Quasilinearization Method Research Articles

A new approach based on shifted Vieta-Fibonacci-quasilinearization technique and its convergence analysis for nonlinear third-order Emden–Fowler equation with multi-singularity

A shifted Vieta-Fibonacci (SVF) matrix technique is presented to find the solution of third-order Emden–Fowler equations with nonlinearity and multi-singularity. To develop an efficient approach, the quasilinearization method (QLM) is first applied to the underlying model problem and then a collocation scheme based on the SVF functions is applied to the resultant sequence of linearized equations. The application of SFV-QLM yields an algebraic system of linear equations to be solved iteratively. A comprehensive error analysis is presented to show that the SVF series solution is convergent in the weighted L2 and L∞ norms. Finally, numerical experiments confirm the theoretical results and the applicability as well as the practicability of SVF-QLM algorithm. The consumed computational time for our proposed algorithm is provided. The proposed algorithm has numerous merits. It is not only effortless to implement, but also highly accurate and computationally efficient.

WITHDRAWN: Generalized Vieta-Fibonacci quasilinearization method for solutions of third-order singular Emden–Fowler type boundary value problems

In this article, we apply a generalized Vieta-Fibonacci (GVF) quasilinearization method to explore the numerical solution of third-order singular and nonlinear boundary value problems. Firstly, we use the quasilinearization technique to convert the governing nonlinear boundary value problem into a class of linearized equations. Then we build a collocation technique based on GVF functions to solve the resultant sequence of linearized equations. Convergence result of the method is established with respect to L 2 norm. Four test problems are considered to illustrate the applicability, efficiency and robustness of the proposed algorithm and to validate the theoretical estimate as well by comparing it with previous work.

The unsteady nonlinear convective flow of a nanofluid due to impulsive motion: Triple diffusion and magnetic effects

• The unsteady nonlinear convection due to impulsive motion is investigated. • Governing equations have been transformed using Mangler's transformations. • Quasi-linearization technique is utilized for solution of the problem. • The surface drag coefficient increases with magnetic parameter. • Heat transport rate rises only when the Eckert number is positive. By a semi-similar approach, the paper aims to study the unsteady nonlinear convection due to impulsive motion with triple diffusion and magnetic effects. Since the mainstream has been set into impulsive motion, the fluid motion is unsteady. Simultaneously, the temperature of the surface is abruptly changed from its ambient fluid temperature. The influence of chemical reactions and convective boundary constraints are also considered in this investigation. The nonlinear coupled partial differential equations (PDEs) with appropriate boundary conditions that govern the given problem are transformed into dimensionless equations by using Mangler's transformations. For mathematical computations, the technique of Quasi-linearization and second order implicit finite difference method are utilized, and so obtained results are semi-similar solutions that are discussed with the aid of graphs. The results reveal that for the case of aiding buoyancy, the surface drag coefficient increases about 5% when the nonlinear convection parameter increases from 0 to 3 and about 61% when the magnetic parameter increases from 0 to 0.5. The heat transfer rate enhances about 65% for variation of the magnetic parameter from 0 to 0.3 for a positive Eckert number. The mass transfer rate is pronounced more for linear chemical reaction than the nonlinear chemical reaction for the constructive chemical reaction parameter. It is found at about 7% for Schmidt number 240. The current findings are compared to previously reported literature and are found to be in good agreement.

Decoupling and quasi-linearization methods for boundary value problems in relative orbital mechanics

Decoupling and quasi-linearization methods for boundary value problems in relative orbital mechanics

Numerical investigation of the non-Newtonian power-law fluid with convective boundary conditions in a non-Darcy porous medium

The boundary layer nanofluid flow over a horizontal plate embedded in a non-Darcy porous medium is investigated in this study. Non-Newtonian power-law fluid is considered the base fluid. The nanofluid model incorporated the effects of Brownian motion and thermophoresis. A mixed convective boundary condition that is more useful, in practice, is employed at the surface instead of Dirichlet or Neumann boundary conditions. The flow and convective heat transfer has engineering and industrial applications such as the thermal design of industrial equipment dealing with molten plastics and polymeric liquids and applications in nuclear plants and thermal energy storage processes. The main aim of this study is to use the spectral quasi-linearization method to study the effects of non-Newtonian nanofluid flow over a horizontal surface for a porous medium. The objective is to explore both the accuracy and the convergence of the method through the evaluation of residual errors and error norms and to investigate the impact of specific parameters on flow behavior and heat transfer characteristics. The velocity profile increases for higher values of the power law index and mixed convection parameter. The influence of the number of active variables on dimensionless temperature profiles increases for different values of the Biot number.

