scholarly journals Multiple Fractional Solutions for Magnetic Bio-Nanofluid Using Oldroyd-B Model in a Porous Medium with Ramped Wall Heating and Variable Velocity

2020 ◽  
Vol 10 (11) ◽  
pp. 3886 ◽  
Author(s):  
Muhammad Saqib ◽  
Ilyas Khan ◽  
Yu-Ming Chu ◽  
Ahmad Qushairi ◽  
Sharidan Shafie ◽  
...  
Keyword(s):  
Porous Medium ◽  
Temperature Field ◽  
Velocity Field ◽  
Fractional Derivatives ◽  
Fluid Velocity ◽  
Fractional Operators ◽  
Caputo Fractional Derivatives ◽  
Wall Heating ◽  
The Laplace Transform ◽  
Fractional Analysis

Three different fractional models of Oldroyd-B fluid are considered in this work. Blood is taken as a special example of Oldroyd-B fluid (base fluid) with the suspension of gold nanoparticles, making the solution a biomagnetic non-Newtonian nanofluid. Based on three different definitions of fractional operators, three different models of the resulting nanofluid are developed. These three operators are based on the definitions of Caputo (C), Caputo–Fabrizio (CF), and Atnagana–Baleanu in the Caputo sense (ABC). Nanofluid is taken over an upright plate with ramped wall heating and time-dependent fluid velocity at the sidewall. The effects of magnetohydrodynamic (MHD) and porous medium are also considered. Triple fractional analysis is performed to solve the resulting three models, based on three different fractional operators. The Laplace transform is applied to each problem separately, and Zakian’s numerical algorithm is used for the Laplace inversion. The solutions are presented in various graphs with physical arguments. Results are computed and shown in various plots. The empirical results indicate that, for ramped temperature, the temperature field is highest for the ABC derivative, followed by the CF and Caputo fractional derivatives. In contrast, for isothermal temperature, the temperature field of C-derivative is higher than the CF and ABC derivatives, respectively. It was noticed that the velocity field for the ABC derivative is higher than the CF and Caputo fractional derivatives for ramped velocity. However, the velocity field for the Caputo fractional derivative is lower than the ABC and CF for isothermal velocity.

scholarly journals On the General Solutions of Some Non-Homogeneous Div-Curl Systems with Riemann–Liouville and Caputo Fractional Derivatives

2021 ◽  
Vol 5 (3) ◽  
pp. 117
Author(s):  
Briceyda B. Delgado ◽  
Jorge E. Macías-Díaz
Keyword(s):  
Fractional Order ◽  
Fractional Derivatives ◽  
Decomposition Theorem ◽  
Helmholtz Decomposition ◽  
Fractional Operators ◽  
Gradient Vector ◽  
Caputo Fractional Derivatives ◽  
General Solutions ◽  
Derivatives Of ◽  
Fractional Space

In this work, we investigate analytically the solutions of a nonlinear div-curl system with fractional derivatives of the Riemann–Liouville or Caputo types. To this end, the fractional-order vector operators of divergence, curl and gradient are identified as components of the fractional Dirac operator in quaternionic form. As one of the most important results of this manuscript, we derive general solutions of some non-homogeneous div-curl systems that consider the presence of fractional-order derivatives of the Riemann–Liouville or Caputo types. A fractional analogous to the Teodorescu transform is presented in this work, and we employ some properties of its component operators, developed in this work to establish a generalization of the Helmholtz decomposition theorem in fractional space. Additionally, right inverses of the fractional-order curl, divergence and gradient vector operators are obtained using Riemann–Liouville and Caputo fractional operators. Finally, some consequences of these results are provided as applications at the end of this work.

scholarly journals On the Lauwerier formulation of the temperature field problem in oil strata

2005 ◽  
Vol 2005 (10) ◽  
pp. 1577-1588 ◽  
Author(s):  
Lyubomir Boyadjiev ◽  
Ognian Kamenov ◽  
Shyam Kalla
Keyword(s):  
Porous Medium ◽  
Temperature Field ◽  
Boundary Value Problem ◽  
Heat Equation ◽  
Laplace Transform ◽  
Special Functions ◽  
Integral Form ◽  
Field Problem ◽  
Fractional Heat Equation ◽  
The Laplace Transform

The paper is concerned with the fractional extension of the Lauwerier formulation of the problem related to the temperature field description in a porous medium (sandstone) saturated with oil (strata). The boundary value problem for the fractional heat equation is solved by means of the Caputo differintegration operatorD∗(α)of order0<α≤1and the Laplace transform. The solution is obtained in an integral form, where the integrand is expressed in terms of a convolution of two special functions of Wright type.

scholarly journals Influence of Inclined MHD on Unsteady Flow of Generalized Maxwell Fluid with Fractional Derivative between Two Inclined Coaxial Cylinders through a Porous Medium

2020 ◽  
pp. 1705-1714
Author(s):  
Sundos Bader ◽  
Suad Naji Kadhim ◽  
Ahmed. M. Abdulhadi
Keyword(s):  
Porous Medium ◽  
Velocity Field ◽  
Fractional Derivative ◽  
Maxwell Fluid ◽  
Hankel Transform ◽  
Analytic Solutions ◽  
Circular Cylinders ◽  
Physical Parameters ◽  
The Laplace Transform ◽  
Generalized Maxwell Fluid

"This paper presents a study of inclined magnetic field on the unsteady rotating flow of a generalized Maxwell fluid with fractional derivative between two inclined infinite circular cylinders through a porous medium. The analytic solutions for velocity field and shear stress are derived by using the Laplace transform and finite Hankel transform in terms of the generalized G functions. The effect of the physical parameters of the problem on the velocity field is discussed and illustrated graphically.

