# Sumudu transform and the solution of integral equation of convolution type

01 Nov 2001-International Journal of Mathematical Education in Science and Technology (INTERNATIONAL JOURNAL OF MATHEMATICAL EDUCATION IN SCIENCE AND TECHNOLOGY)-Vol. 32, Iss: 6, pp 906-910

TL;DR: The convolution theorem for the Sumudu transform of a function which can be expressed as a polynomial or a convergent infinite series is proved and its applicability demonstrated in solving convolution type integral equations as mentioned in this paper.

Abstract: The convolution theorem for the Sumudu transform of a function which can be expressed as a polynomial or a convergent infinite series is proved and its applicability demonstrated in solving convolution type integral equations.

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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

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TL;DR: The fundamental purpose of the present paper is to apply an effective numerical algorithm based on the mixture of homotopy analysis technique, Sumudu transform approach and homotopes to obtain the approximate solution of a nonlinear fractional Drinfeld–Sokolov–Wilson equation.

122 citations

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TL;DR: In this paper, the Sumudu transform of a special function f (t) with a corresponding sumudu transformation F (u) has been studied and the effect of shifting the parameter t in the function f(t) by τ on the transform F(u) is analyzed.

Abstract: This note discusses the general properties of the Sumudu transform and the Sumudu transform of special functions. For any function f (t) with corresponding Sumudu transform F (u), the effect of shifting the parameter t in f (t) by τ on the Sumudu transform F (u) is found. Also obtained are the effect of the multiplication of any function f (t) by a power of t and the division of the function f (t) by t on the Sumudu transform F (u). For any periodic function f (t) with periodicity T > 0 the Sumudu transform is easily derived. Illustrations are provided with Abel's integral equation, an integro-differential equation, a dynamic system with delayed time signals and a differential dynamic system.

109 citations

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Fırat University

^{1}TL;DR: In this paper, the Sumudu transform was used to solve nonhomogeneous fractional ordinary differential equations (FODEs) and then the solutions were used to form two-dimensional (2D) graphs.

Abstract: We introduce the rudiments of fractional calculus and the consequent applications of the Sumudu transform on fractional derivatives. Once this connection is firmly established in the general setting, we turn to the application of the Sumudu transform method (STM) to some interesting nonhomogeneous fractional ordinary differential equations (FODEs). Finally, we use the solutions to form two-dimensional (2D) graphs, by using the symbolic algebra package Mathematica Program 7.

105 citations

### Cites methods from "Sumudu transform and the solution o..."

...The applications followed in three consecutive papers by Asiru dealing with the convolution-type integral equations and the discrete dynamic systems [8, 9]....

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JECRC University

^{1}TL;DR: The discussed fractional blood alcohol model yields important and useful results to interpolate new information in the direction of medical environment and shows the new features of composite fractional derivative in the discussed model.

Abstract: The present article deals with certain new and interesting features of fractional blood alcohol model associated with powerful Hilfer fractional operator. The solution of the model depends on three parameters such as (i) the initial concentration of alcohol in stomach after ingestion (ii) the rate of alcohol absorption into the blood stream (ii) the rate at which the alcohol is metabolized by the liver. By employing Sumudu transform algorithm the analytic results of the concentration of alcohol in stomach and the concentration of alcohol in the blood are analyzed. The general solution of concentration of alcohol in stomach and the concentration of alcohol in the blood are demonstrated in the form of extended Mittag-Leffler function. The effect of fractional parameter on concentration of alcohol in stomach and the concentration of alcohol in the blood are shown in graphical form. The comparative study for both the concentrations shows the new features of composite fractional derivative in the discussed model. The discussed fractional blood alcohol model yields important and useful results to interpolate new information in the direction of medical environment.

95 citations

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01 Jan 1992

TL;DR: u and F (u) are no longer dummies but can be treated as replicas of t and f (t) and can be expressed using same respective units, and therefore one can check the consistency of units of a differential equation even after the Sumudu transform.

Abstract: It is possible to solve differential equations, integral equations, and control engineering problems by a transformation in which the differentiation and integration of f(t) in the t-domain is made equivalent to division and multiplication of F(u) by u in the u-domain. The new transformation which is called the Sumudu transformation possesses many interesting properties which make the visualization of the transformation process easier to a newcomer. Some of the properties of the Sumudu transformation are: (1) The unit-step function in t-domain is transformed to unity in u-domain. (2) Scaling of f (t) in t-domain is equivalent to the scaling of F (u) by the same scale factor, and this is true even for negative scale factors. (3) The limit of f (t) as t tends to zero is equal to the limit of F (u) as u tends to zero. (4) The slope of f (t) at t=0 is equal to the slope of F (u) at u = 0. Thus u and F (u) are no longer dummies but can be treated as replicas of t and f (t) and can be expressed using same respective units, and therefore one can check the consistency of units of a differential equation even after the Sumudu transform.

440 citations

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TL;DR: The Sumudu transform as discussed by the authors is a new integral transform that makes its visualization easier and has many interesting properties, such as: (1) the differentiation and integration in the tdomain is equivalent to division and multiplication of the transformed function F(u) by uin the udomain.

Abstract: A new integral transform called the Sumudu transform is introduced. This transform possesses many interesting properties which make its visualization easier. Some of these properties are: (1) The differentiation and integration in the t‐domain is equivalent to division and multiplication of the transformed function F(u)by uin the u‐domain. (2) The unit‐step function in the t‐domain is transformed to unity in the u‐domain. (3) Scaling of the function f(t)in the t‐domain is equivalent to scaling of F(u) in the u‐domain by the same scale factor. (4) The limit of f(t) as ttends to zero is equal to the limit of F(u)as utends to zero. (5) For several cases, the limit of F(t)as ttends to infinity is the same as the limit of F(u)as u tends to infinity. (6) The slope of the function f(t) at t =0is the same as the slope of F(u) at u = 0. Hence uand F(u)are no longer dummies but can be treated as replicas of tand f(t).It is even possible to express uand F(u)using the units of tand f(t) respectively.

400 citations

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TL;DR: In this paper, the Sumudu transform of partial derivatives is derived, and its applicability demonstrated using three different partial differential equations (PDEs) is demonstrated with respect to three different PDEs.

Abstract: The Sumudu transform of partial derivatives is derived, and its applicability demonstrated using three different partial differential equations.

114 citations