption that it is true), if equations (5,6) are originated from some external derivation, for example, by the change of variables in Fourier transform to polar coordinates. 806 Chapter IX The Integral Transform Methods IX.4 The Hankel Transform November 24, 2020 Properties of Hankel transform Both, the Hankel transform and its inverse are linear operators. 1 λ Hν { f( ar)} = fν scaling a2 a δλ( ) H1{ } = 0 λ e−ar H0 r δ (rc− ) H0 r H0 { Ha( − r)} −λ2 2 1 He−cr = e 4c 0 { } 2c −ar λ He1 = { } 3 (a22+ λ ) e−ar H1 r H0 { rf′( r) } =−−2f00(λλ) f′( λ) H00{ fr( ) } = f (λ ) reciprocity Have to be checked! Chapter IX The Integral Transform Methods IX.4 The Hankel Transform November 24, 2020 807 Operational Property Consider application of the Hankel trnsform of order ν to differential operator (1), in which the additional term defines the order of the applied Hankel transform: 1u∂∂ν 2 ∞ 1 ∂ ∂u ν 2 − = λ − Hrν 2 u ∫ rJν ( r) r 2 udr rr∂∂ r r 0 r ∂r ∂r r ∞ ∂ ∂u ∞ ν 2 = Jν (λr) r dr − rJν (λ r) u dr ∫ ∂r ∂r ∫ 2 0 0 r ∞ ∂u ∞ ν 2 = λ − λ ∫ Jν ( r)d r ∫ rJν ( r) 2 u dr 0 ∂r 0 r ∞ ∂u ∞ ∂u ∞ ν 2 = r Jν (λr) − r d[Jν (λr)] − rJν (λ r) u dr ∂ ∫ ∂ ∫ 2 r 0 0 r 0 r ∞ ∂u ′ ∞ ν 2 = − λ − λ ∫ r [Jν ( r)] dr ∫ rJν ( r) 2 u dr 0 ∂r 0 r ∞ ′ ∞ ν 2 = − λ − λ From self-adjoint form of ∫ r[Jν ( r)] du ∫ rJν ( r) 2 u dr 0 0 r the Bessel Equation: ∞ ∞ ∞ 2 ν 2 ′′ν ′ 2 = ruJr(λλ) + ud rJr ( ) − rJ(λ r) u dr (ry′) +− +λ r y = 0 { νν} ∫ { } ∫ ν 2 r 0 0 0 r write ∞ ′ ∞ 2 2 ′ ν ′ ν 2 = λ − λ ′ =−+ λ ∫ u r Jν ( r) dr rJν ( r) 2 u dr (ry) r 2 y { } ∫ r 0 0 r Then for y= J(λ r) ν ∞ ν 2 ∞ ν 2 = − λ2 λ − λ ∫ur 2 Jν ( r)dr ∫ rJν ( r) 2 u dr 0 r 0 r ν 2 ′ λλ′ =2 − λ rJrrνν( ) 2 Jr( ) r ∞ ∞ ν 2 ∞ ν 2 = −λ2 (λ ) + λ − λ ∫ rJν r udr ∫ rJν ( r) 2 udr ∫ rJν ( r) 2 u dr 0 0 r 0 r ∞ 2 = −λ ∫ rJν (λr)udr 0 2 = −λλuν ( ) 2 ∞ 2 2 ν 1u∂∂ν 2 Hν ∇− ur( ) = rJν (λ r) r− u dr = −λλuν ( ) (9) 2 ∫ ∂∂2 r 0 rr r r 0th order and 1st order Hankel Transforms 1u∂∂ ∞ 1u∂∂ ∞ = λ = −λλ2 λ λ th H0 r ∫ rJ0 ( r) r dr u 0 ( ) u0 ( ) = ∫ rJ0 ( r) u( r) dr 0 order rr∂∂ r 0 rr∂∂ r 0 1∂∂ uu ∞ 1∂∂ uu ∞ −= λ − = −λλ2 λ λ st Hr1 2 ∫ rJ1 ( r) r2 dr u1 ( ) u1 ( ) = ∫ rJ1 ( r) u( r) dr 1 order rr∂∂ r r 0 rr∂∂ r r 0 We will use mostly the 0th order Hankel transform (in this case, index 0 in the transformed function is usually omitted) 808 Chapter IX The Integral Transform Methods IX.4 The Hankel Transform November 24, 2020 EXAMPLE TRANSIENT HEAT TRANSFER Cooling of the bell-shaped temperature profile Consider the axisymmetric case of the Heat Equation in cylindrical coordinates 2 1∂∂ u 1u ∂ ∇≡ur = r∈∞[ 0, ) t > 0 rr∂∂ rα ∂ t −br2 −br2 ur0 ( ) = e u( r,0) = u0 ( r) = e (initial condition) There are no boundary conditions, but we assume that the function u and its derivative has radial decaying at infinity: ∂u infinite metal = = ru r→∞ 0 r 0 plate ∂r r→∞ Physically it can be treated as a heated to bell shaped temperature profile infinite flat plate thermoinsulated from both sides. Heat is conducted along the plate. 