Mathematics

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17 pages, 683 KiB

Open AccessArticle

Mixed Convection Flow of Powell–Eyring Nanofluid near a Stagnation Point along a Vertical Stretching Sheet

by Nadhirah Abdul Halim and Noor Fadiya Mohd Noor

Mathematics 2021, 9(4), 364; https://0-doi-org.brum.beds.ac.uk/10.3390/math9040364 - 11 Feb 2021

Cited by 28 | Viewed by 2056

A stagnation-point flow of a Powell–Eyring nanofluid along a vertical stretching surface is examined. The buoyancy force effect due to mixed convection is taken into consideration along with the Brownian motion and thermophoresis effect. The flow is investigated under active and passive controls [...] Read more.

A stagnation-point flow of a Powell–Eyring nanofluid along a vertical stretching surface is examined. The buoyancy force effect due to mixed convection is taken into consideration along with the Brownian motion and thermophoresis effect. The flow is investigated under active and passive controls of nanoparticles at the surface. The associating partial differential equations are converted into a set of nonlinear, ordinary differential equations using similarity conversions. Then, the equations are reduced to first-order differential equations before further being solved using the shooting method and

b v p 4 c

function in MATLAB. All results are presented in graphical and tabular forms. The buoyancy parameter causes the skin friction coefficient to increase in opposing flows but to decrease in assisting flows. In the absence of buoyancy force, there is no difference in the magnitude of the skin friction coefficient between active and passive controls of the nanoparticles. Stagnation has a bigger influence under passive control in enhancing the heat transfer rate as compared to when the fluid is under active control. Assisting flows have better heat and mass transfer rates with a lower magnitude of skin friction coefficient as compared to opposing flows. In this case, the nanofluid parameters, the Brownian motion, and thermophoresis altogether reduce the overall heat transfer rates of the non-Newtonian nanofluid. Full article

(This article belongs to the Special Issue Numerical Modeling and Analysis)

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12 pages, 1462 KiB

Open AccessArticle

Effective Conductivity of Densely Packed Disks and Energy of Graphs

by Wojciech Nawalaniec, Katarzyna Necka and Vladimir Mityushev

Mathematics 2020, 8(12), 2161; https://0-doi-org.brum.beds.ac.uk/10.3390/math8122161 - 4 Dec 2020

Cited by 4 | Viewed by 1474

Abstract

The theory of structural approximations is extended to two-dimensional double periodic structures and applied to determination of the effective conductivity of densely packed disks. Statistical simulations of non-overlapping disks with the different degrees of clusterization are considered. The obtained results shows that the [...] Read more.

The theory of structural approximations is extended to two-dimensional double periodic structures and applied to determination of the effective conductivity of densely packed disks. Statistical simulations of non-overlapping disks with the different degrees of clusterization are considered. The obtained results shows that the distribution of inclusions in a composite, as an amount of geometrical information, remains in the discrete corresponding Voronoi tessellation, hence, precisely determines the effective conductivity for random composites. Full article

(This article belongs to the Special Issue Numerical Modeling and Analysis)

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21 pages, 960 KiB

Open AccessArticle

A Numerical Method for the Solution of the Two-Phase Fractional Lamé–Clapeyron–Stefan Problem

by Marek Błasik

Mathematics 2020, 8(12), 2157; https://0-doi-org.brum.beds.ac.uk/10.3390/math8122157 - 3 Dec 2020

Cited by 6 | Viewed by 2087

Abstract

In this paper, we present a numerical solution of a two-phase fractional Stefan problem with time derivative described in the Caputo sense. In the proposed algorithm, we use a special case of front-fixing method supplemented by the iterative procedure, which allows us to [...] Read more.