Initialization of Fractional Differential Equations: Theory and Application

2007 ◽  
Author(s):  
Carl F. Lorenzo ◽  
Tom T. Hartley
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Fractional Derivatives ◽  
Companion Paper ◽  
Laplace Transforms ◽  
Fractional Integrals ◽  
Fractional Operators ◽  
The Laplace Transform ◽  
Fractional Differential ◽  
Derivatives Of

It has been known that the initialization of fractional operators requires time-varying functions, a complicating factor. This paper simplifies the process of initialization of fractional differential equations by deriving Laplace transforms for the initialized fractional integral and derivative that generalize those for the integer-order operators. A companion paper in this conference determines the Laplace transforms for initialized fractional integrals of any order and fractional derivatives of order less than one. This paper extends the theory for the Laplace transform of the derivative to higher order and provides applications.

scholarly journals A Study of Chemically Reactive Species and Thermal Radiation Effects on an Unsteady MHD Free Convection Flow Through a Porous Medium Past a Flat Plate with Ramped Wall Temperature

2017 ◽  
Vol 22 (4) ◽  
pp. 945-964
Author(s):  
K. K. Pandit ◽  
D. Sarma ◽  
S. I. Singh
Keyword(s):  
Porous Medium ◽  
Free Convection ◽  
Thermal Radiation ◽  
Flat Plate ◽  
Radiation Effects ◽  
Fluid Velocity ◽  
Practical Interest ◽  
Convection Flow ◽  
Laplace Transform Technique ◽  
The Laplace Transform

Abstract An investigation of the effects of a chemical reaction and thermal radiation on unsteady MHD free convection heat and mass transfer flow of an electrically conducting, viscous, incompressible fluid past a vertical infinite flat plate embedded in a porous medium is carried out. The flow is induced by a general time-dependent movement of the vertical plate, and the cases of ramped temperature and isothermal plates are studied. An exact solution of the governing equations is obtained in closed form by the Laplace Transform technique. Some applications of practical interest for different types of plate motions are discussed. The numerical values of fluid velocity, temperature and species concentration are displayed graphically whereas the numerical values of skin friction, Nusselt number and Sherwood number are presented in a tabular form for various values of pertinent flow parameters for both ramped temperature and isothermal plates.

scholarly journals A Novel Attractive Algorithm for Handling Systems of Fractional Partial Differential Equations

2021 ◽  
Vol 20 ◽  
pp. 524-539
Author(s):  
Mohammad Alaroud ◽  
Yousef Al-Qudah
Keyword(s):  
Fractional Derivatives ◽  
Applied Mathematics ◽  
Convergent Series ◽  
Initial Value ◽  
Caputo Fractional Derivatives ◽  
Approximate Analytical Solutions ◽  
The Laplace Transform ◽  
Residual Power Series ◽  
Analysis Of Results ◽  
Residual Power

The purpose of this work is to provide and analyzed the approximate analytical solutions for certain systems of fractional initial value problems (FIVPs) under the time-Caputo fractional derivatives by means of a novel attractive algorithm, called the Laplace residual power series (LRPS) algorithm. It combines the Laplace transform operator and the RPS algorithm. The proposed algorithm produces the fractional series solutions in the Laplace space based upon basically on the limit concept and then transforming bake them to original spaces to get a rapidly convergent series approximate solution. To validate the efficiency, accuracy, and applicability of the proposed algorithm, two illustrative examples are performed. Obtained solutions are simulated graphically and numerically. The analysis of results reached shows that the proposed algorithm is applicable, effective, and very fast in determining the solutions for many fractional problems arising in the various areas of applied mathematics

Solution of inverse coefficient problem for fractional anomalous diffusion equation

2012 ◽  
Vol 9 (2) ◽  
pp. 65-70
Author(s):  
E.V. Karachurina ◽  
S.Yu. Lukashchuk
Keyword(s):  
Anomalous Diffusion ◽  
Fractional Derivatives ◽  
Descent Method ◽  
Extremum Problem ◽  
Steepest Descent Method ◽  
Coefficient Problem ◽  
Caputo Fractional Derivatives ◽  
Inverse Coefficient Problem ◽  
Residual Functional ◽  
Inverse Coefficient

An inverse coefficient problem is considered for time-fractional anomalous diffusion equations with the Riemann-Liouville and Caputo fractional derivatives. A numerical algorithm is proposed for identification of anomalous diffusivity which is considered as a function of concentration. The algorithm is based on transformation of inverse coefficient problem to extremum problem for the residual functional. The steepest descent method is used for numerical solving of this extremum problem. Necessary expressions for calculating gradient of residual functional are presented. The efficiency of the proposed algorithm is illustrated by several test examples.

Numerical study of the flow of a mixture of reacting gases in an inhomogeneous catalyst layer

2003 ◽  
Vol 3 ◽  
pp. 246-254
Author(s):  
C.I. Mikhaylenko ◽  
S.F. Urmancheev
Keyword(s):  
Velocity Field ◽  
Chemical Reactions ◽  
Porous Layer ◽  
Computational Experiment ◽  
Numerical Study ◽  
Fluid Velocity ◽  
Catalyst Layer ◽  
Granular Catalyst ◽  
Fluid Velocity Field

The behavior of a liquid flowing through a fixed bulk porous layer of a granular catalyst is considered. The effects of the nonuniformity of the fluid velocity field, which arise when the surface of the layer is curved, and the effect of the resulting inhomogeneity on the speed and nature of the course of chemical reactions are investigated by the methods of a computational experiment.

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