1) Transformed equation Apply 0th order Hankel transform ∞ λ = λ ut( , ) ∫ rJ0 ( r) u( r,t) dr 0 Then equation is transformed to 1u∂ −=λ 2u α ∂t With transformed initial condition (see p.806) λ2 − ∞∞ 4b 2 e λ λλ= −br = u( ,0) = ∫∫rJ00( r) u dr rJ 0( r) e dr 00 2b That is the 1st order linear ODE ∂u +=αλ 2u0 ∂t The solution is 1 −+αλ2 2 1 t u(λ ,t) = u(λ ,0) e−αλ t = e 4b 2b 2) Inverse transform Solution of the problem (use table p.806) 1 2 −+αλt −1 1 4b u(r,t) = He0 2b λ2 − 4 1 + α 1 − 4t 1 = He1 b denote c = 0 1 2b + 4tα b r2 − λ2 1 − +4tα 2c− 1 c − 2 1 Hankel example p.808.mws u( r,t ) = He1 4c ` = e cr ` = e b 0 + α 2b 2c b (1 4 bt) r2 − 1 1 +4tα u(r,t) = e b (1+ 4α bt) Chapter IX The Integral Transform Methods IX.4 The Hankel Transform November 24, 2020 809 IX.4.2 INSTANTENEOUS ENERGY SOURCE We consider the volumetric heat source S(r,t) which releases its entire energy spontaneously at the time t = t0 at the points r = r0 [Ozisik, HC, p.220]. g Ws⋅ In the axisymmetric case (does not depend on θ ) of cylindrical S( r,t) =δδ( r −− r) ( t t ) g 2rπ 00 m coordinates and no dependence on the z coordinate (the same source strength for any plane perpendicular to the z axis), this is a cylindrical surface heat source which can be represented with the help of the Dirac delta function g W = δ − δ − S(r,t) (r r0 ) (t t0 ) 3 2πr m W ⋅ s where g is the source strength per unit length m g W ⋅ s 2 is the source strength per unit area 2πr m r [m] the variable r appears in the denominator as a scale factor of the transformation to polar coordinates 1 δ (r − r0 ) defines the radius of the source r = r0 m 1 δ (t − t0 ) defines the moment of time t = t0 s Ws⋅ Integrated over the entire time-space domain (in the plane g m perpendicular to the z coordinate), it should yield the strength of g the heat source (conservation of energy): S( r,t) = δδ( r) ( t− t ) gW[ ⋅ s] 2rπ 0 ∞ 2π ∞ ∞ 2π ∞ g g S(r,t)rdrdθdt = δ (r − r )δ (t − t ) rdrdθdt =δδ −− δ ∫ ∫ ∫ ∫ ∫ ∫ 0 0 S( r,t) ( r) ( z z00) ( t t ) π 2rπ 0 0 0 0 0 0 2 r g ∞ 2π ∞ = δ − δ − θ ∫ ∫ ∫ (r r0 ) (t t0 )drd dt 2π 0 0 0 g ∞ 2π = δ − θ ∫ ∫ (t t0 )d dt 2π 0 0 g 2π = ∫ dθ 2π 0 g 2π = [θ ] 2π 0 = g gW[ ⋅ s] In the 3-dimensional case, the heat source is a circular ring in g S( r,t) =δδ( r −−− r0) ( z z 00) δ( t t ) = 2rπ the plane z z0 defined by g S(r,t) = δ (r − r )δ (z − z )δ (t − t ) 2πr 0 0 0 An instantaneous source is used for finding the Green function for the non-homogeneous PDE.
