scholarly journals Numerical Investigation on the Swimming of Gyrotactic Microorganisms in Nanofluids through Porous Medium over a Stretched Surface

Mathematics ◽  
2020 ◽  
Vol 8 (3) ◽  
pp. 380 ◽  
Author(s):  
Anwar Shahid ◽  
Hulin Huang ◽  
Muhammad Mubashir Bhatti ◽  
Lijun Zhang ◽  
Rahmat Ellahi
Keyword(s):  
Relaxation Method ◽  
Linearization Method ◽  
Variable Coefficients ◽  
Darcy Law ◽  
Gyrotactic Microorganisms ◽  
Chebyshev Spectral Collocation ◽  
Linear Differential ◽  
Simple Notion ◽  
Chebyshev Spectral Collocation Method ◽  
Local Linearization Method

In this article, the effects of swimming gyrotactic microorganisms for magnetohydrodynamics nanofluid using Darcy law are investigated. The numerical results of nonlinear coupled mathematical model are obtained by means of Successive Local Linearization Method. This technique is based on a simple notion of the decoupling systems of equations utilizing the linearization of the unknown functions sequentially according to the order of classifying the system of governing equations. The linearized equations, that developed a sequence of linear differential equations along with variable coefficients, were solved by employing the Chebyshev spectral collocation method. The convergence speed of the SLLM technique can be willingly upgraded by successive applying over relaxation method. The comparison of current study with available published literature has been made for the validation of obtained results. It is found that the reported numerical method is in perfect accord with the said similar methods. The results are displayed through tables and graphs.

scholarly journals On the bivariate spectral quasi-linearization method for solving the two-dimensional Bratu problem

Open Physics ◽  
2018 ◽  
Vol 16 (1) ◽  
pp. 554-562
Author(s):  
Hillary Muzara ◽  
Stanford Shateyi ◽  
Gerald Tendayi Marewo
Keyword(s):  
Collocation Method ◽  
Linearization Method ◽  
Spectral Collocation Method ◽  
Two Dimensional ◽  
Weighted Residual Method ◽  
Spectral Collocation ◽  
Finite Differences Method ◽  
Chebyshev Spectral Collocation ◽  
Bratu Problem ◽  
Chebyshev Spectral Collocation Method

AbstractIn this paper, a bivariate spectral quasi-linearization method is used to solve the highly non-linear two dimensional Bratu problem. The two dimensional Bratu problem is also solved using the Chebyshev spectral collocation method which uses Kronecker tensor products. The bivariate spectral quasi-linearization method and Chebyshev spectral collocation method solutions converge to the lower branch solution. The results obtained using the bivariate spectral quasi-linearization method were compared with results from finite differences method, the weighted residual method and the homotopy analysis method in literature. Tables and graphs generated to present the results obtained show a close agreement with known results from literature.

scholarly journals Study of Activation Energy on the Movement of Gyrotactic Microorganism in a Magnetized Nanofluids Past a Porous Plate

Processes ◽  
2020 ◽  
Vol 8 (3) ◽  
pp. 328 ◽  
Author(s):  
Muhammad Mubashir Bhatti ◽  
Anwar Shahid ◽  
Tehseen Abbas ◽  
Sultan Z Alamri ◽  
Rahmat Ellahi
Keyword(s):  
Porous Plate ◽  
Linearization Method ◽  
Computational Technique ◽  
Nanoparticle Concentration ◽  
Nonlinear Functions ◽  
Gyrotactic Microorganisms ◽  
Local Linearization ◽  
Relaxation Parameters ◽  
Gyrotactic Microorganism ◽  
Local Linearization Method

The present study deals with the swimming of gyrotactic microorganisms in a nanofluid past a stretched surface. The combined effects of magnetohydrodynamics and porosity are taken into account. The mathematical modeling is based on momentum, energy, nanoparticle concentration, and microorganisms’ equation. A new computational technique, namely successive local linearization method (SLLM), is used to solve nonlinear coupled differential equations. The SLLM algorithm is smooth to establish and employ because this method is based on a simple univariate linearization of nonlinear functions. The numerical efficiency of SLLM is much powerful as it develops a series of equations which can be subsequently solved by reutilizing the data from the solution of one equation in the next one. The convergence was improved through relaxation parameters in the study. The accuracy of SLLM was assured through known methods and convergence analysis. A comparison of the proposed method with the existing literature has also been made and found an excellent agreement. It is worth mentioning that the successive local linearization method was found to be very stable and flexible for resolving the issues of nonlinear magnetic materials processing transport phenomena.

