Traveling wave solutions of conformable time-fractional Zakharov–Kuznetsov and Zoomeron equations
TL;DR: In this article, the exact traveling wave solutions of the conformable time-fractional Zakharov-Kuznetsov equation and conformable Time-Fractional Zoomeron equation have been obtained and also solutions have been illustrated.
About: This article is published in Chinese Journal of Physics.The article was published on 2020-04-01. It has received 24 citations till now. The article focuses on the topics: Differential equation & Conformable matrix.
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TL;DR: In this article, exact traveling wave solutions of the conformable differential equations have been examined by means of the wave transformation and properties of conformable derivative (CD), conformable nonlinear Schrodinger equation (CNLSE) has been converted into an integer order differential equation.
48 citations
TL;DR: In this paper, several classes of exact analytical solutions for the time-fractional $$(2+1)$$ -dimensional Ito equation are obtained with the aid of the Mathematica package.
Abstract: In this paper, with the aid of the Mathematica package, several classes of exact analytical solutions for the time-fractional $$(2+1)$$
-dimensional Ito equation are obtained. To analytically tackle the above equation, the Kudryashov simple equation approach and its modified form are applied. Rational, exponential-rational, periodic, and hyperbolic functions with a number of free parameters were represented by the obtained soliton solutions. Graphical illustrations with special choices of free constants and different fractional orders are included for certain acquired solutions. Both approaches include the efficiency, applicability and easy handling of the solution mechanism for nonlinear evolution equations that occur in the various real-life problems.
30 citations
TL;DR: In this paper, the authors obtained the approximate numerical solution of space-time fractional-order reaction-diffusion equation using an efficient technique homotopy perturbation technique using Laplace transform method with fractional order derivatives in Caputo sense.
29 citations
TL;DR: The conformable derivative and adequate fractional complex transform are implemented to discuss the fractional higher-dimensional Ito equation analytically in this article, where the Jacobi elliptic function method and the adequate FCT transform are used.
Abstract: The conformable derivative and adequate fractional complex transform are implemented to discuss the fractional higher-dimensional Ito equation analytically. The Jacobi elliptic function method and ...
28 citations
TL;DR: In this paper, the numerical wave solutions of two fractional biomathematical and statistical physics models (the Kolmogorov-Petrovskii - Piskunov (KPP) equation and the (2 + 1)-dimensional Zoomeron (Z) equation) are investigated.
Abstract: The numerical wave solutions of two fractional biomathematical and statistical physics models (the Kolmogorov—Petrovskii - Piskunov (KPP) equation and the (2 + 1)-dimensional Zoomeron (Z) equation) are investigated in this manuscript. Many novel analytical solutions in different mathematical formulations such as trigonometric, hyperbolic, exponential, and so on can be constructed using the generalized Riccati—expansion analytical scheme and the Caputo—Fabrizio fractional derivative. The fractional nonlinear evolution equation is converted into an ordinary differential equation with an integer order using this fractional operator. The obtained solution is used to describe the transmission of a preferred allele and the nonlinear interaction of moving waves, and the relative wave mode’s amplitude dynamic. To illustrate the fractional examined models, several drawings are explained in two dimensions and density plots.
24 citations
References
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Book•
19 May 1993
TL;DR: The Riemann-Liouville Fractional Integral Integral Calculus as discussed by the authors is a fractional integral integral calculus with integral integral components, and the Weyl fractional calculus has integral components.
Abstract: Historical Survey The Modern Approach The Riemann-Liouville Fractional Integral The Riemann-Liouville Fractional Calculus Fractional Differential Equations Further Results Associated with Fractional Differential Equations The Weyl Fractional Calculus Some Historical Arguments.
7,643 citations
TL;DR: A new definition of fractional derivative and fractional integral is given and it is shown that it is the most natural definition, and the most fruitful one.
2,068 citations
Book•
01 Jan 2006TL;DR: Fractional calculus (integral and differential operations of noninteger order) is not often used to model biological systems, which is surprising because the methods of fractional calculus, when defined as a Laplace or Fourier convolution product, are suitable for solving many problems in biomedical research.
Abstract: Fractional calculus (integral and differential operations of noninteger order) is not often used to model biological systems. Although the basic mathematical ideas were developed long ago by the mathematicians Leibniz (1695), Liouville (1834), Riemann (1892), and others and brought to the attention of the engineering world by Oliver Heaviside in the 1890s, it was not until 1974 that the first book on the topic was published by Oldham and Spanier. Recent monographs and symposia proceedings have highlighted the application of fractional calculus in physics, continuum mechanics, signal processing, and electromagnetics, but with few examples of applications in bioengineering. This is surprising because the methods of fractional calculus, when defined as a Laplace or Fourier convolution product, are suitable for solving many problems in biomedical research. For example, early studies by Cole (1933) and Hodgkin (1946) of the electrical properties of nerve cell membranes and the propagation of electrical signals are well characterized by differential equations of fractional order. The solution involves a generalization of the exponential function to the Mittag-Leffler function, which provides a better fit to the observed cell membrane data. A parallel application of fractional derivatives to viscoelastic materials establishes, in a natural way, hereditary integrals and the power law (Nutting/Scott Blair) stress-strain relationship for modeling biomaterials. In this review, I will introduce the idea of fractional operations by following the original approach of Heaviside, demonstrate the basic operations of fractional calculus on well-behaved functions (step, ramp, pulse, sinusoid) of engineering interest, and give specific examples from electrochemistry, physics, bioengineering, and biophysics. The fractional derivative accurately describes natural phenomena that occur in such common engineering problems as heat transfer, electrode/electrolyte behavior, and sub-threshold nerve propagation. By expanding the range of mathematical operations to include fractional calculus, we can develop new and potentially useful functional relationships for modeling complex biological systems in a direct and rigorous manner.
1,609 citations
TL;DR: The basic concepts in this new simple interesting fractional calculus called conformable fractional derivative are set and the fractional versions of chain rule, exponential functions, Gronwall's inequality, integration by parts, Taylor power series expansions, Laplace transforms and linear differential systems are proposed and discussed.
1,331 citations