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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. It is well known that the Fourier transform method is used to solve partial differential equations [12-21]. The transform is applied to PDEs on infinite and semi-infinite spatial domains [1]. Fourier transforms have been used to remove Cartesian coordinates from initial boundary value problems on infinite intervals; Fourier sine and cosine transforms are applicable to Cartesian coordinates on semi-infinite intervals[17]. To quote [23], “In two dimensions phenomena are richer than in one dimension.” True enough, working in two dimensions offers many new and rich possibilities. Contemporary applications of the Fourier transform are just as likely to come from problems in two, three, and even higher dimensions as they are in one Fourier transforms may be extend to functions of two or more variables[14]. We consider the case for functions of two variables primarily to understand an origin for a Hankel transform [5, 14]. Here, we show that radially sampled data can be processed directly using Fourier transforms in polar coordinates[6,19].In many practical problems, data are often acquired in such a form that is desirable to perform a two dimensional polar Fourier transform that is a Hankel transform rather than the Cartesian forms. So, we transform the Cartesian coordinates into the polar coordinates. The Hankel transform arises naturally in the discussion of problems posed in cylindrical coordinates [4, 19]. This transform is more appropriate in solving differential equations with boundary conditions in which there is an axial symmetry [7, 19]. The Hankel transform is frequently used as a tool for solving numerous scientific problems. It is widely used in several fields like, elasticity, optics, fluid mechanics, seismology, astronomy and image processing[4,23].The Hankel transform IJMSET-Advanced Scientific Research Forum (ASRF), All Rights Reserved “ASRF promotes research nature, Research nature enriches the world’s future” 24 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 becomes very useful in analysis of wave fields where it is used in mathematical handling of radiation, diffraction, and field projection. As written in Eq. (9), the transform is evaluated first with respect to 푟, and second with respect to 휃 [20]. We shall refer to this form of the polar Fourier Transform as a radial transform. This equation fits naturally with the interpretation of the 2-D Fourier Transform kernel as a superposition of 1-D cosine and sine waves in all directions [21]; here, each spoke of 푓 (푟, 휃) at fixed 휃 contributes the weighting coefficients for the cosine and sine waves in direction 휃 used in the synthesis of 퐹. 2. BASIC IDEA OF HANKEL TRANSFORM METHOD: 2.1 The Radial Fourier Transform For use in many applications, we’re going to consider one further aspects of the 2-dimensional case. A function on 푅2 is radial(also called radially symmetric or circularly symmetric) if it depends only on the distance from the origin [2]. In polar coordinates the distance from the origin is denoted by 푟, so to say that a function is radial is to say that it depends only on 푟 (and that it does not depend on 휃, writing the usual polar coordinates as (푟, 휃)).The definition of the Fourier transform is set up in Cartesian coordinates, and it’s clear that we’ll be better off writing it in polar coordinates if we work with radial functions. This is actually not so straightforward, or, at least, it involves introducing some special functions to write the formulas in a compact way. We have to convert ∞ ∞ −2휋푖퐱.