Load more

Recommended publications
  • Zero-Order Hankel Transform Method for Partial Differential Equations
    Naol Tufa Negero / International Journal of Modern Sciences and Engineering Technology (IJMSET) ISSN 2349-3755; Available at https://www.ijmset.com Volume 3, Issue 10, 2016, pp.24-36 Zero-Order Hankel Transform Method for Partial Differential Equations Naol Tufa Negero Faculty of Natural and Computational Science, Department of Mathematics, Wollega University, Nekemte,Ethiopia. [email protected] Abstract This paper deals with the solution of Partial Differential Equations (PDEs) by the method of Hankel transform. Firstly, Hankel transforms is derived using two dimensional Fourier transform, next its properties are presented because their results are used widely in solving partial differential equations in the polar and axisymmetric cylindrical configurations, finally it applied to partial differential equation with precise formulation of initial and boundary value problems. In this article, we use the Hankel transforms method, namely Hankel transforms of zero order and of order one are introduced, such that PDE reduced to an ODE, which can be subsequently solved using ODE techniques which involves the Bessel functions. Moreover the main result of this paper is the zero-order Hankel transform method for Partial differential equations in unbounded problems with radial symmetry that is appropriate in the polar and cylindrical coordinates, which is analyzed in some detail. Keywords: Radial Fourier transforms, Two Dimensional Fourier Transform, Polar, Cylindrical and spherical Coordinates, Zero-Order Hankel Transform, Partial Differential Equations 1. INTRODUCTION: There are several integral transforms which are frequently used as a tool for solving numerous scientific problems. In many real applications, Fourier transforms as well as Mellin transforms and Hankel transforms are all very useful.
    [Show full text]
  • The Bessel Function, the Hankel Transform and an Application to Differential Equations
    Georgia Southern University Digital Commons@Georgia Southern Electronic Theses and Dissertations Graduate Studies, Jack N. Averitt College of Summer 2017 The Bessel Function, the Hankel Transform and an Application to Differential Equations Isaac C. Voegtle Follow this and additional works at: https://digitalcommons.georgiasouthern.edu/etd Part of the Partial Differential Equations Commons Recommended Citation I. Voegtle, "The Bessel Function, the Hankel Transform and an Application to Differential Equations". Georgia Southern University, 2017. This thesis (open access) is brought to you for free and open access by the Graduate Studies, Jack N. Averitt College of at Digital Commons@Georgia Southern. It has been accepted for inclusion in Electronic Theses and Dissertations by an authorized administrator of Digital Commons@Georgia Southern. For more information, please contact [email protected]. THE BESSEL FUNCTION, THE HANKEL TRANSFORM AND AN APPLICATION TO DIFFERENTIAL EQUATIONS by ISAAC VOEGTLE (Under the Direction of Yi Hu) ABSTRACT In this thesis we explore the properties of Bessel functions. Of interest is how they can be applied to partial differential equations using the Hankel transform. We use an example in two dimensions to demonstrate the properties at work as well as formulate thoughts on how to take the results further. INDEX WORDS: Bessel function, Hankel transform, Schrodinger¨ equation 2009 Mathematics Subject Classification: 35, 42 THE BESSEL FUNCTION, THE HANKEL TRANSFORM AND AN APPLICATION TO DIFFERENTIAL EQUATIONS by ISAAC VOEGTLE B.A., Anderson University, 2015 A Thesis Submitted to the Graduate Faculty of Georgia Southern University in Partial Fulfillment of the Requirements for the Degree MASTER OF SCIENCE STATESBORO, GEORGIA c 2017 ISAAC VOEGTLE All Rights Reserved 1 THE BESSEL FUNCTION, THE HANKEL TRANSFORM AND AN APPLICATION TO DIFFERENTIAL EQUATIONS by ISAAC VOEGTLE Major Professor: Yi Hu Committee: Shijun Zheng Yan Wu Electronic Version Approved: 2017 2 DEDICATION This thesis is dedicated to my wonderful wife.