Chebyshev spectral method for solving a class of local and nonlocal elliptic boundary value problems

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Harendra Singh
Keyword(s):  
Boundary Value Problems ◽  
Elliptic Boundary ◽  
Boundary Value ◽  
High Accuracy ◽  
Elliptic Boundary Value Problems ◽  
Strong Nonlinearity ◽  
Elliptic Boundary Value ◽  
Chebyshev Spectral Collocation ◽  
Chebyshev Spectral Method ◽  
Chebyshev Spectral Collocation Method

Abstract This paper deals with a class of Bratu’s type, Troesch’s and nonlocal elliptic boundary value problems. Due to strong nonlinearity and presence of parameter δ, it is very difficult to solve these problems. Here we solve these classes of important equations using the Chebyshev spectral collocation method. We have provided the convergence of the proposed approximate method. The trueness of the method is shown by applying it to some illustrative examples. Results are compared with some known methods to highlight its neglectable error and high accuracy.

scholarly journals STUDYING THE STABILITY BY USING LOCAL LINEARIZATION METHOD

2020 ◽  
Vol 2 (1) ◽  
pp. 27-31
Author(s):  
Abdulghafoor Jasim Salim ◽  
Kais Ismail Ebrahem ◽  
Suhirman
Keyword(s):  
Singular Point ◽  
Limit Cycle ◽  
Autoregressive Model ◽  
Stable Limit Cycle ◽  
Linearization Method ◽  
Stable Limit ◽  
Local Linearization ◽  
Non Linear ◽  
The Stability ◽  
Local Linearization Method

Abstract: In this paper we study the stability of one of a non linear autoregressive model with trigonometric term  by using local linearization method proposed by Tuhro Ozaki .We find the singular point ,the stability of the singular point and the limit cycle. We conclude  that the proposed model under certain conditions have a non-zero singular point which is  a asymptotically salable ( when  0 ) and have an  orbitaly stable limit cycle . Also we give some examples in order to explain the method. Key Words : Non-linear Autoregressive model; Limit cycle; singular point; Stability.

scholarly journals A special interpretation of the concept constant breadth for a space curve

2019 ◽  
Vol 23 (Suppl. 1) ◽  
pp. 371-382
Author(s):  
Tuba Agirman-Aydin
Keyword(s):  
Variable Coefficients ◽  
Space Curve ◽  
Normal Vector ◽  
Exit Point ◽  
Constant Breadth ◽  
Normal Vectors ◽  
Linear Differential ◽  
Curve Of Constant Breadth ◽  
Definition Of ◽  
Curvature And Torsion

The definition of curve of constant breadth in the literature is made by using tangent vectors, which are parallel and opposite directions, at opposite points of the curve. In this study, normal vectors of the curve, which are parallel and opposite directions are placed at the exit point of the concept of curve of constant breadth. In this study, on the concept of curve of constant breadth according to normal vector is worked. At the conclusion of the study, is obtained a system of linear differential equations with variable coefficients characterizing space curves of constant breadth according to normal vector. The coefficients of this system of equations are functions depend on the curvature and torsion of the curve. Then is obtained an approximate solution of this system by using the Taylor matrix collocation method. In summary, in this study, a different interpretation is made for the concept of space curve of constant breadth, the first time. Then this interpretation is used to obtain a characterization. As a result, this characterization we?ve obtained is solved.

scholarly journals Monte-Carlo for solving large linear systems of ordinary differential equations

2021 ◽  
Vol 8 (1) ◽  
pp. 37-48
Author(s):  
Sergey M. Ermakov ◽  
◽  
Maxim G. Smilovitskiy ◽  
Keyword(s):  
Monte Carlo ◽  
Markov Chain ◽  
Differential Equations ◽  
Integral Equations ◽  
Virtual Machines ◽  
Variable Coefficients ◽  
Fredholm Type ◽  
Type Integral ◽  
Constant Coefficients ◽  
Linear Differential

Monte-Carlo approach towards solving Cauchy problem for large systems of linear differential equations is being proposed in this paper. Firstly, a quick overlook of previously obtained results from applying the approach towards Fredholm-type integral equations is being made. In the main part of the paper, a similar method is being applied towards a linear system of ODE. It is transformed into an equivalent system of Volterra-type integral equations, which relaxes certain limitations being present due to necessary conditions for convergence of majorant series. The following theorems are being stated. Theorem 1 provides necessary compliance conditions that need to be imposed upon initial and transition distributions of a required Markov chain, for which an equality between estimate’s expectation and a desirable vector product would hold. Theorem 2 formulates an equation that governs estimate’s variance, while theorem 3 states a form for Markov chain parameters that minimise the variance. Proofs are given, following the statements. A system of linear ODEs that describe a closed queue made up of ten virtual machines and seven virtual service hubs is then solved using the proposed approach. Solutions are being obtained both for a system with constant coefficients and time-variable coefficients, where breakdown intensity is dependent on t. Comparison is being made between Monte-Carlo and Rungge Kutta obtained solutions. The results can be found in corresponding tables.

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