Quasi-Linearization Analysis for Entropy Generation in MHD Mixed-Convection Flow of Casson Nanofluid over Nonlinear Stretching Sheet with Arrhenius Activation Energy

Thermal performance of magnetically driven Casson nanofluid over a nonlinear stretching sheet under the influence of entropy, activation energy and convective boundary conditions was analyzed numerically, employing the quasi-linearization method (QLM). The collective behavior of thermophoretic diffusion and Brownian motion along with special effects of viscous dissipation, thermal radiation, heat generation and joule heating are considered in the energy equation for the flow problem. The addition of nanoparticles helps to stabilize the flowing of a nanofluid and maintain the symmetry of the flowing structure. The governing highly nonlinear coupled differential equations of velocity, temperature, concentration and entropy are simulated through an iterative scheme encoded with MATLAB programming language. The geometric model is, therefore, described using a symmetry technique. A comparative analysis of linear and nonlinear stretching in sheets is presented via graphs and tables regarding pertinent dimensionless parameters. It is worth noting that the Nusselt number and Sherwood number decrease at relatively higher rates with growing values of activation energy in the case of nonlinear stretching. Moreover, the entropy generation rate near the stretching surface decreases due to the strong effects of Brownian motion and thermophoretic diffusion while it goes on improving far off the stretching surface.

A mathematical model for cancer treatment based on combination of anti-angiogenic and immune cell therapies

In this study, we investigated a mathematical model that integrated mechanisms of tumor angiogenesis and tumor-targeted cytotoxicity in immune cells. The model considered the interaction of cancer cells with the immune system and the treatment that combines unlicensed dendritic cells and anti-vascular endothelial growth factor antibodies. We proposed an unconventional treatment protocol; for now, treatment is purely theoretical and has not been practically attempted. The treatment has been described by analytical functions that depend on time and dose. This was a function of t and q, where q is the amount (dose) of the drug. A customized treatment is a significant advantage of explicit analytical functions. We compared the results obtained from the mathematical model with the proposed treatment to the clinical results and observed an almost complete match between the treatments.In addition, we applied the global quasi-linearization GQL method to the mathematical model, which enabled us to expose the hierarchy of the system and, hence, split the system into fast and slow subsystems. This decomposition enabled us to analytically compute the equilibrium points of the model and their stability, which is crucial for the application.

Spectral quasilinearization method for Sisko nanofluid past a stretching cylinder with activation energy and entropy generation effects subject to variable thermal conductivity

AbstractThis article focuses on how activation energy and entropy generation affect the parameters involved in Sisko nanofluid with the presence of a stretching cylinder. Spectral quasilinearization method is applied to solve the ordinary differential equations representing the flow equations to have a highly accurate solution. Furthermore, all the parameters involved are graphically analyzed to see the behavior of velocity, temperature, and concentration behavior under different circumstances. With the increment of heat generation parameter, Sisko nanofluid shows an increasing nature in concentration unlike base fluids. Thermophoresis and Brownian motion effects are observed to be different than usual, especially in nearby wall zones due to activation energy and entropy terms being introduced on a stretching cylinder medium.

Multi-domain multivariate spectral collocation method for (2+1) dimensional nonlinear partial differential equations

