scholarly journals LOCAL AND NONLOCAL DIFFERENTIAL OPERATORS: A COMPARATIVE STUDY OF HEAT AND MASS TRANSFERS IN MHD OLDROYD-B FLUID WITH RAMPED WALL TEMPERATURE

Fractals ◽  
2020 ◽  
Vol 28 (08) ◽  
pp. 2040033 ◽  
Author(s):  
MUHAMMAD BILAL RIAZ ◽  
ABDON ATANGANA ◽  
THABET ABDELJAWAD
Keyword(s):  
Wall Temperature ◽  
Analytical Solutions ◽  
Fractional Derivatives ◽  
Differential Operators ◽  
Classical Mechanics ◽  
Rate Of Change ◽  
Dynamical Process ◽  
New Class ◽  
Mass Transfers ◽  
And Inversion

Study of heat and mass transfers is carried out for MHD Oldroyd-B fluid (OBF) over an infinite vertical plate having time-dependent velocity and with ramped wall temperature and constant concentration. It is proven in many already published articles that the heat and mass transfers do not really or always follow the classical mechanics process that is known as memoryless process. Therefore, the model using classical differentiation based on the rate of change cannot really replicate such dynamical process very accurately, thus, a different concept of differentiation is needed to capture such process. Very recently, a new class of differential operators were introduced and have been recognized to be efficient in capturing processes following the power-law, the decay law and the crossover behaviors. For the study of heat and mass transfers, we applied the newly introduced differential operators to model such flow and compare the results with integer-order derivative. Laplace transform and inversion algorithms are used for all the cases to find analytical solutions and to predict the influences of different parameters. The obtained analytical solutions were plotted for different values of fractional orders [Formula: see text] and [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text] on the velocity field. In comparison, Atangana–Baleanu (ABC) fractional derivatives are found to be the best to explain the memory effects than the classical, Caputo (C) and Caputo–Fabrizio (CF) fractional derivatives. Some calculated values for Nusselt number and Sherwood number are presented in tables. Moreover, from the present solutions, the already published results were found as limiting cases.

Effect of Magnetic Field with Parabolic Motion on Fractional Second Grade Fluid

2021 ◽  
Vol 5 (4) ◽  
pp. 163
Author(s):  
Nazish Iftikhar ◽  
Muhammad Bilal Riaz ◽  
Jan Awrejcewicz ◽  
Ali Akgül
Keyword(s):  
Mass Transfer ◽  
Heat And Mass Transfer ◽  
Analytical Solutions ◽  
Differential Operators ◽  
Fluid Velocity ◽  
Rate Of Change ◽  
Second Grade ◽  
Dynamical Process ◽  
Second Grade Fluid ◽  
Grade Fluid

This paper is an analysis of the flow of magnetohydrodynamics (MHD) second grade fluid (SGF) under the influence of chemical reaction, heat generation/absorption, ramped temperature and concentration and thermodiffusion. The fluid was made to flow through a porous medium. It has been proven in many already-published articles that heat and mass transfer do not always follow the classical mechanics process that is known as memoryless process. Therefore, the model using classical differentiation based on the rate of change cannot really replicate such a dynamical process very accurately; thus, a different concept of differentiation is needed to capture such a process. Very recently, new classes of differential operators were introduced and have been recognized to be efficient in capturing processes following the power law, the decay law and the crossover behaviors. For the study of heat and mass transfer, we applied the newly introduced differential operators to model such flow. The equations for heat, mass and momentum are established in the terms of Caputo (C), Caputo–Fabrizio (CF) and Atangana–Baleanu in Caputo sense (ABC) fractional derivatives. The Laplace transform, inversion algorithm and convolution theorem were used to derive the exact and semi-analytical solutions for all cases. The obtained analytical solutions were plotted for different values of existing parameters. It is concluded that the fluid velocity shows increasing behavior for κ, Gr and Gm, while velocity decreases for Pr and M. For Kr, both velocity and concentration curves show decreasing behavior. Fluid flow accelerates under the influence of Sr and R. Temperature and concentration profiles increase for Sr and R. Moreover, the ABC fractional operator presents a larger memory effect than C and CF fractional operators.

