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N-Transform Properties and Application
Abstract
A new integral transform similar to Laplace and Sumudu transforms is introduced. It converges to both transforms just by changing variables. A table is presented for existing Laplace and Sumudu transforms. To explain the use of N-Transform in linear differential equation, an example of unsteady fluid flow over a plane wall is presented.
... Later on, many researchers used Adomian decomposition [19][20][21] and its various modifications [22,23] for the solutions of various differential equations. In [24], the authors introduced the concept of natural transform. The natural transform directly produced the Laplace and Sumudu transforms only by providing unity values to the transformation parameters. ...
... Note that H-difference some time does not exist, but if the H-difference and the limit in the Remark 1 exist for a fuzzy function, then the fuzzy function is gH-differentiable. Definition 6. [24] Let N(u(ϑ)) = U(s, r) where N is a natural transform with transform parameters s and r defined as follows, ...
In this manuscript, we will discuss the solutions of Goursat problems with fuzzy boundary conditions involving gH-differentiability. The solutions to these problems face two main challenges. The first challenge is to deal with the two types of fuzzy gH-differentiability: (i)-differentiability and (ii)-differentiability. The sign of coefficients in Goursat problems and gH-differentiability produces sixteen possible cases. The existing literature does not afford a solution method that addresses all the possible cases of this problem. The second challenge is the mixed derivative term in Goursat problems with fuzzy boundary conditions. Therefore, we propose to discuss the solutions of fuzzy Goursat problems with gH-differentiability. We will discuss the solutions of fuzzy Goursat problems in series form with natural transform and Adomian decompositions. To demonstrate the usability of the established solution methods, we will provide some numerical examples.
... For the purpose of solving differential and integral equations, one of the most used methods is the integral transform method. [1][2][3][4][5][6][7][8][9][10][11] In the literature, the Laplace transform is also the most often employed method. 12 Watugala 13 first suggested the Laplace-Carson transform in 1993. ...
... The N-transform can solve the unsteady fluid flow through a flat wall problem by changing the variables to offer both Laplace and Sumudu integral transforms. [5][6][7][8] Most linear and nonlinear approaches that exist in a variety of scientific areas, such as population models, fluid mechanics, solid state physics, plasma physics, and chemical kinetics, can be modeled using differential equations. As a result, it is still very difficult to obtain exact or approximate solutions to linear and nonlinear differential equations in applied mathematics and physics, which calls for new methods. ...
One of the most noteworthy differential equations of the first order is the Riccati differential equation. It is applied in various branches of mathematics, including algebraic geometry, physics, and conformal mapping theory. The J -transform Adomian decomposition method is employed in the current study to find exact solutions for different kinds of nonlinear differential equations. We give thorough detailed proofs for new theorems related to the J -transform technique. The Adomian decomposition method and the J -transform method serve as the foundation for this technique. For certain differential equations, the theoretical analysis of the J -transform Adomian decomposition method is examined and is computed using readily computed terms. Our findings are contrasted with exact solutions found in the literature that were produced using alternative techniques. The significant features of the J -transform Adomian decomposition method are described in the article. It has been shown that the J -transform Adomian decomposition method is very efficient, useful, and adaptable to a broad variety of linear and nonlinear differential equations. Most of the symbolic and numerical calculations were performed using Mathematica.
... The Laplace transform is a well-established and extensively utilized integral transform, particularly in solving problems in engineering sciences. Several new integral transforms have been developed recently, including the Sumudu, Kamal, Mohand, Natural, Elzaki, Aboodh and Jafari Transforms [1][2][3][4][5][6][7]. These newer transforms often employ exponential-type kernels and are considered generalized forms of the Laplace transform. ...
In this study, we introduce a novel modified general integral transform known as the JSN transform, which offers several advantages over the Laplace and other integral transforms with exponential kernels. Fundamental results of the JSN transform of the Caputo fractional derivative are discussed. Furthermore, we develop a novel hybrid technique called the JSN Fractional Residual Power Series Method (JSN FRPSM). This new technique incorporates the JSN transform with the existing Residual Power Series Method. To demonstrate the efficiency of the proposed hybrid technique in solving fractional differential equations, we apply it to various fractional differential equations encountered in science and engineering. Statistical and error analyses are conducted to validate the results obtained through the proposed method. Additionally, the series solutions obtained via the proposed method are illustrated graphically.