The novelty of this work rests upon the use of the domain partitioning technique in time variable when discretizing the domain of solution in spectral collocation algorithm. The single domain multivariate spectral collocation based methods have been proven to be effective in solving time-dependent partial differential equations (PDEs) defined over small time domains. However, there is a significant loss of accuracy as time computational domain proliferates and also when the number of grid nodes approaches a definite particular number. Therefore, the establishment of the new innovative multi-domain multivariate spectral quasilinearisation method (MDMV-SQLM) is described for the purpose of solving (2+1) dimensional nonlinear PDEs defined on large time intervals. The main output of this study is confirmation that minimizing the size of time computational domain at each subinterval assures sufficiently accurate results that are attained using minimal number of nodal points and less computational time. The solution algorithm involves partitioning the time domain into multiple non-overlapping sub-domains, simplification of the nonlinear PDEs using the quasilinearisation method and assumption of approximate solutions using triple Lagrange interpolating polynomials with Chebyshev–Gauss–Lobatto (CGL) points. The multi-domain spectral collocation procedure is executed on the linear systems of algebraic equations, where the subsequent matrix systems are solved separately in every time sub-interval with the continuity equation essentially used in obtaining initial conditions in the next subintervals. MATLAB software is used to implement the solution algorithm and numerical results are demonstrated graphically and in tabular form. To highlight the efficaciousness and accuracy of the MDMV-SQLM, error estimates, condition numbers and computational time are presented for well known (2+1) dimensional nonlinear initial-Dirichlet boundary value problems. The adoption of the domain decomposition technique is efficacious in suppressing the numerical challenges linked to large matrices and ill-conditioned nature of the resulting coefficient matrix. Also, the communicated results confirm that the numerical scheme is computationally cheap, fast and yield extremely accurate and stable results with the aid of fewer number of grid points for large time domains.

Study the Impact of an Exponentially Stretching Rate and Shape Factor of the Axisymmetric Bullet-Shaped Object on the Mixed Convection Boundary Layer Flow and Heat Transfer with Stream-Wise Coordinate and Viscous Dissipation

The current work has been investigated the influence of the exponentially stretching rate and shapes factor of the axisymmetric bullet-shaped object on the mixed convection magnetohydrodynamic boundary layer flow and heat transfer with viscous dissipation, stream-wise coordinate, and internal heat generation. The main goal of this problem is to discuss the effect of the surface shape and size, stream-wise coordinate, and also the exponential stretching factor of the bullet-shaped object on the fluid flow distribution. The novelty of the present work involved in the area of recently developed numerical method to solve these highly nonlinear differential equations. The present analysis has been performed for both of the fixed (ε = 0) and moving (ε > 0) bullet-shaped object in the two cases of thinner (0 < s < 0.1) and thicker (s ≥ 0.3) surfaces of the bullet-shaped object. It is noted that when ε = 0 means for a fixed bullet-shaped object in a moving fluid and while a moving bullet-shaped object in a fixed fluid represents when ε > 1. The governing equations have been converted into a system of ODEs by using suitable local axisymmetric transformations and solved by applying the spectral quasi-linearization method. This method helps to identify the accuracy, validity, and convergence of the present numerical computations. The computations have been investigated by the effects of different parameters on the flow field, wall friction, and heat transfer. The investigation depicts that the flow field and temperature do not converge the free stream condition asymptotically in the case of a thicker bullet-shaped object instead it intersects the axis with a steep angle which is contradictory with the boundary layer theory and the boundary layer structure has no defined shape whereas in the case of a thinner bullet-shaped object (0 < s < 0.1) the ambient condition satisfies asymptotically and formed a definite boundary layer structure. Heat transfer rate at the bullet-shaped object is negatively correlated with the magnetic parameter, Eckert number, heat generation parameter, and surface thickness parameter but positively correlated with the Prandtl number, location parameter, mixed convection parameter, and stretching ratio parameter. The investigation represents that surface thickness parameter (shape and size) and stretching ratio parameter have a prominent effect on fluid flow properties and cannot be neglected. It is also noticed that the thinner bullet-shaped object acts as a good cooling conductor compared to thicker bullet-shaped object and the wall friction can be reduced much when a thinner bullet-shaped object is used rather than the thicker bullet-shaped object in both types of moving or static bullet-shaped object (ε = 0.0 and ε = 0.2). The present analysis reveals that the heat transfer and the friction factor will be helpful in industrial sectors such as a cooling device in nuclear reactors, automotive engineering, electronic engineering, biomedical engineering, control the cooling rate and quality of the final product.

