Numerical study of the nanofluid thin film flow past an unsteady stretching sheet with fractional derivatives using the spectral collocation Chebyshev approximation

2020 ◽  
pp. 2150026
Author(s):  
M. M. Khader
Keyword(s):  
Thin Film ◽  
Fractional Derivatives ◽  
Volume Fraction ◽  
Numerical Study ◽  
Chebyshev Approximation ◽  
Algebraic Equations ◽  
Thin Film Flow ◽  
Spectral Collocation ◽  
Unsteady Stretching Surface ◽  
Fractional Differential

In this work, a mathematical model of fractional-order in fluid will be analyzed numerically to describe and study the influence of thermal radiation on the magnetohydrodynamic flow of nanofluid thin film which moves due to the unsteady stretching surface with viscous dissipation. The set of nonlinear fractional differential equations in the form of velocity, temperature and concentration which describe our proposed problem are tackled through the spectral collocation method based on Chebyshev polynomials of the third-kind. This method reduces the presented model to a system of algebraic equations. The effect of the influence parameters which governs the process of flow and mass heat transfer is discussed. The numerical values of the dimensionless velocity, temperature and concentration are depicted graphically. Also, computations of the values of skin-friction, Nusselt number and Sherwood number have been carried out and presented in the same figures. Finally, our numerical analysis shows that both the magnetic and the unsteadiness parameters can enhance the free surface temperature and nanoparticle volume fraction.

scholarly journals A multiple-step adaptive pseudospectral method for solving multi-order fractional differential equations

2019 ◽  
Vol 8 (1) ◽  
pp. 702-718
Author(s):  
Mahmoud Mashali-Firouzi ◽  
Mohammad Maleki
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Fractional Derivatives ◽  
Pseudospectral Method ◽  
Algebraic Equations ◽  
Computational Accuracy ◽  
Step Size ◽  
Collocation Points ◽  
Fractional Differential ◽  
Multiple Step

Abstract The nonlocal nature of the fractional derivative makes the numerical treatment of fractional differential equations expensive in terms of computational accuracy in large domains. This paper presents a new multiple-step adaptive pseudospectral method for solving nonlinear multi-order fractional initial value problems (FIVPs), based on piecewise Legendre–Gauss interpolation. The fractional derivatives are described in the Caputo sense. We derive an adaptive pseudospectral scheme for approximating the fractional derivatives at the shifted Legendre–Gauss collocation points. By choosing a step-size, the original FIVP is replaced with a sequence of FIVPs in subintervals. Then the obtained FIVPs are consecutively reduced to systems of algebraic equations using collocation. Some error estimates are investigated. It is shown that in the present multiple-step pseudospectral method the accuracy of the solution can be improved either by decreasing the step-size or by increasing the number of collocation points within subintervals. The main advantage of the present method is its superior accuracy and suitability for large-domain calculations. Numerical examples are given to demonstrate the validity and high accuracy of the proposed technique.

Numerical treatment for studying the blood ethanol concentration systems with different forms of fractional derivatives

2020 ◽  
Vol 31 (03) ◽  
pp. 2050044 ◽  
Author(s):  
M. M. Khader ◽  
Khaled. M. Saad
Keyword(s):  
Fractional Derivatives ◽  
Ethanol Concentration ◽  
Algebraic Equations ◽  
Efficient Tool ◽  
Blood Ethanol Concentration ◽  
Fractional Models ◽  
Fractional Differential ◽  
Relative Errors ◽  
The Given ◽  
First Time

The purpose of this paper is to implement an approximate method for obtaining the solution of a physical model called the blood ethanol concentration system. This model can be expressed by a system of fractional differential equations (FDEs). Here, we will consider two forms of the fractional derivative namely, Caputo (with singular kernel) and Atangana–Baleanu–Caputo (ABC) (with nonsingular kernel). In this work, we use the spectral collocation method based on Chebyshev approximations of the third-kind. This procedure converts the given model to a system of algebraic equations. The implementation of the proposed method to solve fractional models in ABC-sense is the first time. We satisfy the efficiency and the accuracy of the given procedure by evaluating the relative errors. The results show that the implemented technique is an easy and efficient tool to simulate the solution of such models.

scholarly journals A New Operational Matrix of Fractional Derivatives to Solve Systems of Fractional Differential Equations via Legendre Wavelets

Mathematics ◽  
2018 ◽  
Vol 6 (11) ◽  
pp. 238 ◽  
Author(s):  
Aydin Secer ◽  
Selvi Altun
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Matrix Method ◽  
Fractional Derivatives ◽  
Operational Matrix ◽  
Operational Matrices ◽  
Algebraic Equations ◽  
Legendre Wavelets ◽  
Operational Matrix Method ◽  
Fractional Differential