휿 −2휋푖 푥푘 +푦푘 푒 푓 퐱 푑퐱 = 푒 1 2 푓 푥, 푦 푑푥푑푦 푅2 −∞ −∞ to polar coordinates. There are several steps: To say that 푓(퐱) is a radial function means that it becomes 푓(푟). To describe all of 푅2 in the limits of integration, we take 푟 going from 0 to ∞ and 휃 going from 0 to 2휋. The area element 푑푥푑푦 becomes 푟푑푟푑휃. Finally, the problem is the inner product 퐱. 휿 = 푥푘1 + 푦푘2 in the exponential and how to write it in polar coordinates. If we identify 푥, 푦 = (푟, 휃) (varying over the 푥, 푦 − plane) and put 푘1, 푘2 = 휌, 휙 (fixed in the integral) then 퐱. 휿 = 퐱. 휿 푐표푠 휃 − 휙 = 푟휌푐표푠 휃 − 휙 . ∞ ∞ −2휋푖.휿 2휋 ∞ −2휋푖푟휌푐표푠 휃 − 휙 The Fourier transform of 푓 is thus −∞ −∞ 푒 푓 퐱 푑퐱 = 0 0 푓 r 푒 푟푑푟푑휃 [9]. There’s more to be done. First of all, because 푒−2휋푖푟휌푐표푠 휃 − 휙 , is periodic (in θ) of period 2휋, 2휋 −2휋푖푟휌푐표푠 휃 − 휙 the integral 0 푒 푑휃 does not depend on 휙. Consequently, 2휋 −2휋푖푟휌푐표푠 휃 − 휙 2휋 −2휋푖푟휌푐표푠휃 0 푒 푑휃 = 0 푒 푑휃. The next step is to define ourselves out of trouble. We 1 2휋 −푖푎휌푐표푠휃 introduce the function 퐽 푎 = 푒 푑휃. We give this integral a name, 퐽 푎 , because, try 0 2휋 0 0 as you might, there is no simple closed form expression for it, so we take the integral as defining a new function. It is called the zero order Bessel function of the first kind. Sorry, but Bessel functions, of whatever order and kind, always seem to come up in problems involving circular symmetry. 2휋 −2휋푖푟휌푐표푠휃 Incorporating 퐽0 into what we’ve done, 0 푒 푑휃 = 2휋퐽0 2휋푟휌 and the Fourier transform of ∞ 푓(푟) is 2휋 0 푓(푟)퐽0 2휋푟휌 푟푑푟. ∞ Generally, If 푓(퐱) is a radial function then its Fourier transform is 퐹 휌 = 2휋 0 푓(푟)퐽0 2휋푟휌 푟푑푟. In words, the important conclusion to take away from this is that the Fourier transform of a radial function is also radial. The formula for 퐹(휌) in terms of 푓(푟) is sometimes called the zero order Hankel transform of 푓(푟) but, again, we understand that it is nothing other than the Fourier transform of a radial function. IJMSET-Advanced Scientific Research Forum (ASRF), All Rights Reserved “ASRF promotes research nature, Research nature enriches the world’s future” 25 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 Definition (Separability in Polar Coordinates) A function ), 푓(푟, 휃) is separable in polar coordinates if it can be written in the form 푓 푟, 휃 = 푓푟 푟 푓휃 휃 .Suppose such an f is circularly symmetric, with 푓휃 휃 = 1. As a specific example, we will consider a 2D circular (or cylinder) function: 1, 푟 < 1 1 푐푖푟푐 푟 = , 푟 = 1 2 0, 푟 > 1 ∞ The Fourier transform of a circularly symmetric function is 퐹(휌, 휙) = 2휋 0 푟퐽0 2휋휌푟 푓푟 푟 푑푟 This is also known as the Hankel transform of order zero and as the Fourier-Bessel transform. 1 2휋 푖푎 휃−휙 The function 퐽 is the zero order Bessel function of the first kind defined as 퐽 푎 = 푒 푑휃 0 0 2휋 0 It oscillates like a damped cosine. Continuing with our specific example, the Fourier transform of 푐푖푟푐 푟 ∞ 1 is Ӻ 푐푖푟푐 푟 = 2휋 0 푟푐푖푟푐 푟 퐽0 2휋휌푟 푑푟 = 2휋 0 푟퐽0 2휋휌푟 푑푟. ′ ′ 1 2휋휌 ′ ′ ′ 퐽1 2휋휌 Substitute 푟 = 2휋휌푟, and 푑푟 = 2휋휌푑푟 to find: Ӻ 푐푖푟푐 푟 = 푟 퐽 푟 푑푟 = , since 2휋 0 0 휌 훼 0 푥퐽0 푥 푑푥 = 훼퐽1 훼 where 퐽1 is the first order Bessel function of the first kind, similar to a damped sinusoid. The function 푠표푚푏 휌 or 푠표푚푏푟푒푟표 (also known as Mexican hat, Bessinc, and jinc) is defined as 푠표푚푏 휌 = 2퐽 2휋휌 1 . Thus, the Fourier transform of 푐푖푟푐 푟 is proportional to a 푠표푚푏푟푒푟표 function of 휌, the radial 2휋휌 coordinate in frequency space: Ӻ 푐푖푟푐 푟 = 휋푠표푚푏 휌 .The first result is that the radial Fourier transform is given by a Hankel transform. Suppose 푓 is a function on 푅푛. Its Fourier transform is −푖휿.퐱 푛 푓 휿 = 푒 푓(퐱)푑 퐱.
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