Influence of Shape Factor and Non-Linear Stretching of the Bullet-Shaped Object on the Mixed Convection Boundary Layer Flow and Heat Transfer with Viscous Dissipation and Internal Heat Generation

The two-dimensional incompressible axisymmetric mixed convection magnetohydrodynamic fluid flow and energy transfer over a bullet-shaped object with a non-linear stretching surface have been investigated. The main goal of this problem is to discuss the effect of the shape and size of the bullet-shaped object on the fluid velocity and temperature distributions. The present analysis has been performed in about two cases ε=0.0 and 2.0. Therefore, fluid velocity and temperature distributions have been investigated in two types of flow geometries such as the thicker surface (s ≥ 2) and the thinner surface (0 < s < 2) of the bullet-shaped object. The equations for momentum and heat transfer have been converted into ODEs by using suitable local similarity transformations. These equations have been performed with a recently developed spectral quasi-linearization method (SQLM). This method helps to identify the accuracy, validity, and convergence of the present solution. The novelty of the present work has been applying the recently developed numerical method to solve these highly nonlinear differential equations. The investigation shows that in the case of a thicker bullet-shaped object (s ≥ 2) the velocity and temperature profiles do not converse the far-field boundary condition asymptotically but cross the axis with an upright angle and the boundary layer structure has no definite shape whereas in the case of a thinner bullet-shaped object (0 < s < 2) the velocity profile converge the ambient condition asymptotically and the boundary layer structure has a definite shape. The innovation of this current work lies in the unification of relevant physical parameters into the governing equations and trying to explain how the flow properties are affected by these parameters.

A Numerical Analysis on the Unsteady Flow of a Thermomagnetic Reactive Maxwell Nanofluid over a Stretching/Shrinking Sheet with Ohmic Dissipation and Brownian Motion

The major objective of this current investigation is to examine the unsteady flow of a thermomagnetic reactive Maxwell nanofluid flow over a stretching/shrinking sheet with Ohmic dissipation and Brownian motion. Suitable similarity transformations were used to reduce the governing non-linear partial differential equations of momentum, energy and species conservation into a set of coupled ordinary differential equations. The reduced similarity ordinary differential equations were solved numerically using the Spectral Quasi-Linearization Method. The influence of some pertinent physical parameters on the velocity, temperature and concentration distributions was studied and analysed graphically. Further investigations were made on the impact of the Eckert number, Prandtl number, Schmidt number, thermal radiation parameter, Brownian motion parameter, thermophoresis parameter and chemical reaction parameter on the skin friction coefficient, surface heat and mass transfer rates. The results were displayed in a tabular form. Obtained results reveal that the Maxwell parameter and the unsteadiness parameter reduce the Maxwell nanofluid velocity and the fluid temperature is increased with an increase in the Eckert number and thermal radiation parameter.

Entropy generation analysis on convective radiative stagnation point flow of a Casson nanofluid in a non-Darcy porous medium with induced magnetic field and activation energy: the multiple quadratic regression model

The present article analyzes the entropy generation on the nonlinear thermally radiating magnetohydrodynamic Casson nanofluid flow closer to the stagnation point inside a non-Darcy porous medium with viscous dissipation, magnetic dissipation, mixed convection, and heat generation. The induced magnetic field's impact and the implications of the chemical reaction and the activation energy are also included in the study. A mathematical model based on the substantive problem is developed, consisting of nonlinear partial differential equations and compatible boundary conditions. The mathematical model is converted to a set of nonlinear ordinary differential equations with appropriate boundary conditions using a suitable similarity transformation, which is then solved using the spectral quasi-linearization method. A complete examination is executed for achieving in-depth information of different varying flow parameters' consequences on the flow field and the significant physical quantities using several graphs and tables. Using the regression models on the reduced Nusselt number NuxRex−1/2 and the reduced Sherwood number ShxRex−1/2, the maximum relative errors are shown as 0.15%–0.30% and 0.016024%–0.016225%, respectively. The present results signify that the larger velocity ratio parameter boosts the velocity along the x-axis and the transverse velocity for M = 3, but the transverse velocity for M = 1, temperature, and induced magnetic field follow a reverse pattern.

