Further distinctive investigations of the Sumudu transform
27 Jan 2017-Vol. 1798, Iss: 1, pp 020025
TL;DR: The Sumudu transform of time function f (t) is computed by making the transform variable u as factor of function f(t) and then integrated against exp(−t).
Abstract: The Sumudu transform of time function f (t) is computed by making the transform variable u of Sumudu as factor of function f (t) and then integrated against exp(−t). Being a factor in the original function f (t), becomes f (ut) preserves units and dimension. This preservation property distinguishes Sumudu from other integral transforms. With obtained definition, the related complete set of properties were derived for the Sumudu transform. Framgment of Symbolic C++ program was given for Sumudu computation as series. Also procedure in Maple was given for Sumudu computation in closed form. The Method proposed herein not depends neither on any of homotopy methods such as HPM, HAM nor any of decomposition methods such as ADM.
Citations
More filters
01 Jan 2017
TL;DR: In this article, the Sumudu transform is used to solve fractional differential equations for various values of fractional degrees (α) and various boundary conditions, followed by Stokes-Ekman boundary thickness problem.
Abstract: The Natural transform is used to solve fractional differential equations for various values of fractional degrees \(\alpha \), and various boundary conditions. Fractional diffusion problems solutions are analyzed, followed by Stokes–Ekman boundary thickness problem. Furthermore, the Sumudu transform is applied for fluid flow problems, such as Stokes, Rayleigh, and Blasius, toward obtaining their solutions and corresponding boundary layer thickness.
14 citations
04 Dec 2018
TL;DR: In this article, the Discrete Inverse Sumudu Transform (DIST) multiple shifting properties are used to design a methodology for solving ordinary differential equations, and a DIST Table for elementary functions is provided.
Abstract: In this research article, the Discrete Inverse Sumudu Transform (DIST) multiple shifting properties are used to design a methodology for solving ordinary differential equations. We say ”Discrete” because it acts on the Taylor or Mclaurin series of the function when any. The algorithm applied to solve the Whittaker and Zettl equations and get their exact solutions. A DIST Table for elementary functions is provided.
5 citations
TL;DR: In this paper, the Sumudu transform of Dixon elliptic functions for higher arbitrary powers was used for Hankel determinants calculations by connecting the formal power series (Maclaurin series) and associated continued fractions.
Abstract: In this work, Sumudu transform of Dixon elliptic functions for higher arbitrary powers smN(x,α);N > 1, smN(x,α)cm(x,α); N > 0 and smN(x,α)cm2(x,α);N > 0 by considering modulus α 6= 0 is obtained as three term recurrences and hence expanded as product of quasi associated continued fractions where the coefficients are functions of α. Secondly the coefficients of quasi associated continued fractions are used for Hankel determinants calculations by connecting the formal power series (Maclaurin series) and associated continued fractions. c ©2017 All rights reserved.
3 citations
Cites methods from "Further distinctive investigations ..."
...Symbolic C++ and Maple procedure for Sumudu transform is given in [6]....
[...]
06 Jun 2018
TL;DR: In this paper, Hankel determinants of the Dixon elliptic function with non zero modulus a ≠ 0 for arbitrary powers are derived by product of Quasi C fractions.
Abstract: Sumudu transform of the Dixon elliptic function with non zero modulus a ≠ 0 for arbitrary powers smN(x,a) ; N ≥ 1 ; smN(x,a)cm(x,a) ; N ≥ 0 and smN(x,a)cm2(x,a) ; N ≥ 0 is given by product of Quasi C fractions. Next by assuming denominators of Quasi C fraction to 1 and hence applying Heliermann correspondance relating formal power series (Maclaurin series of Dixon elliptic functions) and regular C fraction, Hankel determinants are calculated and showed by taking a = 0 gives the Hankel determinants of regular C fraction. The derived results were back tracked to the Laplace transform of sm(x,a) ; cm(x,a) and sm(x,a)cm(x,a).
Cites methods from "Further distinctive investigations ..."
...Symbolic C++ program for Sumudu transform given in [8]....
[...]
...Without using any of decomposition, perturbation (or) analysis techniques Sumudu transform of functions calculated by differentiating the function in [8]....
[...]
17 Apr 2019
TL;DR: In this paper, the Sumudu transform of the Dixon elliptic function with non-zero modulus α ≠ 0 for arbitrary powers N is given by the product of quasi C fractions.
Abstract: The Sumudu transform of the Dixon elliptic function with non-zero modulus α ≠ 0 for arbitrary powers N is given by the product of quasi C fractions. Next, by assuming the denominators of quasi C fractions as one and applying the Heliermanncorrespondence relating formal power series (Maclaurin series of the Dixon elliptic function) and the regular C fraction, the Hankel determinants are calculated for the non-zero Dixon elliptic functions and shown by taking α = 0 to give the Hankel determinants of the Dixon elliptic function with zero modulus. The derived results were back-tracked to the Laplace transform of Dixon elliptic functions.
