Novel technique to investigate the convergence analysis of the tempered fractional natural transform method applied to diffusion equations

Our main goal in the current research work is to explore proofs of newly discovered theorems related to tempered fractional calculus. We use a new mechanism, namely, the natural tempered fractional transformation method, which can be used to solve important tempered fractional differential equations that are important in science, such as the linear and nonlinear tempered fractional differential equations. Indeed, we found new exact solutions to both tempered fractional Langevin and Vasicek differential equations and an exact solution for the famous tempered fractional diffusion equation. The new method makes it easier to do the calculations than with the traditional methods, and all you need is a few simple manipulations. Our new research technique is straightforward to use and highly accurate.

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“…In this section, important facts about the tempered fractional N-transformations will be presented; see previous research. 37,38,39 Fact 1.…”

Section: Tempered Fractional Natural Transform (Tfnt) Theoriesmentioning

confidence: 99%

Theories of tempered fractional calculus applied to tempered fractional Langevin and Vasicek equations

Obeidat

Rawashdeh

2023

Math Methods in App Sciences

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“…As a result of that, obtaining accurate or approximate solutions to different types of equations in physics and applied mathematics is still a significant issue. Various powerful mathematical methods, such as the Adomian decomposition method, the natural decomposition method (Rawashdeh, 8 Rawashdeh and Maitama, 9 Obeidat and Bentil, 10 Obeidat and Bentil, 11 Obeidat and Bentil 12 ), the reduced differential transform method (RDTM) (Rawashdeh, 13 Rawashdeh, 14 Rawashdeh and Obeidat, 15 Rawashdeh 16 ), the natural transform method (Belgacem and Silambarasan, 17 Hussain and Belgacern, 18 Khan and Khan 19 ), The Summudu transform method, (Belgacem et al 20 ) Laplace decomposition method (Spiegel 21 ) are still useful tools for solving these equations. By using the natural Adomian decomposition method, we don't have to worry about discretization, linearization, or thinking of any restrictive assumptions like with differential transform method and Homotopy perturbation method.…”

Section: Introductionmentioning

confidence: 99%

On theories of natural decomposition method applied to system of nonlinear differential equations in fluid mechanics

Obeidat

Rawashdeh

2023

Advances in Mechanical Engineering

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In shallow waters, the Wu-Zhang (WZ) system describes the (1+1)-dimensional dispersive long wave in two horizontal directions, which is important for the engineering community. This paper presents proofs for various theorems and shows that the natural decomposition method (NDM) solves systems of linear and nonlinear ordinary and partial differential equations under proper initial conditions, such as the Wu-Zhang system. We use a combination of two methods, namely the natural transform method to deal with the linear terms and the Adomian decomposition method to deal with the nonlinear terms. Several examples of linear and nonlinear systems (ODEs and PDEs) are given, including the Wu-Zhang (WZ) system. The present approach, which has numerous applications in the science and engineering fields, is a great alternative to the many existing methods for solving systems of differential equations. It also holds great promise for additional real-world applications.

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“…Fractional differential equations (FDE) have been successfully employed over the last few centuries to describe a variety of structures that can be used to create their mathematical models. A few of these models are also used in engineering and technology, including fractional calculus, which has been used in fluid dynamics [5], bioengineering [6], electromagnetism [7], modeling the transfer of heat in heterogeneous media [8], anomalous diffusion in [9][10][11], and the crossover phenomena is characterized by the manipulative control activities of human operators in their interactions with fractional order plants; for more information, see Martínez-García and Gordon [12]. Additionally, boundary value problems (BVPs) of FDEs are crucial and have an impact on many contemporary applied science procedures.…”

Section: Introductionmentioning

confidence: 99%

Using the decomposition method to solve the fractional order temperature distribution equation: A new approach

Rawashdeh

Obeidat

Ababneh

2023

Math Methods in App Sciences

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Due to its importance in science, finding both exact and approximate solutions to fractional partial differential equations with boundary conditions is important for the research community. The natural decomposition method (NDM), which is based on the natural transformation method (NTM) and the Adomian decomposition method, is modified in this study to produce exact and approximate solutions for boundary value problems (BVPs) of partial differential equations (PDEs) with fractional coefficients. In addition, we present an exact solution to the temperature distribution in a slab constructed of materials with variable thermal conductivity's combined convection–radiation lumped system. We present these findings as numerical tables and graphs that show the convergence and stability rates. The study demonstrates that this approach is effective since it is simple to apply and produces reliable findings. We are the first to use this approach for such applications, as far as we are aware. Additionally, this method is applicable to a sizable class of BVPs for ordinary differential equations (ODEs) and PDEs.

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Novel technique to investigate the convergence analysis of the tempered fractional natural transform method applied to diffusion equations

Cited by 14 publications

References 18 publications

Theories of tempered fractional calculus applied to tempered fractional Langevin and Vasicek equations

Theories of tempered fractional calculus applied to tempered fractional Langevin and Vasicek equations

On theories of natural decomposition method applied to system of nonlinear differential equations in fluid mechanics

Using the decomposition method to solve the fractional order temperature distribution equation: A new approach

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