In this paper, we present a numerical solution of a two-phase fractional Stefan problem with time derivative described in the Caputo sense. In the proposed algorithm, we use a special case of front-fixing method supplemented by the iterative procedure, which allows us to determine the position of the moving boundary. The presented method is an extension of a front-fixing method for the one-phase problem to the two-phase case. The novelty of the method is a new discretization of the partial differential equation dedicated to the second phase, which is carried out by introducing a new spatial variable immobilizing the moving boundary. Then, the partial differential equation is transformed to an equivalent integro-differential equation, which is discretized on a homogeneous mesh of nodes with a constant spatial and time step. A new convergence criterion is also proposed in the iterative algorithm determining the location of the moving boundary. The motivation for the development of the method is that the analytical solution of the considered problem is impossible to calculate in some cases, as can be seen in the figures in the paper. Moreover, the change of the boundary conditions makes obtaining a closed analytical solution very problematic. Therefore, creating new numerical methods is very valuable. In the final part, we also present some examples illustrating the comparison of the analytical solution with the results received by the proposed numerical method. Full article

(This article belongs to the Special Issue Numerical Modeling and Analysis)

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19 pages, 523 KiB

Open AccessArticle

Meshless Analysis of Nonlocal Boundary Value Problems in Anisotropic and Inhomogeneous Media

by Zaheer-ud-Din, Muhammad Ahsan, Masood Ahmad, Wajid Khan, Emad E. Mahmoud and Abdel-Haleem Abdel-Aty

Mathematics 2020, 8(11), 2045; https://0-doi-org.brum.beds.ac.uk/10.3390/math8112045 - 17 Nov 2020

Cited by 15 | Viewed by 2160

Abstract

In this work, meshless methods based on a radial basis function (RBF) are applied for the solution of two-dimensional steady-state heat conduction problems with nonlocal multi-point boundary conditions (NMBC). These meshless procedures are based on the multiquadric (MQ) RBF and its modified version [...] Read more.

In this work, meshless methods based on a radial basis function (RBF) are applied for the solution of two-dimensional steady-state heat conduction problems with nonlocal multi-point boundary conditions (NMBC). These meshless procedures are based on the multiquadric (MQ) RBF and its modified version (i.e., integrated MQ RBF). The meshless method is extended to the NMBC and compared with the standard collocation method (i.e., Kansa’s method). In extended methods, the interior and the boundary solutions are approximated with a sum of RBF series, while in Kansa’s method, a single series of RBF is considered. Three different sorts of solution domains are considered in which Dirichlet or Neumann boundary conditions are specified on some part of the boundary while the unknown function values of the remaining portion of the boundary are related to a discrete set of interior points. The influences of NMBC on the accuracy and condition number of the system matrix associated with the proposed methods are investigated. The sensitivity of the shape parameter is also analyzed in the proposed methods. The performance of the proposed approaches in terms of accuracy and efficiency is confirmed on the benchmark problems. Full article

(This article belongs to the Special Issue Numerical Modeling and Analysis)

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21 pages, 1014 KiB

Open AccessArticle

Numerical Analysis of Convection–Diffusion Using a Modified Upwind Approach in the Finite Volume Method

by Arafat Hussain, Zhoushun Zheng and Eyaya Fekadie Anley

Mathematics 2020, 8(11), 1869; https://0-doi-org.brum.beds.ac.uk/10.3390/math8111869 - 28 Oct 2020

Cited by 14 | Viewed by 5141

Abstract

The main focus of this study was to develop a numerical scheme with new expressions for interface flux approximations based on the upwind approach in the finite volume method. Our new proposed numerical scheme is unconditionally stable with second-order accuracy in both space [...] Read more.

The main focus of this study was to develop a numerical scheme with new expressions for interface flux approximations based on the upwind approach in the finite volume method. Our new proposed numerical scheme is unconditionally stable with second-order accuracy in both space and time. The method is based on the second-order formulation for the temporal approximation, and an upwind approach of the finite volume method is used for spatial interface approximation. Some numerical experiments have been conducted to illustrate the performance of the new numerical scheme for a convection–diffusion problem. For the phenomena of convection dominance and diffusion dominance, we developed a comparative study of this new upwind finite volume method with an existing upwind form and central difference scheme of the finite volume method. The modified numerical scheme shows highly accurate results as compared to both numerical schemes. Full article

(This article belongs to the Special Issue Numerical Modeling and Analysis)

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26 pages, 7274 KiB

Open AccessArticle

Three-Dimensional Volume Integral Equation Method for Solving Isotropic/Anisotropic Inhomogeneity Problems

by Jungki Lee and Mingu Han

Mathematics 2020, 8(11), 1866; https://0-doi-org.brum.beds.ac.uk/10.3390/math8111866 - 26 Oct 2020

Cited by 4 | Viewed by 2063

Abstract

In this paper, the volume integral equation method (VIEM) is introduced for the analysis of an unbounded isotropic solid composed of multiple isotropic/anisotropic inhomogeneities. A comprehensive examination of a three-dimensional elastostatic VIEM is introduced for the analysis of an unbounded isotropic solid composed [...] Read more.

