On New Solutions of Time-Fractional Wave Equations Arising in Shallow Water Wave Propagation

Jena, Rajarama Mohan; Chakraverty, Snehashish; Băleanu, Dumitru

doi:10.3390/math7080722

Cited by 36 publications

(12 citation statements)

References 21 publications

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“…Recently, the presence of temporal memory kernels on descriptions of physical systems has been an interesting mathematical tool to investigate complexity in nature [55][56][57][58][59]. In this sense, a series of news kernels was proposed to approach mathematical models that have become limited.…”

Section: Preliminaries Concepts: Fractional Derivativesmentioning

confidence: 99%

Mittag–Leffler Memory Kernel in Lévy Flights

Santos

2019

Mathematics

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In this article, we make a detailed study of some mathematical aspects associated with a generalized Lévy process using fractional diffusion equation with Mittag–Leffler kernel in the context of Atangana–Baleanu operator. The Lévy process has several applications in science, with a particular emphasis on statistical physics and biological systems. Using the continuous time random walk, we constructed a fractional diffusion equation that includes two fractional operators, the Riesz operator to Laplacian term and the Atangana–Baleanu in time derivative, i.e., a A B D t α ρ ( x , t ) = K α , μ ∂ x μ ρ ( x , t ) . We present the exact solution to model and discuss how the Mittag–Leffler kernel brings a new point of view to Lévy process. Moreover, we discuss a series of scenarios where the present model can be useful in the description of real systems.

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Section: Preliminaries Concepts: Fractional Derivativesmentioning

confidence: 99%

Mittag–Leffler Memory Kernel in Lévy Flights

Santos

2019

Mathematics

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“…In this section, we will illustrate the necessary steps for the FRDTM. [41][42][43][44][45] First, let us consider a function of (n + 1) variables…”

Section: Fractional Reduced Differential Transform Methodsmentioning

confidence: 99%

On the wave solutions of time‐fractional Sawada‐Kotera‐Ito equation arising in shallow water

Jena

Sedighi

2020

Math Methods in App Sciences

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A widescale description of various phenomena in science and engineering such as physics, chemical, acoustics, control theory, finance, economics, mechanical engineering, civil engineering, and social sciences is well described by nonlinear fractional differential equations (NLFDEs). In turbulence, fluid dynamics, and nonlinear biological systems, applications of NLFDEs can also be found. NLFDEs are believed to be powerful tools to describe real-world problems more precisely than the differential equation of the integer-order. In this research, we have used the fractional reduced differential transform method (FRDTM) to find the solution of the time-fractional Sawada-Kotera-Ito seventh-order equation. The novelty of the FRDTM is that it does not require any discretization, transformation, perturbation, or any restrictive conditions. In addition, compared to other methods, this approach needs less calculation. For special cases of an integer and noninteger orders, computed results are compared with existing results. Present results are in good agreement with the existing solutions. Here, the fractional derivatives are considered in the Caputo sense. Convergence analysis of the results has also been studied with the increasing number of terms of the solution.

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“…In order to demonstrate the elementary idea of RDTM, let us take the nonlinear PDE of the form as follows with initial condition (IC) [35,38,39] Original function Transformed form…”

Section: Fractional Reduced Differential Transform Methodsmentioning

confidence: 99%

On the solution of time-fractional coupled system of partial differential equations

Jena

Chakraverty

2019

SN Appl. Sci.

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In this research, a newly proposed Laplace Padé reduced differential transform method (LPRDTM) has been applied for solving a coupled system of partial differential equations with time-fractional derivatives. Here the fractional derivatives are defined in the Caputo sense. This is a hybrid technique which is the coupling of laplace transform, Padé approximant, and the reduced differential transform method. Two examples are solved using the present method, and the obtained results are compared with the analytical solutions in terms of the tables, which recommend that LPRDTM provides exact solutions with a few iterations. The main benefit of applying this method is that it does not require any assumption, perturbation, and discretization for solving the governing fractional PDEs. Also, the computation time of this method is less compared to other techniques.

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On New Solutions of Time-Fractional Wave Equations Arising in Shallow Water Wave Propagation

Cited by 36 publications

References 21 publications

Mittag–Leffler Memory Kernel in Lévy Flights

Mittag–Leffler Memory Kernel in Lévy Flights

On the wave solutions of time‐fractional Sawada‐Kotera‐Ito equation arising in shallow water

On the solution of time-fractional coupled system of partial differential equations

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