... During this time, Laplace introduced the Laplace Transform, a very effective tool for finding the solution of ordinary or partial differential equations with suitable initial and boundary value problems [12]. In 1993, G.K. Watugala introduced the Sumudu Transform [4,3], which is a simple variant of the Laplace Transform, and in 2008, Zafar Hayat Khan [8] introduced an integral transform named the Natural Transform, which is similar to the Laplace and Sumudu Transforms. The Natural Transform converges to both the Laplace Transform and the Sumudu Transform by changing variables. ...
... Different kinds of integral transformations include the Soham Transform [11], Kushare Transform [12], Fourier transformation [5], Natural transformation [14], and new general integral transform [4], among others. Many fields, especially engineering, physics, applied mathematics, and most other sciences, may profit from these kinds of changes. ...
This is an Open Access Journal / article distributed under the terms of the Creative Commons Attribution License (CC BY-NC-ND 3.0) which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. All rights reserved. Partial differential equations can be solved using many different kinds of methods. Among these, the integral transform proves to be an extremely efficient method for solving partial differential equations. Thus, a double-Formable transform shall be used to solve boundary value problems in this paper. Here, we'll solve Klein-Gordon equations, homogeneous and non-homogeneous telegraph equations, and partial differential equations of first and second order etc.
... In 2019, a new transform has been introduced and it is called Shehu transform, it is derived from Fourier, Laplace, Sumudu and Natural transform. Shehu transform applied to solve ODEs and PDEs and defined as (5) Historically, there are many other transforms also obtained from Fourier, Laplace and Sumudu integral transform. However, most of the existing integral transforms have some limitations and cannot be used directly to solve nonlinear problems or many complex mathematical models. ...
In this paper, we introduced a new Laplace-type integral transform called RAHMOH transform which is generalized of Laplace and Sumudu transforms for solving ordinary and partial differential equations. We presented its existence, inverse transform and some essential properties with some theorems and applications.
... The Adomian natural decomposition method (ANDM) offers infinite series solutions, such that if the exact solution exists, it may converge to a closed-form solution. It is based on the natural transform technique (NTM) [23][24][25][26] and the Adomian decomposition method (ADM) [27]. We solve fractional nonlinear as well as linear differential equations using the Adomian decomposition natural method (ANDM) [8], which was inspired and motivated by the ongoing research in this field. ...
In the current study, we solve the nonlinear time fractional‐order enzyme kinematic system and the linear time fractional blood alcohol level using an effective method called the Adomian natural decomposition method (ANDM), which is a combination of the Adomian decomposition method (ADM) and the natural transform method (NTM). On the basis of the Banach fixed point theorem, we offer proofs for the existence and uniqueness theorems as they apply to nonlinear ordinary differential equations. Using the current method, approximate analytical solutions of nonlinear time fractional‐order enzyme kinematic system and exact solutions to the linear time fractional‐order blood alcohol level problem have been obtained. The outcomes indicate that the ANDM is highly efficient and useful. The method outlined in this paper is intended to be applied in subsequent work to handle comparable nonlinear problems connected to fractional calculus.
... In recent years, transform has been shown not only in different fields of mathematics but also in other majors such as physics, chemistry, and technology, [2]. The following decades produced a lot of new transformations, for example, the Shehu transformation was also created and applied to find the approximate solutions, [3]; the Sumudu transformation is used to solve delay fractional Bagley-Torvik equations shown combining efficient method, [4]; the Elzaki transformation is created from Laplace transform and applied in finding solutions of partial differential equations, [5], [6]; the natural transform (N-transform) was invented by, [7], and was performed in solving unstable fluid flow problems; the Aboodh transforms, [8], and new α−Integral Laplace transform, [9], derived from Laplace transform are useful tools combining the Homotopy method for finding the exact solution of various differential equations. Following this stream, Pourreza transforms, [10], and Mohand transforms, [11], set a new expansion to solve higher differential equations with constant coefficients. ...
The development of technology has supported effective tools in industrial machines and set up the remarkable phase that serves well-being such as kinetic energy, kinetic movement, and nuclear energy. Applied mathematics has also contributed valuable procedures in various fields of these sciences, especially the creation of transformation. With practical relevance, a new general integral (NGI) transform has also shown a crucial role in the same pragmatic methods. In this paper, the NGI transform using the combination of Padé approximation including continued fraction expansions (CFE) has been used to attain approximate solutions of space-time fractional telegraph equations by directly getting the inverse transform.
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