In this paper, the volume integral equation method (VIEM) is introduced for the analysis of an unbounded isotropic solid composed of multiple isotropic/anisotropic inhomogeneities. A comprehensive examination of a three-dimensional elastostatic VIEM is introduced for the analysis of an unbounded isotropic solid composed of isotropic/anisotropic inhomogeneity of arbitrary shape. The authors hope that the volume integral equation method can be used to compute critical values of practical interest in realistic models of composites composed of strong anisotropic and/or heterogeneous inhomogeneities of arbitrary shapes. Full article

(This article belongs to the Special Issue Numerical Modeling and Analysis)

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34 pages, 1562 KiB

Open AccessEditor’s ChoiceArticle

On Semi-Analytical Solutions for Linearized Dispersive KdV Equations

by Appanah Rao Appadu and Abey Sherif Kelil

Mathematics 2020, 8(10), 1769; https://0-doi-org.brum.beds.ac.uk/10.3390/math8101769 - 14 Oct 2020

Cited by 15 | Viewed by 2091

Abstract

The most well-known equations both in the theory of nonlinearity and dispersion, KdV equations, have received tremendous attention over the years and have been used as model equations for the advancement of the theory of solitons. In this paper, some semi-analytic methods are [...] Read more.

The most well-known equations both in the theory of nonlinearity and dispersion, KdV equations, have received tremendous attention over the years and have been used as model equations for the advancement of the theory of solitons. In this paper, some semi-analytic methods are applied to solve linearized dispersive KdV equations with homogeneous and inhomogeneous source terms. These methods are the Laplace-Adomian decomposition method (LADM), Homotopy perturbation method (HPM), Bernstein-Laplace-Adomian Method (BALDM), and Reduced Differential Transform Method (RDTM). Three numerical experiments are considered. As the main contribution, we proposed a new scheme, known as BALDM, which involves Bernstein polynomials, Laplace transform and Adomian decomposition method to solve inhomogeneous linearized dispersive KdV equations. Besides, some modifications of HPM are also considered to solve certain inhomogeneous KdV equations by first constructing a newly modified homotopy on the source term and secondly by modifying Laplace’s transform with HPM to build HPTM. Both modifications of HPM numerically confirm the efficiency and validity of the methods for some test problems of dispersive KdV-like equations. We also applied LADM and RDTM to both homogeneous as well as inhomogeneous KdV equations to compare the obtained results and extended to higher dimensions. As a result, RDTM is applied to a 3D-dispersive KdV equation. The proposed iterative schemes determined the approximate solution without any discretization, linearization, or restrictive assumptions. The performance of the four methods is gauged over short and long propagation times and we compute absolute and relative errors at a given time for some spatial nodes. Full article

(This article belongs to the Special Issue Numerical Modeling and Analysis)

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15 pages, 330 KiB

Open AccessArticle

Quasilinearized Semi-Orthogonal B-Spline Wavelet Method for Solving Multi-Term Non-Linear Fractional Order Equations

by Can Liu, Xinming Zhang and Boying Wu

Mathematics 2020, 8(9), 1549; https://0-doi-org.brum.beds.ac.uk/10.3390/math8091549 - 10 Sep 2020

Viewed by 1527

Abstract

In the present article, we implement a new numerical scheme, the quasilinearized semi-orthogonal B-spline wavelet method, combining the semi-orthogonal B-spline wavelet collocation method with the quasilinearization method, for a class of multi-term non-linear fractional order equations that contain both the Riemann–Liouville fractional integral [...] Read more.

