scholarly journals Integer and Non-Integer Order Study of the GO-W/GO-EG Nanofluids Flow by Means of Marangoni Convection

Symmetry ◽  
2019 ◽  
Vol 11 (5) ◽  
pp. 640 ◽  
Author(s):  
Taza Gul ◽  
Haris Anwar ◽  
Muhammad Altaf Khan ◽  
Ilyas Khan ◽  
Poom Kumam
Keyword(s):  
Differential Equation ◽  
Graphene Oxide ◽  
Ethylene Glycol ◽  
Fractional Order ◽  
Marangoni Convection ◽  
Traditional Approach ◽  
Stable Dispersion ◽  
Fractional Differential ◽  
Predictor Corrector ◽  
The Impact

Characteristically, most fluids are not linear in their natural deeds and therefore fractional order models are very appropriate to handle these kinds of marvels. In this article, we studied the base solvents of water and ethylene glycol for the stable dispersion of graphene oxide to prepare graphene oxide-water (GO-W) and graphene oxide-ethylene glycol (GO-EG) nanofluids. The stable dispersion of the graphene oxide in the water and ethylene glycol was taken from the experimental results. The combined efforts of the classical and fractional order models were imposed and compared under the effect of the Marangoni convection. The numerical method for the non-integer derivative that was used in this research is known as a predictor corrector technique of the Adams–Bashforth–Moulton method (Fractional Differential Equation-12) or shortly (FDE-12). The impact of the modeled parameters were analyzed and compared for both GO-W and GO-EG nanofluids. The diverse effects of the parameters were observed through a fractional model rather than the traditional approach. Furthermore, it was observed that GO-EG nanofluids are more efficient due to their high thermal properties compared with GO-W nanofluids.

Influence of Dynamics Viscosity on the Water Base Graphene Oxide–Ethylene Glycol/Graphene Oxide–Water Nanofluid Flow Over a Stretching Cylinder

2019 ◽  
Vol 8 (8) ◽  
pp. 1661-1667
Author(s):  
Ali Rehman ◽  
Zabidin Salleh ◽  
Taza Gul
Keyword(s):  
Differential Equation ◽  
Graphene Oxide ◽  
Ethylene Glycol ◽  
Order Approximation ◽  
Nonlinear Differential Equations ◽  
Physical Parameters ◽  
Stretching Cylinder ◽  
Homotopy Analysis ◽  
Water Base ◽  
The Impact

This research paper presents the impact of dynamics viscosity of water base GO–EG (graphene oxide–ethylene glycol)/GO–W (graphene oxide–water) nanofluid over a stretching cylinder with non-porous medium. The impact of different parameter for both velocity and temperature profile are displayed and discussed graphically. The similarity transformation are used to convert the partial differential equation to nonlinear ordinary differential equation. The solution of the problem is obtained using the optimal homotopy analysis method (OHAM). (Liao, S. J., 2010. An optimal homotopy-analysis approach for strongly nonlinear differential equations. Communications in Nonlinear Science and Numerical Simulation, 15, pp.2003–2016) used this method for the solution of nonlinear problem and show that this method is quickly convergent to the approximate solution. This method gives us series solution in the form of function, and all the physical parameters of the problem involved in this method. The stability of the problems is also obtained up to the 30th order approximation using the BVPh 2.0 package. The outputs are displayed graphically and discussed. The effects of various parameters on the skin friction coefficient and Nusselt number coefficient of base GO–EG/GO–Ware displayed and discussed.

scholarly journals Mathematical modelling of the mass-spring-damper system - A fractional calculus approach

2012 ◽  
Vol 22 (5) ◽  
pp. 5-11 ◽  
Author(s):  
José Francisco Gómez Aguilar ◽  
Juan Rosales García ◽  
Jesus Bernal Alvarado ◽  
Manuel Guía
Keyword(s):  
Differential Equation ◽  
Fractional Calculus ◽  
Mathematical Modelling ◽  
Fractional Differential Equation ◽  
Fractional Order ◽  
Time Derivative ◽  
System A ◽  
Fractional Differential ◽  
Mass Spring ◽  
Derivatives Of

In this paper the fractional differential equation for the mass-spring-damper system in terms of the fractional time derivatives of the Caputo type is considered. In order to be consistent with the physical equation, a new parameter is introduced. This parameter char­acterizes the existence of fractional components in the system. A relation between the fractional order time derivative and the new parameter is found. Different particular cases are analyzed

Solitary Wave and other Solutions of Nonlinear Space-Time Fractional Differential Equation Systems

2018 ◽  
pp. 515-522
Keyword(s):  
Differential Equation ◽  
Fractional Differential Equation ◽  
Fractional Order ◽  
Expansion Method ◽  
Travelling Wave ◽  
Boussinesq System ◽  
Liouville Derivative ◽  
Fractional Differential ◽  
Fractional System ◽  
Partial Fractional Differential Equation

In this study, we have successfully found some travelling wave solutions of the variant Boussinesq system and fractional system of two-dimensional Burgers' equations of fractional order by using the -expansion method. These exact solutions contain hyperbolic, trigonometric and rational function solutions. The fractional complex transform is generally used to convert a partial fractional differential equation (FDEs) with modified Riemann-Liouville derivative into ordinary differential equation. We showed that the considered transform and method are very reliable, efficient and powerful in solving wide classes of other nonlinear fractional order equations and systems.