    [Show full text]
  • International Journal of Pure and Applied Mathematics ————————————————————————– Volume 55 No
    International Journal of Pure and Applied Mathematics ————————————————————————– Volume 55 No. 2 2009, 247-256 THE HANKEL CONVOLUTION OF ARBITRARY ORDER C.K. Li Department of Mathematics and Computer Science Brandon University Brandon, Manitoba, R7A 6A9, CANADA e-mail: [email protected] Abstract: Let µ> 1/2. The classical Hankel transform is defined by − ∞ (hµφ)(t)= xJµ(xt)φ(x)dx t (0, ), (1) Z0 ∈ ∞ where Jµ(x) denotes the Bessel function of the first kind and order µ. The goal of this paper is to construct the Hankel transform of arbitrary order hµ,k based −1 on the two differential operators and show that hµ,k = h on H 1 for µ R. µ,k µ− 2 ∈ Furthermore, the Hankel convolution of arbitrary order is introduced with the following identity −µ (hµ,kh)(t)= t (hµ,kφ)(t)(hµ,kψ)(t) on the spaces (H 1 , H 1 ) and (Sµ, H 1 ) respectively. µ− 2 µ− 2 µ− 2 AMS Subject Classification: 46F12 Key Words: Hankel transform, Bessel function, Hankel convolution and automorphism 1. Introduction Let R+ = (0, ) and a weight function ν(x) > 0 in R+. We define a set of ∞ functions L(R+,ν(x)) as ∞ L(R+,ν(x)) = f(x) f(x) ν(x)dx < . { | Z0 | | ∞} Different types of the Hankel transforms as well as their convolutions, including modified ones, have been investigated over the decades (for example, see [7]- Received: August 1, 2009 c 2009 Academic Publications 248 C.K. Li [5] and [1]) both in the classical sense and in spaces of generalized functions.
    [Show full text]
  • A Fast Transform for Spherical Harmonics
    A Fast Transform for Spherical Harmonics Martin J. Mohlenkamp The Journal of Fourier Analysis and Applications 5(2/3):159{184, 1999. A±liation. Department of Mathematics, Yale University; and Department of Applied Mathemat- ics, University of Colorado, Campus Box 526, Boulder CO 80309-0526, [email protected]. Math Subject Classi¯cation. Primary 65T20; Secondary 42C10, 33C55. Keywords and Phrases. Spherical Harmonics, Fast Transforms, Associated Legendre Functions. Acknowledgements and Notes. I would like to thank my thesis advisor, R.R. Coifman, for his help and guidance. Abstract Spherical Harmonics arise on the sphere S2 in the same way that the (Fourier) exponential functions ik e k2Z arise on the circle. Spherical Harmonic series have many of the same wonderful properties asf Fourierg series, but have lacked one important thing: a numerically stable fast transform analogous to the Fast Fourier Transform. Without a fast transform, evaluating (or expanding in) Spherical Harmonic series on the computer is slow|for large computations prohibitively slow. This paper provides a fast transform. For a grid of (N 2) points on the sphere, a direct calculation has computational complexity (N 4), but a simpleO separation of variables and Fast Fourier Transform reduce it to (N 3) time. HereO we present algorithms with times (N 5=2 log N) and (N 2(log N)2). O The problem quickly reduces to the fastO application of matricesO of Associated Legendre Functions of certain orders. The essential insight is that although these matrices are dense and oscillatory, locally they can be represented e±ciently in trigonometric series. 1 Introduction Spherical Harmonics arise on the sphere S2 in the same way that the (Fourier) exponential functions ik e k2Z arise on the circle.
    [Show full text]
  • Hankel Transform, Langlands Functoriality and Functional Equation of Automorphic L-Functions
    Hankel transform, Langlands functoriality and functional equation of automorphic L-functions Ngô Bảo Châu dedicated with affection to the memory of Hiroshi Saito 1 Introduction The functoriality conjecture of Langlands, as stated in [33], offers a general organizational scheme for all automorphic representations. This conjecture can be roughly stated as follows: if H and G are reductive groups defined over a global field k, ξ : L H L G a homomorphism between their Langlands dual groups, then it is possible to attach to! each automorphic rep- H H resentation π = vπv of H a packet of automorphic representations π = vπv of G. In ⊗ ⊗ the formula π = vπv, local components πv are irreducible admissible representations of ⊗ G(kv), kv being the completion of k at a place v k . The automorphic functoriality is expected to be of local nature in the sense that it should2 j j be compatible with the local func- toriality principle that consists in a map from the set of packets of irreducible admissible representations of H(kv) to the set of packets of irreducible admissible representations of G(kv). Since unramified representations of H(kv) at an unramified nonarchimedean place v are parametrized by certain semi-simple conjugacy classes in L H, the local functoriality is required to be compatible with the transfer of conjugacy classes from L H to L G. A crucial part of the functoriality program, known as endoscopy, is now almost completely understood thanks to cumulative efforts of many mathematicians over the last 40 years, see [5]. Its main ingredients include in particular the construction of the general trace formula, mainly due to Arthur [4] and the proof of the transfer conjecture and the fundamental lemma, L see [54] and [39].