In the present article, we implement a new numerical scheme, the quasilinearized semi-orthogonal B-spline wavelet method, combining the semi-orthogonal B-spline wavelet collocation method with the quasilinearization method, for a class of multi-term non-linear fractional order equations that contain both the Riemann–Liouville fractional integral operator and the Caputo fractional differential operator. The quasilinearization method is utilized to convert the multi-term non-linear fractional order equation into a multi-term linear fractional order equation which, subsequently, is solved by means of semi-orthogonal B-spline wavelets. Herein, we investigate the operational matrix and the convergence of the proposed scheme. Several numerical results are delivered to confirm the accuracy and efficiency of our scheme. Full article

(This article belongs to the Special Issue Numerical Modeling and Analysis)

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15 pages, 8749 KiB

Open AccessArticle

Finite Element Validation of an Energy Attenuator for the Design of a Formula Student Car

by José A. López-Campos, Jacobo Baldonedo, Sofía Suárez, Abraham Segade, Enrique Casarejos and José R. Fernández

Mathematics 2020, 8(3), 416; https://0-doi-org.brum.beds.ac.uk/10.3390/math8030416 - 14 Mar 2020

Cited by 3 | Viewed by 2321

Abstract

Passive safety systems of cars include parts on the structure that, in the event of an impact, can absorb a large amount of the kinetic energy by deforming and crushing in a design-controlled way. One such energy absorber part, located in the front [...] Read more.

Passive safety systems of cars include parts on the structure that, in the event of an impact, can absorb a large amount of the kinetic energy by deforming and crushing in a design-controlled way. One such energy absorber part, located in the front structure of a Formula Student car, was measured under impact in a test bench. The test is modeled within the Finite Element (FE) framework including the weld characteristics and weld failure description. The continuous welding feature is almost always disregarded in parts included in impact test models. In this work, the FE model is fully defined to reproduce the observed results. The test is used for the qualitative and quantitative validation of the crushing model. On the one hand, the acceleration against time curve is reproduced, and on the other hand, the plying shapes and welding failure observed in the test are also correctly described. Finally, a model that includes additional elements of the car structure is also simulated to verify that the energy absorption system is adequate according to the safety regulations. Full article

(This article belongs to the Special Issue Numerical Modeling and Analysis)

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22 pages, 4332 KiB

Open AccessArticle

A Novel Meshfree Approach with a Radial Polynomial for Solving Nonhomogeneous Partial Differential Equations

by Cheng-Yu Ku, Jing-En Xiao and Chih-Yu Liu

Mathematics 2020, 8(2), 270; https://0-doi-org.brum.beds.ac.uk/10.3390/math8020270 - 18 Feb 2020

Cited by 1 | Viewed by 2504

Abstract

In this article, a novel radial–based meshfree approach for solving nonhomogeneous partial differential equations is proposed. Stemming from the radial basis function collocation method, the novel meshfree approach is formulated by incorporating the radial polynomial as the basis function. The solution of the [...] Read more.

In this article, a novel radial–based meshfree approach for solving nonhomogeneous partial differential equations is proposed. Stemming from the radial basis function collocation method, the novel meshfree approach is formulated by incorporating the radial polynomial as the basis function. The solution of the nonhomogeneous partial differential equation is therefore approximated by the discretization of the governing equation using the radial polynomial basis function. To avoid the singularity, the minimum order of the radial polynomial basis function must be greater than two for the second order partial differential equations. Since the radial polynomial basis function is a non–singular series function, accurate numerical solutions may be obtained by increasing the terms of the radial polynomial. In addition, the shape parameter in the radial basis function collocation method is no longer required in the proposed method. Several numerical implementations, including homogeneous and nonhomogeneous Laplace and modified Helmholtz equations, are conducted. The results illustrate that the proposed approach may obtain highly accurate solutions with the use of higher order radial polynomial terms. Finally, compared with the radial basis function collocation method, the proposed approach may produce more accurate solutions than the other. Full article

(This article belongs to the Special Issue Numerical Modeling and Analysis)

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18 pages, 310 KiB

Open AccessArticle

Homotopy Analysis Method for a Fractional Order Equation with Dirichlet and Non-Local Integral Conditions

by Said Mesloub and Saleem Obaidat

Mathematics 2019, 7(12), 1167; https://0-doi-org.brum.beds.ac.uk/10.3390/math7121167 - 2 Dec 2019

Cited by 1 | Viewed by 2164

Abstract

The main purpose of this paper is to obtain some numerical results via the homotopy analysis method for an initial-boundary value problem for a fractional order diffusion equation with a non-local constraint of integral type. Some examples are provided to illustrate the efficiency [...] Read more.

