Numerical analysis for the fractional diffusion and fractional Buckmaster equation by the two-step Laplace Adam-Bashforth method

Jain²,

et al. 2018

Front. Phys.

Self Cite

In this paper, the operators of fractional integration introduced by Marichev-Saigo-Maeda involving Appell's function F 3 (•) are applied, and several new image formulas of generalized Lommel-Wright function are established. Also, by implementing some integral transforms on the resulting formulas, few more image formulas have been presented. We can conclude that all derived results in our work generalize numerous well-known results and are capable of yielding a number of applications in the theory of special functions.

Section: Introductionmentioning

confidence: 99%

A Remark on the Fractional Integral Operators and the Image Formulas of Generalized Lommel–Wright Function

Jain²,

et al. 2018

Front. Phys.

Self Cite

“…To find the answer of this question, the mathematicians have managed to open a new window of opportunities to improve the mathematical modeling of real world problems, which has given birth to many new questions and intriguing results. These newly established results have numerous implementation in many areas of engineering [1,2], such as fractional-order Buck master and diffusion problems [3], fractional-order telegraph model [4,5], fractional KdV-Burger-Kuramoto equation [6], fractal vehicular traffic flow [7], fractional Drinfeld-Sokolov-Wilson equation [8], fractional-order anomalous sub-diffusion model [9], fractional design of hepatitis B virus [10], fractional modeling chickenpox disease [11], fractional blood ethanol concentration model [12], fractional model for tuberculosis [13], fractional vibration equation [14], fractional Black-Scholes option pricing equations [15], fractionally damped beams [16], fractionally damped coupled system [17], fractional-order heat, wave and diffusion equations [18,19], fractional order pine wilt disease model [20], fractional diabetes model [21] etc.…”

Section: Introductionmentioning

confidence: 95%

Analytical Solutions of (2+Time Fractional Order) Dimensional Physical Models, Using Modified Decomposition Method

et al. 2019

In this article, a new analytical technique based on an innovative transformation is used to solve (2+time fractional-order) dimensional physical models. The proposed method is the hybrid methodology of Shehu transformation along with Adomian decomposition method. The series form solution is obtained by using the suggested method which provides the desired rate of convergence. Some numerical examples are solved by using the proposed method. The solutions of the targeted problems are represented by graphs which have confirmed closed contact between the exact and obtained solutions of the problems. Based on the novelty and straightforward implementation of the method, it is considered to be one of the best analytical techniques to solve linear and non-linear fractional partial differential equations.

“…(19) and (20) the Laplace transform method L{f (t)}(q) =f (q) = ∞ 0 f (t)e -qt dt, will be applied by using the corresponding initial and boundary conditions from Eqs. (17) and (18) to develop exact analytical solutions for the velocity and temperature profiles.…”

Section: Solution Of the Problemmentioning

confidence: 99%

Application of fractional differential equations to heat transfer in hybrid nanofluid: modeling and solution via integral transforms

2019

This article deals with the generalization of natural convection flow of Cu -Al 2 O 3 -H 2 O hybrid nanofluid in two infinite vertical parallel plates. To demonstrate the flow phenomena in two parallel plates of hybrid nanofluids, the Brinkman type fluid model together with the energy equation is considered. The Caputo-Fabrizio fractional derivative and the Laplace transform technique are used to developed exact analytical solutions for velocity and temperature profiles. The general solutions for velocity and temperature profiles are brought into light through numerical computation and graphical representation. The obtained results show that the velocity and temperature profiles show dual behaviors for 0 < α < 1 and 0 < β < 1 where α and β are the fractional parameters. It is noticed that, for a shorter time, the velocity and temperature distributions decrease with increasing values of the fractional parameters, whereas the trend reverses for a longer time. Moreover, it is found that the velocity and temperature profiles oppositely behave for the volume fraction of hybrid nanofluids. MSC: 34A08; 76R10