Integral Transforms
Total Page:16
File Type:pdf, Size:1020Kb
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. -
Some Schemata for Applications of the Integral Transforms of Mathematical Physics
mathematics Review Some Schemata for Applications of the Integral Transforms of Mathematical Physics Yuri Luchko Department of Mathematics, Physics, and Chemistry, Beuth University of Applied Sciences Berlin, Luxemburger Str. 10, 13353 Berlin, Germany; [email protected] Received: 18 January 2019; Accepted: 5 March 2019; Published: 12 March 2019 Abstract: In this survey article, some schemata for applications of the integral transforms of mathematical physics are presented. First, integral transforms of mathematical physics are defined by using the notions of the inverse transforms and generating operators. The convolutions and generating operators of the integral transforms of mathematical physics are closely connected with the integral, differential, and integro-differential equations that can be solved by means of the corresponding integral transforms. Another important technique for applications of the integral transforms is the Mikusinski-type operational calculi that are also discussed in the article. The general schemata for applications of the integral transforms of mathematical physics are illustrated on an example of the Laplace integral transform. Finally, the Mellin integral transform and its basic properties and applications are briefly discussed. Keywords: integral transforms; Laplace integral transform; transmutation operator; generating operator; integral equations; differential equations; operational calculus of Mikusinski type; Mellin integral transform MSC: 45-02; 33C60; 44A10; 44A15; 44A20; 44A45; 45A05; 45E10; 45J05 1. Introduction In this survey article, we discuss some schemata for applications of the integral transforms of mathematical physics to differential, integral, and integro-differential equations, and in the theory of special functions. The literature devoted to this subject is huge and includes many books and reams of papers. -
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. -
Integral Equation for the Number of Integer Points in a Circle
ITALIAN JOURNAL OF PURE AND APPLIED MATHEMATICS { N. 41{2019 (522{525) 522 INTEGRAL EQUATION FOR THE NUMBER OF INTEGER POINTS IN A CIRCLE Magomed A. Chakhkiev Gagik S. Sulyan Manya A. Ziroyan Department of Applied Mathematics Social State University of Russia Wilhelm Pieck St. 4, Moscow Russian Federation Nikolay P. Tretyakov Department of Applied Mathematics Social State University of Russia Wilhelm Pieck St. 4, Moscow Russian Federation and School of Public Policy The Russian Presidential Academy of National Economy and Public Administration Prospect Vernadskogo, 84, Moscow 119571 Russian Federation Saif A. Mouhammad∗ Department of Physics Faculty of Science Taif University Taif, AL-Haweiah Kingdom of Saudi Arabia [email protected] Abstract. The problem is to obtain the most accurate upper estimate for the absolute value of the difference between the number of integer points in a circle and its area (when the radius tends to infinity). In this paper we obtain an integral equation for the function expressing the dependence of the number of integer points in a circle on its radius. The kernel of the equation contains the Bessel functions of the first kind, and the equation itself is a kind of the Hankel transform. Keywords: Gauss circle problem, integral equation, Hankel transform. 1. The problem and calculations The Gauss circle problem is the problem of determining how many integer lattice points there are in a circle centered at the origin andp with given radius. Let us consider the circle K(R): x2 + y2 ≤ R and let A( R) be the number of ∗. Corresponding author INTEGRAL EQUATION FOR THE NUMBER OF INTEGER POINTS IN A CIRCLE 523 p points with integer coordinates within this circle. -
Laplace Transforms: Theory, Problems, and Solutions
Laplace Transforms: Theory, Problems, and Solutions Marcel B. Finan Arkansas Tech University c All Rights Reserved 1 Contents 43 The Laplace Transform: Basic Definitions and Results 3 44 Further Studies of Laplace Transform 15 45 The Laplace Transform and the Method of Partial Fractions 28 46 Laplace Transforms of Periodic Functions 35 47 Convolution Integrals 45 48 The Dirac Delta Function and Impulse Response 53 49 Solving Systems of Differential Equations Using Laplace Trans- form 61 50 Solutions to Problems 68 2 43 The Laplace Transform: Basic Definitions and Results Laplace transform is yet another operational tool for solving constant coeffi- cients linear differential equations. The process