A Numerical Scheme for a Class of Parametric Problem of Fractional Variational Calculus

2012 ◽  
Vol 7 (2) ◽  
Author(s):  
Om P. Agrawal ◽  
M. Mehedi Hasan ◽  
X. W. Tangpong
Keyword(s):  
Analytical Solutions ◽  
Numerical Scheme ◽  
Fractional Derivatives ◽  
Optimization Problems ◽  
Practical Interest ◽  
Variational Calculus ◽  
Functional Minimization ◽  
Orthogonal Functions ◽  
Order Problem ◽  
Parametric Problem

Fractional derivatives (FDs) or derivatives of arbitrary order have been used in many applications, and it is envisioned that in the future they will appear in many functional minimization problems of practical interest. Since fractional derivatives have such properties as being non-local, it can be extremely challenging to find analytical solutions for fractional parametric optimization problems, and in many cases, analytical solutions may not exist. Therefore, it is of great importance to develop numerical methods for such problems. This paper presents a numerical scheme for a linear functional minimization problem that involves FD terms. The FD is defined in terms of the Riemann-Liouville definition; however, the scheme will also apply to Caputo derivatives, as well as other definitions of fractional derivatives. In this scheme, the spatial domain is discretized into several subdomains and 2-node one-dimensional linear elements are adopted to approximate the solution and its fractional derivative at point within the domain. The fractional optimization problem is converted to an eigenvalue problem, the solution of which leads to fractional orthogonal functions. Convergence study of the number of elements and error analysis of the results ensure that the algorithm yields stable results. Various fractional orders of derivative are considered, and as the order approaches the integer value of 1, the solution recovers the analytical result for the corresponding integer order problem.

FOURIER ANALYSIS OVER HYPERBOLIC ALGEBRA, PSEUDO-DIFFERENTIAL OPERATORS, AND HYPERBOLIC DEFORMATION OF CLASSICAL MECHANICS

2007 ◽  
Vol 10 (03) ◽  
pp. 421-438 ◽  
Author(s):  
ANDREI KHRENNIKOV
Keyword(s):  
Quantum Mechanics ◽  
Fourier Analysis ◽  
Poisson Bracket ◽  
Commutative Algebra ◽  
Differential Operators ◽  
Classical Mechanics ◽  
Complex Numbers ◽  
Two Dimensional ◽  
Quantum Deformation ◽  
Pseudo Differential Operators

We develop Fourier analysis over hyperbolic algebra (the two-dimensional commutative algebra with the basis e1 = 1, e2 = j, where j2 = 1). We demonstrated that classical mechanics has, besides the well-known quantum deformation over complex numbers, another deformation — so-called hyperbolic quantum mechanics. The classical Poisson bracket can be obtained as the limit h → 0 not only of the ordinary Moyal bracket, but also a hyperbolic analogue of the Moyal bracket.

scholarly journals Generalized fractional differential ring

2021 ◽  
Vol 5 (1) ◽  
pp. 279-287
Author(s):  
Zeinab Toghani ◽  
◽  
Luis Gaggero-Sager ◽  
Keyword(s):  
Fractional Derivative ◽  
Infinite Number ◽  
Fractional Derivatives ◽  
Differential Operators ◽  
Differential Ring ◽  
Fractional Differential

There are many possible definitions of derivatives, here we present some and present one that we have called generalized that allows us to put some of the others as a particular case of this but, what interests us is to determine that there is an infinite number of possible definitions of fractional derivatives, all are correct as differential operators each of them must be properly defined its algebra. We introduce a generalized version of fractional derivative that extends the existing ones in the literature. To those extensions it is associated a differentiable operator and a differential ring and applications that shows the advantages of the generalization. We also review the different definitions of fractional derivatives and it is shown how the generalized version contains the previous ones as a particular cases.

scholarly journals New Analytical Solutions of Fractional Symmetric Regularized-Long-Wave Equation

2020 ◽  
Vol 66 (3 May-Jun) ◽  
pp. 297
Author(s):  
Mehmet Senol
Keyword(s):  
Wave Equation ◽  
Analytical Solutions ◽  
Fractional Derivatives ◽  
Algebraic Method ◽  
Symbolic Calculation ◽  
Solution Sets ◽  
Flow Models ◽  
Long Wave ◽  
Regularized Long Wave Equation ◽  
Fractional Differential

In this study, new extended direct algebraic method is successfully implemented to acquire new exact wave solution sets for symmetric regularized-long-wave (SRLW) equation which arise in long water flow models. By the help of Mathematica symbolic calculation package, the method produced a great number of analytical solutions. We also presented a few graphical illustrations for some surfaces. The fractional derivatives are considered in the conformable sense. All of the solutions were checked by substitution to ensure the reliability of the method. Obtained results confirm that the method is straightforward, powerful and effective method to attain exact solutions for nonlinear fractional differential equations. Therefore, the method is a good candidate to take part in the existing literature.