The main purpose of this paper is to obtain some numerical results via the homotopy analysis method for an initial-boundary value problem for a fractional order diffusion equation with a non-local constraint of integral type. Some examples are provided to illustrate the efficiency of the homotopy analysis method (HAM) in solving non-local time-fractional order initial-boundary value problems. We also give some improvements for the proof of the existence and uniqueness of the solution in a fractional Sobolev space. Full article

(This article belongs to the Special Issue Numerical Modeling and Analysis)

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12 pages, 318 KiB

Open AccessArticle

A New Explicit Four-Step Symmetric Method for Solving Schrödinger’s Equation

by Saleem Obaidat and Said Mesloub

Mathematics 2019, 7(11), 1124; https://0-doi-org.brum.beds.ac.uk/10.3390/math7111124 - 17 Nov 2019

Cited by 5 | Viewed by 2049

Abstract

In this article we have developed a new explicit four-step linear method of fourth algebraic order with vanished phase-lag and its first derivative. The efficiency of the method is tested by solving effectively the one-dimensional time independent Schrödinger’s equation. The error and stability [...] Read more.

In this article we have developed a new explicit four-step linear method of fourth algebraic order with vanished phase-lag and its first derivative. The efficiency of the method is tested by solving effectively the one-dimensional time independent Schrödinger’s equation. The error and stability analysis are studied. Also, the new method is compared with other methods in the literature. It is found that this method is more efficient than these methods. Full article

(This article belongs to the Special Issue Numerical Modeling and Analysis)

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16 pages, 4256 KiB

Open AccessArticle

Blended Root Finding Algorithm Outperforms Bisection and Regula Falsi Algorithms

by Chaman Lal Sabharwal

Mathematics 2019, 7(11), 1118; https://0-doi-org.brum.beds.ac.uk/10.3390/math7111118 - 16 Nov 2019

Cited by 15 | Viewed by 8761

Abstract

Finding the roots of an equation is a fundamental problem in various fields, including numerical computing, social and physical sciences. Numerical techniques are used when an analytic solution is not available. There is not a single algorithm that works best for every function. [...] Read more.

Finding the roots of an equation is a fundamental problem in various fields, including numerical computing, social and physical sciences. Numerical techniques are used when an analytic solution is not available. There is not a single algorithm that works best for every function. We designed and implemented a new algorithm that is a dynamic blend of the bisection and regula falsi algorithms. The implementation results validate that the new algorithm outperforms both bisection and regula falsi algorithms. It is also observed that the new algorithm outperforms the secant algorithm and the Newton–Raphson algorithm because the new algorithm requires fewer computational iterations and is guaranteed to find a root. The theoretical and empirical evidence shows that the average computational complexity of the new algorithm is considerably less than that of the classical algorithms. Full article

(This article belongs to the Special Issue Numerical Modeling and Analysis)

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15 pages, 374 KiB

Open AccessFeature PaperArticle

Homotopy Approach for Integrodifferential Equations

by Damian Słota, Edyta Hetmaniok, Roman Wituła, Krzysztof Gromysz and Tomasz Trawiński

Mathematics 2019, 7(10), 904; https://0-doi-org.brum.beds.ac.uk/10.3390/math7100904 - 27 Sep 2019

Cited by 11 | Viewed by 1953

Abstract

In this paper, we present the application of the homotopy analysis method for solving integrodifferential equations. In this method, a series is created, the successive elements of which are determined by calculating the appropriate integral of the previous element. In this elaboration, we [...] Read more.

In this paper, we present the application of the homotopy analysis method for solving integrodifferential equations. In this method, a series is created, the successive elements of which are determined by calculating the appropriate integral of the previous element. In this elaboration, we prove that, if this series is convergent, then its sum is the solution of the objective equation. We formulate and prove the sufficient condition of this convergence, and we give also the estimation of error of an approximate solution obtained by taking the partial sum of the considered series. Moreover, we present in this paper the example of using the investigated method for determining the vibrations of the freely supported reinforced concrete beam as well as for solving the equation of movement of the electromagnet jumper mechanical system. Full article

(This article belongs to the Special Issue Numerical Modeling and Analysis)

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