Unsteady magneto-convective heat-mass transport passing in a vertical permeable sheet with internal heat generation effect

The unsteady magneto-convective heat-mass transport passing in a vertical permeable sheet has been discussed with the internal heat generation/absorption effect. The governing partial differential equations (PDEs) have been transferred into ordinary differential equations (ODEs) by applying the local similarity transformation. The dimensionless ODEs are solved numerically by exerting the Quasilinearization Method with the help of ‘MATLAB ODE45’software. The influence of emerging dimensionless parameters/numbers for example the internal generation/absorption (Q), the suction parameter (v0), Prandtl number (Pr), Soret number (So), Dufour number (Df), Schmidt number (Sc), magnetic force parameter (M), local modified Grashof number (Gm) and local Grashof number (Gr) on the flow, temperature, and concentration profiles are presented and discussed. The fluid velocity and temperature enhance rising values of the internal heat generation and reduces the concentration but the reverse tends to be found for absorption. The velocity field reduces for increasing values of the suction parameter, Magnetic parameter, Prandtl number, and Schmidt number. The local skin friction increases by about 18% and reduces by 6% due to an increase in the internal heat generation parameter (1.0-3.0) and heat absorption parameter (3.0-10.0), respectively. Uplifting values of the heat absorption parameter (3.0-10.0) and internal heat parameter (1.0-3.0) enhance the heat transfer rate by 56% and decrease by about 42%, respectively. The mass transfer rate enhances by about 59% and decreases by 85% due to an increase in the internal heat generation parameter (1.0-3.0) and heat absorption parameter (3.0-10.0), respectively. Finally, we compared our numerical results with previously published paper and observed them be a excellent agreement.

Numerical Approximation of Nonlinear Duffing Oscillator Using a Coupled Approach

Duffing equation can describe many important nonlinear physical systems. In this paper a coupled approach based on quasilinearization and Bessel polynomial collocation method has been suggested to solve nonlinear duffing oscillator equation. The nonlinearity in duffing oscillator can be of various variety. This approach is very efficient, stable and reliable to deal with any kind of nonlinearity. Numerical examples demonstrate the validity and applicability of the approach on various types of nonlinear duffing oscillator equation.

Trivariate Spectral Collocation Approach for the Numerical Solution of Three-Dimensional Elliptic Partial Differential Equations

This article is concerned with the numerical solution of three-dimensional elliptic partial differential equations (PDEs) using the trivariate spectral collocation approach based on the Kronecker tensor product. By using the quasilinearization method, the nonlinear elliptic PDEs are simplified to a linear system of algebraic equations that can be discretized using the spectral collocation method. The method is based on approximating the solutions using the triple Lagrange interpolating polynomials, which interpolate the unknown functions at selected Chebyshev–Gauss–Lobatto (CGL) grid points. The CGL points are preferred to ensure simplicity in the conversion of continuous derivatives to discrete derivatives at the collocation points. The collocation process is carried out at the interior points to reduce the size of differentiation matrices. This work is aimed at verifying that the algorithm based on the method is simple and easily implemented in any scientific software to produce more accurate and stable results. The effectiveness and spectral accuracy of the numerical algorithm is checked through the determination and analysis of errors, condition numbers and computational time for various classes of single or system of elliptic PDEs including those with singular behavior. The communicated results indicate that the proposed method is more accurate, stable and effective for solving elliptic PDEs. This good accuracy becomes possible with the usage of few grid points and less memory requirements for numerical computation.

A Chebyshev Spectral Collocation Method-Based Series Approach for Boundary Layer Flow and Heat Transfer in a Micropolar Fluid past a Permeable Flat Plate

This paper demonstrates the applicability of the large parameter spectral perturbation method (LSPM) to a coupled system of partial differential equations that cannot be solved exactly. The LSPM is a numerical method that employs the Chebyshev spectral collocation method in the solution of a sequence of ordinary differential equations (ODEs) that are derived from decomposing coupled systems of nonlinear partial differential equations (PDEs) using series expansion about a large parameter. The validity of the LSPM is applied to the problem of boundary layer flow and heat transfer in a micropolar fluid past a permeable flat plate in the presence of heat generation and thermal radiation. The coupled nature of the PDEs that define the problem under investigation precludes the option of using series-based methods that seek to generate analytical solutions even in the presence of small or large parameters. The present study demonstrates that the LSPM can easily overcome this limitation while giving very accurate results in a computationally efficient manner. For qualitative validation of the results and the numerical method used, calculations were carried out to graphically obtain the velocity, microrotation, and temperature profiles for selected physical parameter values. The results obtained were found to correlate with the results from a published literature. For quantitative confirmation of our findings, the LSPM numerical solutions were again validated against known results from the literature and against results obtained using the multidomain bivariate spectral quasilinearisation method (MD-BSQLM), and the results were observed to be in perfect agreement. Further accuracy validation is displayed by using residual error and solution error analysis on the governing PDEs and their underlying solutions. This study’s findings indicate that the heat generation and thermal radiation parameters have related effects on the temperature profile, enhancing both the fluid temperature and the thermal boundary layer thickness.