References
More filters
TL;DR: In this article, the Sumudu transform was used to solve an integral production-depreciation problem, where the Laplace transform was applied to solve the problem without resorting to a new frequency domain.
Abstract: The Sumudu transform, whose fundamental properties are
presented in this paper, is little known and not widely used
However, being the theoretical dual to the Laplace
transform, the Sumudu transform rivals it in problem solving
Having scale and unit-preserving properties, the Sumudu transform
may be used to solve problems without resorting to a new
frequency domain Here, we use it to solve an integral
production-depreciation problem
311 citations
TL;DR: In this paper, the Sumudu transform is used to solve problems without resorting to a new frequency domain, which is the theoretical dual to the Laplace transform, and hence ought to rival it in problem solving.
Abstract: The Sumudu transform, whose fundamental properties are presented in this paper, is still not widely known, nor used. Having scale and unit-preserving properties, the Sumudu transform may be used to solve problems without resorting to a new frequency domain. In 2003, Belgacem et al. have shown it to be the theoretical dual to the Laplace transform, and hence ought to rival it in problem solving. Here, using the Laplace-Sumudu duality (LSD), we avail the reader with a complex formulation for the inverse Sumudu transform. Furthermore, we generalize all existing Sumudu differentiation, integration, and convolution theorems in the existing literature. We also generalize all existing Sumudu shifting theorems, and introduce new results and recurrence results, in this regard. Moreover, we use the Sumudu shift theorems to introduce a paradigm shift into the thinking of transform usage, with respect to solving differential equations, that may be unique to this transform due to its unit-preserving properties. Finally, we provide a large and more comprehensive list of Sumudu transforms of functions than is available in the literature.
299 citations
TL;DR: In this article, an application of He's homotopy perturbation method is proposed to compute Laplace transform and the results reveal that the method is very effective and simple.
Abstract: In this paper, an application of He’s homotopy perturbation method is proposed to compute Laplace transform. The results reveal that the method is very effective and simple.
115 citations
Journal Article•
TL;DR: This paper attempts to be the single most comprehensive source about the Sumudu Transform properties, up to date.
Abstract: The Sumudu Transform, herein simply referred to as the Sumudu, was previously firmly established by the author et al.[2003/2005] as the theoretical dual to the Laplace Transform, where from the Laplace-Sumudu Duality (LSD). In fact, due to its units and scale preserving properties, in many instances, the Sumudu may be preferred to its dual for solving problems in engineering mathematics, without leaving the initial argument domain. Many fundamental Sumudu properties were presented in the literature, by this author and others. Aside from reestablishing these with alternative tools, essentially deeper Sumudu properties and connections are analyzed, and new results are presented. As such, this paper attempts to be the single most comprehensive source about the Sumudu Transform properties, up to date.
95 citations
TL;DR: In this article, an application of He's homotopy perturbation method is proposed to compute Sumudu transform, in contrast of usual methods which need integration, requires simple differentiation.
Abstract: In this work, an application of He’s homotopy perturbation method is proposed to compute Sumudu transform. The method, in contrast of usual methods which need integration, requires simple differentiation. The results reveal that the method is very effective and simple. Keywords: Perturbation methods; Homotopy perturbation method; Sumudu transform 1. Introduction Various perturbation methods [1,2] have been widely applied by scientists and engineers to solve nonlinear problems. The traditional perturbation techniques are based on the existence of small parameter. These techniques are so powerful that sometimes small parameter is introduced artificially into a problem having no parameter and then finally set equal to unity to recover the solution of the original problem. He [3-6] proposed a new method called homotopy perturbation method (HPM) in 1998. The HPM, in fact, is a coupling of the traditional perturbation method and homotopy in topology. The HPM method, without demanding a small parameter in equations, deforms continuously to a simple problem which is easily solved. This method yields a very rapid convergence of the solution series in most cases, usually only a few iterations leading to very accurate solutions. This new method was further developed and improved by He and applied to non-linear oscillators with discontinuity [7], asymptotology [8], non-linear wave equations [9], bifurcation of nonlinear problems [10], limit cycle [11], delay-differential equations [[12], and boundary values problems [13]. He’s method is a universal one which can solve various types of non-linear problems. For example, it was applied to the non-Newtonian flow by Siddiqui et al. [14-15], to Volterra’s integro-differential equation by El-Shahed [16], to Helmholtz equation and fifth-order KdV equation by Rafei et al. [17], to nonlinear oscillator by Cai et al. [18], and to compute the Laplace transform by Abbasbandy [19]. A complete review on HPM’s applications is given in [20-21]. Sumudu transform was probably first time introduced by Watugala in his work [22]. Its simple formulation and direct applications to ordinary differential equations immediately sparked interest in this new tool. This new transform was further developed and applied to many problems by various workers. Asiru [23,24] applied to integro-differential equations, Watugala [25,26] extended the transform to two variables with emphasis on solution to partial differential equations and applications to engineering control problem, and its fundamentals properties were established by Belgacem et al. [27-29]. The Sumudu transform has very special and useful properties and can
80 citations