of solution consists of three main steps: • The given \hard" problem is transformed into a \simple" equation. • This simple equation is solved by purely algebraic manipulations. • The solution of the simple equation is transformed back to obtain the so- lution of the given problem. In this way the Laplace transformation reduces the problem of solving a dif- ferential equation to an algebraic problem. The third step is made easier by tables, whose role is similar to that of integral tables in integration. The above procedure can be summarized by Figure 43.1 Figure 43.1 In this section we introduce the concept of Laplace transform and discuss some of its properties. The Laplace transform is defined in the following way. Let f(t) be defined for t ≥ 0: Then the Laplace transform of f; which is denoted by L[f(t)] or by F (s), is defined by the following equation Z T Z 1 L[f(t)] = F (s) = lim f(t)e−stdt = f(t)e−stdt T !1 0 0 The integral which defined a Laplace transform is an improper integral. -
Solution to Volterra Singular Integral Equations and Non Homogenous Time Fractional Pdes
Gen. Math. Notes, Vol. 14, No. 1, January 2013, pp. 6-20 ISSN 2219-7184; Copyright © ICSRS Publication, 2013 www.i-csrs.org Available free online at http://www.geman.in Solution to Volterra Singular Integral Equations and Non Homogenous Time Fractional PDEs A. Aghili 1 and H. Zeinali 2 1,2 Department of Applied Mathematics Faculty of Mathematical Sciences, University of Guilan, P.O. Box- 1841, Rasht – Iran 1E-mail: [email protected] 2E-mail: [email protected] (Received: 8-10-12 / Accepted: 19-11-12) Abstract In this work, the authors implemented Laplace transform method for solving certain partial fractional differential equations and Volterra singular integral equations. Constructive examples are also provided to illustrate the ideas. The result reveals that the transform method is very convenient and effective. Keywords : Non-homogeneous time fractional heat equations; Laplace transform; Volterra singular integral equations. 1 Introduction In this work, the authors used Laplace transform for solving Volterra singular integral equations and PFDEs. Solution to Volterra Singular Integral… 7 The Laplace transform is an alternative method for solving different types of PDEs. Also it is commonly used to solve electrical circuit and systems problems. In this work, the authors implemented transform method for solving the partial fractional heat equation which arise in applications. Several methods have been introduced to solve fractional differential equations, the popular Laplace transform method, [ 1 ] , [ 2 ] , [ 3 ], [ 4 ] , and operational method [ 10]. However, most of these methods are suitable for special types of fractional differential equations, mainly the linear with constant coefficients. More detailed information about some of these results can be found in a survey paper by Kilbas and Trujillo [10]. -
Methods of Integral Transforms - V
COMPUTATIONAL METHODS AND ALGORITHMS – Vol. I - Methods of Integral Transforms - V. I. Agoshkov, P. B. Dubovski METHODS OF INTEGRAL TRANSFORMS V. I. Agoshkov and P. B. Dubovski Institute of Numerical Mathematics, Russian Academy of Sciences, Moscow, Russia Keywords: Integral transform, Fourier transform, Laplace transform, Mellin transform, Hankel transform, Meyer transform, Kontorovich-Lebedev transform, Mehler-Foque transform, Hilbert transform, Laguerre transform, Legendre transform, convolution transform, Bochner transform, chain transform, wavelet, wavelet transform, problems of the oscillation theory, heat conductivity problems, problems of the theory of neutron slowing-down, problems of hydrodynamics, problems of the elasticity theory, Boussinesq problem, coagulation equation, physical kinetics. Contents 1. Introduction 2. Basic integral transforms 2.1. The Fourier Transform 2.1.1. Basic Properties of the Fourier Transform 2.1.2. The Multiple Fourier Transforms 2.2. The Laplace Transform 2.2.1. The Laplace Integral 2.2.2. The Inversion Formula for the Laplace Transform 2.2.3. Limit Theorems 2.3. The Mellin Transform 2.4. The Hankel Transform 2.5. The Meyer Transform 2.6. The Kontorovich-Lebedev Transform 2.7. The Mehler-Foque Transform 2.8. The Hilbert Transform 2.9. The Laguerre and Legendre Transforms 2.10. The Bochner Transform, the Convolution Transform, the Wavelet and Chain Transforms 3. The application of integral transforms to problems of the oscillation theory 3.1. Electric Oscillations 3.2. TransverseUNESCO Oscillations of a String – EOLSS 3.3. Transverse Oscillations of an Infinite Round Membrane 4. The application of integral transforms to heat conductivity problems 4.1. The SolutionSAMPLE of the Heat Conductivity CHAPTERS Problem by the use of the Laplace Transform 4.2. -
