Stokes' first problem for some non-Newtonian fluids: Results and mistakes

The well-known problem of unidirectional plane flow of a fluid in a half-space due to the impulsive motion of the plate it rests upon is discussed in the context of the second-grade and the Oldroyd-B non-Newtonian fluids. The governing equations are derived from the conservation laws of mass and momentum and three correct known representations of their exact solutions given. Common mistakes made in the literature are identified. Simple numerical schemes that corroborate the analytical solutions are constructed.Comment: 10 pages, 2 figures; accepted for publication in Mechanics Research Communications; v2 corrects a few typo

Numerical approximations of fractional differential equations: a Chebyshev pseudo-spectral approach.

Doctoral Degree. University of KwaZulu-Natal, Pietermaritzburg.This study lies at the interface of fractional calculus and numerical methods. Recent studies suggest that fractional differential and integral operators are well suited to model physical phenomena with intrinsic memory retention and anomalous behaviour. The global property of fractional operators presents difficulties in fnding either closed-form solutions or accurate numerical solutions to fractional differential equations. In rare cases, when analytical solutions are available, they often exist only in terms of complex integrals and special functions, or as infinite series. Similarly, obtaining an accurate numerical solution to arbitrary order differential equation is often computationally demanding. Fractional operators are non-local, and so it is practicable that when approximating fractional operators, non-local methods should be preferred. One such non-local method is the spectral method. In this thesis, we solve problems that arise in the ow of non-Newtonian fluids modelled with fractional differential operators. The recurrent theme in this thesis is the development, testing and presentation of tractable, accurate and computationally efficient numerical schemes for various classes of fractional differential equations. The numerical schemes are built around the pseudo{spectral collocation method and shifted Chebyshev polynomials of the first kind. The literature shows that pseudo-spectral methods converge geometrically, are accurate and computationally efficient. The objective of this thesis is to show, among other results, that these features are true when the method is applied to a variety of fractional differential equations. A survey of the literature shows that many studies in which pseudo-spectral methods are used to numerically approximate the solutions of fractional differential equations often to do this by expanding the solution in terms of certain orthogonal polynomials and then simultaneously solving for the coefficients of expansion. In this study, however, the orthogonality condition of the Chebyshev polynomials of the first kind and the Chebyshev-Gauss-Lobatto quadrature are used to numerically find the coefficients of the series expansions. This approach is then applied to solve various fractional differential equations, which include, but are not limited to time{space fractional differential equations, two{sided fractional differential equations and distributed order differential equations. A theoretical framework is provided for the convergence of the numerical schemes of each of the aforementioned classes of fractional differential equations. The overall results, which include theoretical analysis and numerical simulations, demonstrate that the numerical method performs well in comparison to existing studies and is appropriate for any class of arbitrary order differential equations. The schemes are easy to implement and computationally efficient

Accurate Spectral Algorithms for Solving Variable-order Fractional Percolation Equations

A high accurate spectral algorithm for one-dimensional variable-order fractional percolation equations (VO-FPEs) is considered.We propose a shifted Legendre Gauss-Lobatto collocation (SL-GLC) method in conjunction with shifted Chebyshev Gauss-Radau collocation (SC-GR-C) method to solve the proposed problem. Firstly, the solution and its space fractional derivatives are expanded as shifted Legendre polynomials series. Then, we determine the expansion coefficients by reducing the VO-FPEs and its conditions to a system of ordinary differential equations (SODEs) in time. The numerical approximation of SODEs is achieved by means of the SC-GR-C method. The under-study’s problem subjected to the Dirichlet or non-local boundary conditions is presented and compared with the results in literature, which reveals wonderful results

Approximate solutions for solving nonlinear variable-order fractional Riccati differential equations

In this manuscript, we introduce a spectral technique for approximating the variable-order fractional Riccati equation (VO-FRDEs). Firstly, the solution and its space fractional derivatives is expanded as shifted Chebyshev polynomials series. Then we determine the expansion coefficients by reducing the VO-FRDEs and its conditions to a system of algebraic equations. We show the accuracy and applicability of our numerical approach through four numerical examples. &nbsp

MHD free convective flow past a vertical plate

The free convective flow in incompressible viscous fluid past a vertical plate is studied under the presence of magnetic field. The flow is considered along the vertical plate at x-axis in upward direction and y-axis is taken normal to it. The governing equations are written in vector form. Afterwards, the equations are solved numerically using finite element method with automated solution techniques. Later, the effects of magnetic field strength to the velocity and temperature of the fluid are obtained. It is found that for heated plate, the velocity and the temperature of the fluid decreases when the magnetic field strength increases. Meanwhile for cooled plate, the velocity decreases but the temperature increases when the magnetic field strength increases

MHD effects on Casson fluid flow squeezing between parallel plates

We introduce this work by studying the non-Newtonian fluids, which have huge applications in different science fields. We decided to concentrate on taking the time-dependent Casson fluid, which is non-Newtonian, compressed between two flat plates. in fractional form and the magnetohydrodynamic and Darcian flow effects in consideration using the semi-analytical iterative method created by Temimi and Ansari, known as TAM, this method is carefully selected to be suitable for studying the Navier-Stokes model in the modified form to express the studied case mathematically. To simplify the partial differential equations of the system to the nonlinear ordinary differential equation of order four the similarity transformations suggested by Wang (1976) are used. The TAM approach demonstrates a high degree of accuracy, efficiency, and convergence when applied to the resolution of both linear and nonlinear problems, and the results in this article are used to study the effect of the related factors like squeeze number Sq, Casson parameterβ, magnetohydrodynamic parameter Mg and permeability constant Mp and examining the skin friction coefficient effect. The velocity profile is studied numerically, which is tabulated and graphically represented to show and confirm the theoretical study. We can conclude that the success of the proposed method in studying time-dependent Casson fluid, which is non-Newtonian, compressed between two flat plates provides opportunities for additional study and advancements in fluid mechanics using the techniques

International Conference on Mathematical Analysis and Applications in Science and Engineering – Book of Extended Abstracts

The present volume on Mathematical Analysis and Applications in Science and Engineering - Book of Extended Abstracts of the ICMASC’2022 collects the extended abstracts of the talks presented at the International Conference on Mathematical Analysis and Applications in Science and Engineering – ICMA2SC'22 that took place at the beautiful city of Porto, Portugal, in June 27th-June 29th 2022 (3 days). Its aim was to bring together researchers in every discipline of applied mathematics, science, engineering, industry, and technology, to discuss the development of new mathematical models, theories, and applications that contribute to the advancement of scientific knowledge and practice. Authors proposed research in topics including partial and ordinary differential equations, integer and fractional order equations, linear algebra, numerical analysis, operations research, discrete mathematics, optimization, control, probability, computational mathematics, amongst others. The conference was designed to maximize the involvement of all participants and will present the state-of- the-art research and the latest achievements.info:eu-repo/semantics/publishedVersio

An Efficient Numerical Method for Solving Volterra-Fredholm Integro-Differential Equations of Fractional Order by Using Shifted Jacobi-Spectral Collocation Method

The aim of this article is to solve the Volterra-Fredholm integro-differential equations of fractional order numerically by using the shifted Jacobi polynomial collocation method. The Jacobi polynomial and collocation method properties are presented. This technique is used to convert the problem into the solution of linear algebraic equations. The fractional derivatives are considered in the Caputo sense. Numerical examples are given to show the accuracy and reliability of the proposed technique