scholarly journals Modified Decomposition Method with New Inverse Differential Operators for Solving Singular Nonlinear IVPs in First- and Second-Order PDEs Arising in Fluid Mechanics

2014 ◽  
Vol 2014 ◽  
pp. 1-7
Author(s):  
Nemat Dalir
Keyword(s):  
Fluid Mechanics ◽  
Analytical Solutions ◽  
Initial Value Problems ◽  
Decomposition Method ◽  
Differential Operators ◽  
Second Order ◽  
Nonlinear Pdes ◽  
Initial Value ◽  
Exact Analytical Solutions ◽  
First Order

Singular nonlinear initial-value problems (IVPs) in first-order and second-order partial differential equations (PDEs) arising in fluid mechanics are semianalytically solved. To achieve this, the modified decomposition method (MDM) is used in conjunction with some new inverse differential operators. In other words, new inverse differential operators are developed for the MDM and used with the MDM to solve first- and second-order singular nonlinear PDEs. The results of the solutions by the MDM together with new inverse operators are compared with the existing exact analytical solutions. The comparisons show excellent agreement.

A Universal Predictor-corrector Algorithm for Numerical Simulation of Generalized Fractional Differential Equations

2021 ◽  
Author(s):  
Zaid Odibat
Keyword(s):  
Numerical Simulation ◽  
Fractional Derivatives ◽  
Fractional Integral ◽  
Differential Operators ◽  
Numerical Solutions ◽  
Fractional Operators ◽  
Initial Value ◽  
Generalized Fractional Integral ◽  
Fractional Differential ◽  
Predictor Corrector

Abstract This study introduces some remarks on generalized fractional integral and differential operators, that generalize some familiar fractional integral and derivative operators, with respect to a given function. We briefly explain how to formulate representations of generalized fractional operators. Then, mainly, we propose a predictor-corrector algorithm for the numerical simulation of initial value problems involving generalized Caputo-type fractional derivatives with respect to another function. Numerical solutions of some generalized Caputo-type fractional derivative models have been introduced to demonstrate the applicability and efficiency of the presented algorithm. The proposed algorithm is expected to be widely used and utilized in the field of simulating fractional-order models.

Fractional Derivative Consideration on Nonlinear Viscoelastic Dynamical Behavior Under Statical Pre-Displacement

2005 ◽  
Author(s):  
Masataka Fukunaga ◽  
Nobuyuki Shimizu ◽  
Hiroshi Nasuno
Keyword(s):  
Fractional Derivative ◽  
Fractional Derivatives ◽  
Dynamical Behavior ◽  
Variable Coefficients ◽  
Rate Of Change ◽  
Order Derivative ◽  
Nonlinear Viscoelastic ◽  
Higher Order Derivative ◽  
The Difference ◽  
Different Order

Nonlinear fractional calculus model for the viscoelastic material is examined for oscillation around the off-equilibrium point. The model equation consists of two terms of different order fractional derivatives. The lower order derivative characterizes the slow process, and the higher order derivative characterizes the process of rapid oscillation. The measured difference in the order of the fractional derivative of the material, that the order is higher when the material is rapidly oscillated than when it is slowly compressed, is partly attributed to the difference in the frequency dependence between the two fractional derivatives. However, it is found that there could be possibility for the variable coefficients of the two terms with the rate of change of displacement.

scholarly journals Local Fractional Series Expansion Method for Solving Wave and Diffusion Equations on Cantor Sets

2013 ◽  
Vol 2013 ◽  
pp. 1-5 ◽  
Author(s):  
Ai-Min Yang ◽  
Xiao-Jun Yang ◽  
Zheng-Biao Li
Keyword(s):  
Differential Equations ◽  
Series Expansion ◽  
Analytical Solutions ◽  
Fractional Derivatives ◽  
Expansion Method ◽  
Cantor Sets ◽  
Diffusion Equations ◽  
And Diffusion ◽  
Series Expansion Method

We proposed a local fractional series expansion method to solve the wave and diffusion equations on Cantor sets. Some examples are given to illustrate the efficiency and accuracy of the proposed method to obtain analytical solutions to differential equations within the local fractional derivatives.

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