    [Show full text]
  • Linear Restriction Estimates for the Wave Equation with Inverse Square Potential
    LINEAR RESTRICTION ESTIMATES FOR THE WAVE EQUATION WITH INVERSE SQUARE POTENTIAL JUNYONG ZHANG AND JIQIANG ZHENG Abstract. In this paper, we study a modified linear restriction estimates associated with the wave equation with inverse square potential. In particular, we show the classical linear restriction estimates hold in their almost sharp range when the initial data is radial. Key Words: Linear restriction estimate, Spherical harmonics, Bessel func- tion, Inverse square potential AMS Classification: 42B37, 35Q40, 47J35. Contents 1. Introduction and Statement of Main Result 1 2. Preliminary 4 2.1. Spherical harmonic expansions and the Bessel functions 4 3. Proof of the Main Theorem 6 3.1. Hankel transform and the solution 7 3.2. Estimates of Hankel transforms 8 3.3. Conclude the Proof of Theorem 1.2 13 4. Appendix: Vector valued inequality 15 References 17 References 17 1. Introduction and Statement of Main Result In this paper, we study a modified restriction estimate associated with the wave equation perturbed by an inverse square potential. More precisely, we consider the following wave equation with a singular potential: ( @2u − ∆u + a u = 0 (t; x) 2 R × Rn; a 2 R; t jxj2 (1.1) u(t; x)jt=0 = 0;@tu(t; x)jt=0 = f(x): − a The scale-covariance elliptic operator Pa := ∆ + jxj2 appearing in (1.1) plays a key role in many problems of physics and geometry. The heat and Schr¨odingerflows for the elliptic operator Pa have been studied in the theory of combustion (see [8]), and in quantum mechanics (see [6]). The wave equation (1.1) arises in the study of the wave propagation on conic manifolds [5].
    [Show full text]
  • Asymptotic Expansions of Some Integral Transforms by Using Generalized Functions by Ahmed I, Zayed
    TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIElY Volume 272, Number 2, August 1982 ASYMPTOTIC EXPANSIONS OF SOME INTEGRAL TRANSFORMS BY USING GENERALIZED FUNCTIONS BY AHMED I, ZAYED ABSTRACT. The technique devised by Wong to derive the asymptotic expansions of multiple Fourier transforms by using the theory of Schwartz distributions is ex- tended to a large class of integral transforms. The extension requires establishing a general procedure to extend these integral transforms to generalized functions. Wong's technique is then applied to some of these integral transforms to obtain their asymptotic expansions. This class of integral transforms encompasses, among others, the Laplace, the Airy, the K and the Hankel transforms. 1. Introduction. Recently, the theory of generalized functions has been introduced into the field of asymptotic expansions of integral transforms; see [18, 19,6]. One of the advantages of this approach is to be able to interpret and to assign values to some divergent integrals. This approach goes back to Lighthill [10] who was the first to use generalized functions to study the asymptotic expansions of Fourier transforms. Borrowing Lighthill's idea, several people were able to derive asymptotic expan- sions for other types of integral transforms such as Laplace [2], Stieltjes [11], Riemann-Liouville fractional integral transforms [12] and Mellin convolution of two algebraically dominated functions [17]. The use of generalized functions in the theory of asymptotic expansions has another advantage; that is, in some cases it can provide explicit expressions for the error terms in shorter and more direct ways than the classical methods [9,11]. However, one should not overemphasize the importance of the generalized functions approach since, for the most part, it is an alternative but not an exclusive technique.