Spectral Simulation on the Flow Patterns and Thermal Control of Radiative Nanofluid Spraying on an Inclined Revolving Disk Considering the Effect of Nanoparticle Diameter and Solid–Liquid Interfacial Layer

Abstract The current work enlightens the flow pattern and thermal scenario of a nanofluid spraying on an inclined permeable rotating disk. The whirling disk is assumed to revolve with the angular speed Ω. The water-based alumina (Al2O3) nanofluid is considered as a functioning liquid. The nanofluid spraying is treated to be magnetically influenced and thermally radiative. The perception of the nanoparticles' diameter and solid–liquid interfacial layer is incorporated precisely at the nanolevel to observe the thermal variations of the nanofluidic motion. How the magnetic effect, permeability, and nanolayer affect nanofluidic transportation is revealed in detail. The leading flow equations are altered nondimensional using apposite similarity translation, and the spectral quasi-linearization method (SQLM) is instigated to tackle those multi-ordered nonlinear equations. Various three-dimensional figures, graphs, and tables are described to detect and analyze the hydrothermal variations. The linear regression slope technique is addressed to extract the reduction or enhancement rate of heat transference. Also, the probable error is estimated statistically to assure that hydrothermal characteristic is correlated with physical parameters. The consequences indicate enhanced heat transport for nanolayers, but reduced heat transmission for nanoparticles' diameter. Thermal profile intensifies for thickness parameter and impermeable surface, whereas decreases for nanoparticles' diameter.

Effect on Entropy Generation Analysis for Heat Transfer Nanofluid Near a Rotating Disk Using Quasilinearization Method

The study considers gold-water nanofluid flow past a porous rotating disk while accounting for prescribed heat flux and suction at the boundary layer of the disk. The physical parameters of the nanoparticle volume fraction, magnetic parameter and entropy generation are investigated and presented in this paper. The numerically solved nonlinear equations by the spectral quasilinearization technique. The main findings are presented in graphical form and discussed for variations of the flow parameters. The findings indicate that increased nanoparticle volume concentration fall in velocity but a overshoot in temperature, while enhancing the magnetic parameter is associated with reduced velocity distribution and increased skin friction. Among other findings, the results also show that increasing the Brinkman number leads to increased entropy generation but reduced Bejan number, while the Reynolds number increasing in the generation of elevated levels of entropy production. The reliability, error analysis and accuracy are checked through convergence of the method. The accuracy is further tested through a comparison of results for limiting cases with those in the literature. The findings of this study have significant applications in engineering, science and technology.

Entropy Generation in Double Diffusive Convective Magnetic Nanofluid Flow in Rotating Sphere with Viscous Dissipation

We scrutinize the numerically investigates entropy generation for unsteady magnetic nanofluid with stagnation region of an impulsively rotating sphere with viscous dissipation and chemical reaction effects. Employing the similarity technique, the equations modeling are switched into highly nonlinear differential equations with boundary layer flow. These equations are numerically executed with bivariate spectral quasilinearization method. The entropy generation effect of varying various pertinent different parameters are taken into account and the results analyzed graphically. The rate of entropy generation and energy related temperature profiles increased with increased of thermal radiation parameter, chemical reaction and rotation parameters while increased values of the magnetic number had a negative impact impact on rate of entropy generation. Variables of engineering interest which includes the skin friction coefficient, the Nusselt number and the Sherwood number were also analyzed in a table. These numerical results will be of great value to chemists, biologists, physicists, and engineers interested in the theory and applications of entropy generation.