Math 551 Lecture Notes Fredholm Integral Equations (A Brief Introduction)
MATH 551 LECTURE NOTES FREDHOLM INTEGRAL EQUATIONS (A BRIEF INTRODUCTION) Topics covered • Fredholm integral operators ◦ Integral equations (Volterra vs. Fredholm) ◦ Eigenfunctions for separable kernels ◦ Adjoint operator, symmetric kernels • Solution procedure (separable) ◦ Solution via eigenfunctions (first and second kind) ◦ Shortcuts: undetermined coefficients ◦ An example (separable kernel, n = 2) • Non-separable kernels (briefly) ◦ Hilbert-Schmidt theory Preface Read the Fredholm alternative notes before proceeding. This is covered in the book (Section 9.4), but the material on integral equations is not. For references on integral equa- tions (and other topics covered in the book too!), see: • Riley and Hobson, Mathematical methods for physics and engineering (this is an extensive reference, also for other topics in the course) • Guenther and Lee, Partial differential equations of mathematical physics and integral equations (more technical; not the best first reference) • J.D. Logan, Applied mathematics (more generally about applied mathematics tech- niques, with a good section on integral equations) 1. Fredholm integral equations: introduction Differential equations Lu = f are a subset of more general equations involving linear op- erators L. Here, we give a brief treatment of a generalization to integral equations. To motivate this, every ODE IVP can be written as an `integral equation' by integrating. For instance, consider the first order IVP du = f(x; u(x)); u(a) = u : (1.1) dx 0 Integrate both sides from a to x to get the integral equation Z x u(x) = u0 + f(s; u(s)) ds: (1.2) a If u solves (1.2) then it also solves (1.1); they are `equivalent' in this sense. -
Partial Derivative Approach to the Integral Transform for the Function Space in the Banach Algebra
entropy Article Partial Derivative Approach to the Integral Transform for the Function Space in the Banach Algebra Kim Young Sik Department of Mathematics, Hanyang University, Seoul 04763, Korea; [email protected] Received: 27 July 2020; Accepted: 8 September 2020; Published: 18 September 2020 Abstract: We investigate some relationships among the integral transform, the function space integral and the first variation of the partial derivative approach in the Banach algebra defined on the function space. We prove that the function space integral and the integral transform of the partial derivative in some Banach algebra can be expanded as the limit of a sequence of function space integrals. Keywords: function space; integral transform MSC: 28 C 20 1. Introduction The first variation defined by the partial derivative approach was defined in [1]. Relationships among the Function space integral and transformations and translations were developed in [2–4]. Integral transforms for the function space were expanded upon in [5–9]. A change of scale formula and a scale factor for the Wiener integral were expanded in [10–12] and in [13] and in [14]. Relationships among the function space integral and the integral transform and the first variation were expanded in [13,15,16] and in [17,18] In this paper, we expand those relationships among the function space integral, the integral transform and the first variation into the Banach algebra [19]. 2. Preliminaries Let C0[0, T] be the class of real-valued continuous functions x on [0, T] with x(0) = 0, which is a function space. Let M denote the class of all Wiener measurable subsets of C0[0, T] and let m denote the Wiener measure. -
Chapter 1 Introduction to the Integral Equation(IE) and Construction Of
Chapter 1 Introduction to the Integral Equation(IE) and Construction of the IE. 1.1. Introduction: Integral Equation began to appear since the mid-seventeenth century , when some scientists were not able to solve the differential equation . The integral equation developed with appear Abel kernel after that Volterra integral equation lastly Fredholm integral equation. In this time we find numerical method played a great role to solve integral equation. Therefore of great progressing in basic science whether physical or engineering has played essential role. Topics on integral equations has grown and evolved to its direct association lists the large branches of mathematics, such as account differential and integrative and questions of boundary conditions. During the twenty-five last year, there is a marked increase in the use of integral equations and formulations for finding scientific solutions to engineering problems