    [Show full text]
  • Orthogonal Fast Spherical Bessel Transform on Uniform Grid
    Orthogonal fast spherical Bessel transform on uniform grid Vladislav V. Serov Department of Theoretical Physics, Saratov State University, 83 Astrakhanskaya, Saratov 410012, Russia Abstract We propose an algorithm for the orthogonal fast discrete spherical Bessel transform on an uniform grid. Our approach is based upon the spherical Bessel transform factorization into the two subsequent orthogonal transforms, namely the fast Fourier transform and the orthogonal transform founded on the derivatives of the discrete Legendre orthogonal polynomials. The method utility is illustrated by its implementation for the numerical solution of the three-dimensional time-dependent Schr¨odinger equation. Keywords: Spherical Bessel functions, Hankel transforms, time-dependent Schr¨odinger equation PACS: 02.30.Uu, 31.15.-p arXiv:1509.07115v3 [math.NA] 12 May 2016 1. Introduction The discrete spherical Bessel transform (DSBT) arises in a number of ap- plications, such as, e.g., the analysis of the cosmic microwave background [1], the numerical solution of the differential equations [2, 3, 4], and the numeri- ∗Corresponding author. E-mail address: vladislav [email protected] Preprint submitted to Computer Physics Communications May 13, 2016 cal evaluation of multi-center integrals [5, 6]. Many different SBT algorithms have been proposed so far [7, 8, 9, 10]. But none of them possess all of the advantages of their trigonometric progenitor, namely the fast Fourier trans- form (FFT). These advantages are the performance fastness, the uniform coordinate grid, and the orthogonality. An example of the problem requiring the simultaneous presence of all the advantages is the solving of the Schr¨odinger-type equation (SE) by means of the pseudospectral approach [9].
    [Show full text]
  • A Study on Henkel Transform and Its Relation to the Fourier Transform
    International Journal of Multidisciplinary Research and Development Volume: 2, Issue: 5, 555-558 May 2015 A study on Henkel transform and its relation to the www.allsubjectjournal.com e-ISSN: 2349-4182 Fourier transform p-ISSN: 2349-5979 Impact Factor: 3.762 Parmod Kumar Parmod Kumar Abstract Designation... Assistant In mathematics, the Hankel transform expresses any given function f(r) as the weighted sum of an Prof Rajiv Gandhi S.D. infinite number of Bessel functions of the first kind Jν (kr). The Bessel functions in the sum are all of the Commerce and Science same order ν, but differ in a scaling factor k along the r-axis. The necessary coefficient Fν of each Bessel College Narwana (Jind) function in the sum, as a function of the scaling factor k constitutes the transformed function. Hrayana-126115 Keywords: Hankel Transform, Fourier Transform. Introduction The Hankel transform is an integral transform and was first developed by the mathematician Hermann Hankel. It is also known as the Fourier–Bessel transform. Just as the Fourier transform for an infinite interval is related to the Fourier series over a finite interval, so the Hankel transform over an infinite interval is related to the Fourier–Bessel series over a finite interval. The Hankel transform of order ν of a function f(r) is given by: Where is the Bessel function of the first kind of order with The inverse Hankel transform of Fν(k) is defined as: Which can be readily verified using the orthogonality relationship described below. Inverting a Hankel transform of a function f(r) is valid at every point at which f(r) is continuous provided that the function is defined in (0, ∞), is piecewise continuous and of bounded variation in every finite subinterval in (0, ∞), and However, like the Fourier Transform, the domain can be extended by a density argument to include some functions whose above integral is not finite, for example .