This paper introduces a new numerical approach to solving a system of fractional differential equations (FDEs) using the Legendre wavelet operational matrix method (LWOMM). We first formulated the operational matrix of fractional derivatives in some special conditions using some notable characteristics of Legendre wavelets and shifted Legendre polynomials. Then, the system of fractional differential equations was transformed into a system of algebraic equations by using these operational matrices. At the end of this paper, several examples are presented to illustrate the effectivity and correctness of the proposed approach. Comparing the methodology with several recognized methods demonstrates that the advantages of the Legendre wavelet operational matrix method are its accuracy and the understandability of the calculations.

scholarly journals A Thin Film Flow of Nanofluid Comprising Carbon Nanotubes Influenced by Cattaneo-Christov Heat Flux and Entropy Generation

Coatings ◽  
2019 ◽  
Vol 9 (5) ◽  
pp. 296 ◽  
Author(s):  
Dianchen Lu ◽  
Muhammad Ramzan ◽  
Mutaz Mohammad ◽  
Fares Howari ◽  
Jae Dong Chung
Keyword(s):  
Thin Film ◽  
Carbon Nanotubes ◽  
Heat Flux ◽  
Entropy Generation ◽  
Film Flow ◽  
Volume Fraction ◽  
Solid Volume Fraction ◽  
Numerical Approach ◽  
Thin Film Flow ◽  
Dependent Flow

This study aims to scrutinize the thin film flow of a nanofluid comprising of carbon nanotubes (CNTs), single and multi-walled i.e., (SWCNTs and MWCNTs), with Cattaneo-Christov heat flux and entropy generation. The time-dependent flow is supported by thermal radiation, variable source/sink, and magneto hydrodynamics past a linearly stretched surface. The obtained system of equations is addressed by the numerical approach bvp4c of the MATLAB software. The presented results are validated by comparing them to an already conducted study and an excellent synchronization in both results is achieved. The repercussions of the arising parameters on the involved profiles are portrayed via graphical illustrations and numerically erected tables. It is seen that the axial velocity decreases as the value of film thickness parameter increases. It is further noticed that for both types of CNTs, the velocity and temperature distributions increase as the solid volume fraction escalates.

scholarly journals The thin film flow of Al2O3 nanofluid particle over an unsteady stretching surface

2021 ◽  
pp. 101695
Author(s):  
Ramzan Ali ◽  
Azeem Shahzad ◽  
Kaif us Saher ◽  
Zaffar Ellahi ◽  
Tasawar Abbas
Keyword(s):  
Thin Film ◽  
Film Flow ◽  
Thin Film Flow ◽  
Stretching Surface ◽  
Unsteady Stretching Surface ◽  
Al2o3 Nanofluid

Bernstein polynomials for solving fractional heat- and wave-like equations

2012 ◽  
Vol 15 (4) ◽  
Author(s):  
Davood Rostamy ◽  
Kobra Karimi
Keyword(s):  
Numerical Analysis ◽  
Numerical Solution ◽  
Fractional Derivatives ◽  
Bernstein Polynomials ◽  
Operational Matrix ◽  
Algebraic Equations ◽  
Numerical Examples ◽  
Fractional Differential

AbstractIn this paper, a novel numerical analysis is introduced and performed to obtain the numerical solution of the fractional heat- and wave-like equations. A general formulation for the Bernstein fractional derivatives operational matrix is given. In this approach, a truncated Bernstein series together with the Bernstein operational matrix of fractional derivatives are used to reduce the solution of fractional differential problems to the solution of a system of algebraic equations. Numerical examples are considered aiming to demonstrate the validity and applicability of the proposed techniques and to compare with the existing results.

Soret and Dufour effect on the thin film flow over an unsteady stretching surface

2016 ◽  
Author(s):  
Qayyum Shah ◽  
Taza Gul ◽  
Mustafa Bin Mamat ◽  
Waris Khan ◽  
Novan Tofany
Keyword(s):  
Thin Film ◽  
Film Flow ◽  
Thin Film Flow ◽  
Stretching Surface ◽  
Dufour Effect ◽  
Unsteady Stretching Surface

scholarly journals On the Existence and Uniqueness of the Solution of Linear Fractional Differential-Algebraic System

2016 ◽  
Vol 2016 ◽  
pp. 1-9 ◽  
Author(s):  
Zaiyong Feng ◽  
Ning Chen
Keyword(s):  
Existence And Uniqueness ◽  
Fractional Derivatives ◽  
Differential Algebraic Equations ◽  
Equivalent Transformation ◽  
Algebraic Equations ◽  
Differential Algebraic System ◽  
Uniqueness Of The Solution ◽  
Fractional Differential ◽  
Matrix Pair ◽  
Kronecker Canonical Form

The existence and uniqueness of the solution of a new kind of system—linear fractional differential-algebraic equations (LFDAE)—are investigated. Fractional derivatives involved are under the Caputo definition. By using the tool of matrix pair, the LFDAE in which coefficients matrices are both square matrices have unique solution under the condition that coefficients matrices make up a regular matrix pair. With the help of equivalent transformation and Kronecker canonical form of the coefficients matrices, the sufficient condition for existence and uniqueness of the solution of the LFDAE in which coefficients matrices are both not square matrices is proposed later. Two examples are given to justify the obtained theorems in the end.

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