and solving differential equations that are difficult to solve by normal methods. In the recent period found that the integral equations give a better solution than give differential equations. The explosive growth in industry and technology requires constructive adjustments in mathematics text researches. The integral equation it equation that appear in the unknown function under signal or more, from the signals of integrity. There more types of integral equation from it linear and nonlinear integral equation. The general formula linear integral equation it is: ١ () = () + (, )() (1 − 1) Where () unknown function, () known function and (, )known function are called kernel integral equation. We say that integral equation it is linear if that which operations on unknown function in equation it linear operations. And the general formula nonlinear integral equation it is : () = () + (, )(()) (1 − 2) Where unknown function it is nonlinear . -
Basic Magnetic Resonance Imaging
Basic Magnetic Resonance Imaging Mike Tyszka, Ph.D. Magnetic Resonance Theory Course 2004 Caltech Brain Imaging Center California Institute of Technology Spatial Encoding Spatial Localization of Signal Va r y B sp a t i a l l y ω=γB Linear Field Gradients Linear Magnetic Field Gradients One-Dimensional Tw o -Dimensional Gradient direction Bz(x,z) Bz z Distance (x) x Deviation from B0 typically very small (~0.1%) Frequency Encoding Total signal from spins in plane perpendicular to gradient direction Signal Gradient direction Frequency (ω) Distance (ω/γG) The FID in a Field Gradient • FID contains total signal from all spins at all frequencies/locations. • Each frequency component represents the total signal from a plane perpendicular to the gradient direction. • Frequency analysis of FID reveals “projection” of object. MRI from Projections • Can projections be turned into an image? Back Projection • Sweep gradient direction through 180° • Acquire projections of object for each direction Sinogram Back Projection or Inverse Radon Transform Frequency Analysis for MRI MR Signal as a Complex Quantity • Both the magnitude and phase of the MR signal are measured. y’ • Two signal channels are commonly referred to as “real” and “imaginary”. A M φ y x’ Mx The Fourier Transform (FT) The Fourier Transform is an Integral Transform which, for MRI, relates a time varying waveform to the corresponding frequency spectrum. FORWARD TRANSFORM (time -> frequency) REVERSE TRANSFORM (frequency -> time) The Fast Fourier Transform (FFT) • Numerical algorithm for efficient computation of the discrete Fourier Transform. • Requires discrete complex valued samples. • Both forward and reverse transforms can be calculated. -
Study of the Mellin Integral Transform with Applications in Statistics and Probability
Available online a t www.scholarsresearchlibrary.com Scholars Research Library Archives of Applied Science Research, 2012, 4 (3):1294-1310 (http://scholarsresearchlibrary.com/archive.html) ISSN 0975-508X CODEN (USA) AASRC9 Study of the mellin integral transform with applications in statistics And probability S. M. Khairnar 1, R. M. Pise 2 and J. N. Salunkhe 3 1Department of Mathematics, Maharashtra Academy of Engineering, Alandi, Pune, India 2Department of Mathematics, (A.S.& H.), R.G.I.T. Versova, Andheri (W), Mumbai, India 3Department of Mathematics, North Maharastra University, Jalgaon, Maharashtra, India ______________________________________________________________________________ ABSTRACT The Fourier integral transform is well known for finding the probability densities for sums and differences of random variables. We use the Mellin integral transforms to derive different properties in statistics and probability densities of single continuous random variable. We also discuss the relationship between the Laplace and Mellin integral transforms and use of these integral transforms in deriving densities for algebraic combination of random variables. Results are illustrated with examples. Keywords: Laplace, Mellin and Fourier Transforms, Probability densities, Random Variables and Applications in Statistics. AMS Mathematical Classification: 44F35, 44A15, 44A35, 44A12, 43A70. ______________________________________________________________________________ INTRODUCTION The aim of this paper , we define a random variable (RV) as a value in some domain , say ℜ , representing the outcome of a process based on a probability laws .By the information above the probability distribution ,we integrate the probability density function (p d f)in the case of the Gaussian , the p d f is 1 X −π − ( )2 1 σ − ∞ < < ∞ f (x)= E 2 , x σ 2π when µ is mean and σ 2 is the variance.