    [Show full text]
  • Discrete Two-Dimensional Fourier Transform in Polar Coordinates Part I: Theory and Operational Rules
    Article Discrete Two-Dimensional Fourier Transform in Polar Coordinates Part I: Theory and Operational Rules Natalie Baddour 1,* 1 Department of Mechanical Engineering, University of Ottawa, 161 Louis Pasteur, , Ottawa, Canada * Correspondence: [email protected]; Tel.: (+1(613)5625800 X2324; Received: 9 July 2019; Accepted: 26 July 2019; Published: 2 August 2019 Abstract: The theory of the continuous two-dimensional (2D) Fourier transform in polar coordinates has been recently developed but no discrete counterpart exists to date. In this paper, we propose and evaluate the theory of the 2D discrete Fourier transform (DFT) in polar coordinates. This discrete theory is shown to arise from discretization schemes that have been previously employed with the 1D DFT and the discrete Hankel transform (DHT). The proposed transform possesses orthogonality properties, which leads to invertibility of the transform. In the first part of this two- part paper, the theory of the actual manipulated quantities is shown, including the standard set of shift, modulation, multiplication, and convolution rules. Parseval and modified Parseval relationships are shown, depending on which choice of kernel is used. Similar to its continuous counterpart, the 2D DFT in polar coordinates is shown to consist of a 1D DFT, DHT and 1D inverse DFT. Keywords: Fourier Theory, DFT in polar coordinates, polar coordinates, multidimensional DFT, discrete Hankel Transform, discrete Fourier Transform, Orthogonality. 1. Introduction The Fourier transform (FT) in continuous and discrete forms has seen much application in various disciplines [1]. It easily expands to multiple dimensions, with all the same rules of the one- dimensional (1D) case carrying into the multiple dimensions.
    [Show full text]
  • Integral Equations Associated with Hankel
    INTEGRALEQUATIONS ASSOCIATED WITH HANKEL CONVOLUTIONS( ) BY DEBORAH TEPPER HAIMO Chapter I. Introduction 1. Objectives. We denote by V[au ■■ -,an] the number of variations of sign of the sequence alt--,an of real numbers. Hence, for example, V[-1,0,1 ]= 1, V[1,0,1] = 0, V[0,0,0]=-1. If / is a real function defined for 0 < x < <*>,then V[f(x)] = l.u.b. V[/(x,), ■• - ,/(xn)], taken over all sets 0 < X! < x2 < • • ■ < co. Let H be a real-valued function of L'( — °°, «), <ba real-valued, continu- ous function of L °°(— °°, °°) and let (1) H *<b(x) = j~ H(x - t)<fi(t)dt, -co<x<oo. H is said to be a variation diminishing *-kernel if and only if, for every such <b, (2) V[H*<Kx)]£V[<p(x)]. Let (3) 1 C~ C* a ~2t)_„E(s) ' where (4) E(s) = ecs2+1*"f](l - i^) e"/8*, the aA's being real, with Z*"=i(l/oit) < co> and 6 and c real, with c ^ 0. In 1947, I. J. Schoenberg proved that the kernel G defined by (3) is vari- Presented to the Society, November 4, 1963 under the title Inversion for Hankel convolu- tions, and January 25, 1964 under the title Representation for Hankel convolutions; received by the editors November 5, 1963. ( ) This paper includes the major part of the author's doctoral dissertation at Harvard University. Thanks are due to Professor I. I. Hirschman, Jr., of Washington University who suggested the problem and who gave invaluable guidance.
    [Show full text]
  • Application of the Generalized Shift Operator to the Hankel Transform Natalie Baddour
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Crossref Baddour SpringerPlus 2014, 3:246 http://www.springerplus.com/content/3/1/246 a SpringerOpen Journal SHORT REPORT Open Access Application of the generalized shift operator to the Hankel transform Natalie Baddour Abstract It is well known that the Hankel transform possesses neither a shift-modulation nor a convolution-multiplication rule, both of which have found many uses when used with other integral transforms. In this paper, the generalized shift operator, as defined by Levitan, is applied to the Hankel transform. It is shown that under this generalized definition of shift, both convolution and shift theorems now apply to the Hankel transform. The operation of a generalized shift is compared to that of a simple shift via example. Introduction Background It is well known that the Hankel transform does not satisfy Several definitions of the Hankel transform appear in the a convolution or shift theorem in the simple way as that literature. In this paper, we use the definition of the nth the Fourier and Laplace transforms (Piessens 2000), re- order Hankel transform as defined by Piessens in (Piessens ducing its apparent utility. This follows because the 2000) to define the Hankel transform as Bessel functions do not possess a simple addition formula in the same way that the exponentials satisfy ei(x + y) = eixeiy. Z∞ The operation of convolution between two functions FðÞ¼ρ ℍnðÞ¼frðÞ frðÞJ nðÞρr rdr; ð1Þ consists of a shift in one of the functions, a multiplica- 0 tion with the other function, followed by an integration (or summation for discrete transforms) over all allow- where J (z)isthenth order Bessel function.
    [Show full text]