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

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Analytical investigations of the sumudu transform and applications to integral production equations

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.

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Sumudu transform fundamental properties investigations and applications.

Mumps is the most common cause of acquired unilateral deafness in children, in which hearing loss occurs at all auditory frequencies. We use a box model to model hearing loss in children caused by the mumps virus, and since the fractional-order derivative retains the effect of system memory, we use the Caputo–Fabrizio fractional derivative in this modeling. In the beginning, we compute the basic reproduction number R 0 and equilibrium points of the system and investigate the stability of the system at the equilibrium point. By utilizing the Picard–Lindelof technique, we prove the existence an unique solution for given fractional CF -system of hearing loss model and investigate the stability of iterative method by fixed point theory. The optimal control of the system is determined by considering the treatment as a control strategy to reduce the number of infected people. Using the Euler method for the fractional-order Caputo–Fabrizio derivative, the approximate solution of the system is calculated. We present a numerical simulation for the transmission of disease with respect to the transmission rate and the basic reproduction number in two cases R 0 1 and R 0 > 1 . To investigate the effect of fractional order derivative on the behavior and value of each of the variables in Model 2, we calculate the results for several fractional order derivatives and compare the results. Also, considering the importance of reproduction number in the continuation of disease transmission, we analyze the sensitivity of R 0 respect to each of the model parameters and determine the impact of each parameter.

A theoretical study of the Caputo-Fabrizio fractional modeling for hearing loss due to Mumps virus with optimal control

By using the fractional Caputo–Fabrizio derivative, we investigate a new version for the mathematical model of HIV. In this way, we review the existence and uniqueness of the solution for the model by using fixed point theory. We solve the equation by a combination of the Laplace transform and homotopy analysis method. Finally, we provide some numerical analytics and comparisons of the results.

/pdf/analysis-of-the-model-of-hiv-1-infection-of-cd4-cd4-t-cell-3hbj0uhjoi.pdf

Analysis of the model of HIV-1 infection of CD4+$CD4^{+}$ T-cell with a new approach of fractional derivative

In this paper a new integral transform namely Elzaki transform was applied to solve linear ordinary differential equations with constant coefficients.

The new integral transform elzaki transform

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.

Sumudu Transform - a New Integral Transform to Solve Differential Equations and Control Engineering Problems

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.

G. K. Watugala

Papers

Sumudu Transform - a New Integral Transform to Solve Differential Equations and Control Engineering Problems

Sumudu transform: a new integral transform to solve differential equations and control engineering problems