scholarly journals Analytical solutions of incommensurate fractional differential equation systems with fractional order $ 1 < \alpha, \beta < 2 $ via bivariate Mittag-Leffler functions

2021 ◽  
Vol 7 (2) ◽  
pp. 2281-2317
Author(s):  
Yong Xian Ng ◽  
◽  
Chang Phang ◽  
Jian Rong Loh ◽  
Abdulnasir Isah ◽  
...  
Keyword(s):  
Differential Equation ◽  
Analytical Solution ◽  
Fractional Differential Equation ◽  
Fractional Order ◽  
Successive Approximations ◽  
Explicit Analytical Solution ◽  
Special Cases ◽  
Fractional Differential ◽  
Alpha Beta ◽  
Published Paper

<abstract><p>In this paper, we derive the explicit analytical solution of incommensurate fractional differential equation systems with fractional order $ 1 &lt; \alpha, \beta &lt; 2 $. The derivation is extended from a recently published paper by Huseynov et al. in <sup>[<xref ref-type="bibr" rid="b1">1</xref>]</sup>, which is limited for incommensurate fractional order $ 0 &lt; \alpha, \beta &lt; 1 $. The incommensurate fractional differential equation systems were first converted to Volterra integral equations. Then, the Mittag-Leffler function and Picard's successive approximations were used to obtain the analytical solution of incommensurate fractional order systems with $ 1 &lt; \alpha, \beta &lt; 2 $. The solution will be simplified via some combinatorial concepts and bivariate Mittag-Leffler function. Some special cases will be discussed, while some examples will be given at the end of this paper.</p></abstract>

scholarly journals Use of fractional calculus in modeling of heat transfer process through external building partitions

2018 ◽  
pp. 61-70
Author(s):  
Ewa Szymanek
Keyword(s):  
Heat Transfer ◽  
Differential Equation ◽  
Mathematical Model ◽  
Heat Transfer Process ◽  
Heat Distribution ◽  
Experiment Data ◽  
Numerical Studies ◽  
External Conditions ◽  
Fractional Differential ◽  
The Impact

This paper is devoted to experimental and numerical studies of heat distribution in an external building bulkhead. It analyzes the variation of temperature across the width of the bulkheads including the impact of changing external conditions. Mathematical model used in the research is formulated based on a fractional differential equation, which was proven to be a useful tool for describing this type of process in previous paper. Numerical results are compared with experiment data for different bulkhead configurations.

The Impact of Viscous Dissipation on the Thin Film Unsteady Flow of GO-EG/GO-W Nanofluids

Mathematics ◽  
2019 ◽  
Vol 7 (7) ◽  
pp. 653 ◽  
Author(s):  
Ali Rehman ◽  
Zabidin Salleh ◽  
Taza Gul ◽  
Zafar Zaheer
Keyword(s):  
Heat Transfer ◽  
Graphene Oxide ◽  
Ethylene Glycol ◽  
Unsteady Flow ◽  
Heat Transfer Enhancement ◽  
Nonlinear Differential Equations ◽  
Physical Parameters ◽  
Transfer Enhancement ◽  
Flexible Surface ◽  
The Impact

The unsteady flow of nanoliquid film over a flexible surface has been inspected. Water and ethylene glycol are used as the base liquids for the graphene oxide platelets. The comparison of two sorts of nanoliquids has been used for heat transfer enhancement applications. The thickness of the nanoliquid film is kept as a variable. The governing equations for the flow problem have been altered into the set of nonlinear differential equations. The BVP 2.0 package has been used for the solution of the problem. The sum of the square residual error has been calculated up to the 10th order approximations. It has been observed that the graphene oxide ethylene glycol based nanofluid (GO-EG) is more efficient for heat transfer enhancement as compared to the graphene oxide water based nanofluid (GO-W). The impact of the physical parameters has been plotted and discussed.

scholarly journals Solution of Some Types for Composition Fractional Order Differential Equations Corresponding to Optimal Control Problems

2018 ◽  
Vol 2018 ◽  
pp. 1-12
Author(s):  
Sameer Qasim Hasan ◽  
Moataz Abbas Holel
Keyword(s):  
Differential Equation ◽  
Optimal Control ◽  
Fractional Differential Equation ◽  
Fractional Order ◽  
Optimal Control Problems ◽  
Chebyshev Approximation ◽  
Approximate Solutions ◽  
Control Problems ◽  
Fractional Differential ◽  
Working Method

The approximate solution for solving a class of composition fractional order optimal control problems (FOCPs) is suggested and studied in detail. However, the properties of Caputo and Riemann-Liouville derivatives are also given with complete details on Chebyshev approximation function to approximate the solution of fractional differential equation with different approach. Also, the relation between Caputo and Riemann-Liouville of fractional derivative took a big role for simplifying the fractional differential equation that represents the constraints of optimal control problems. The approximate solutions are defined on interval [0,1] and are compared with the exact solution of order one which is an important condition to support the working method. Finally, illustrative examples are included to confirm the efficiency and accuracy of the proposed method.

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