Quasilinearization Method Research Articles

A combination of the quasilinearization method and linear barycentric rational interpolation to solve nonlinear multi-dimensional Volterra integral equations

In this paper, an iterative scheme including a combination of the quasilinearization technique and multi-dimensional linear barycentric rational interpolation is applied to solve nonlinear multi-dimensional Volterra integral equations. First, employing the quasilinearization method, the nonlinear multi-dimensional Volterra integral equation is reduced to a sequence of linear Volterra integral equations. Under appropriate assumptions, the constructed iterative sequence is uniformly convergent to the unique solution of the problem. In general, finding an analytical solution for linear Volterra integral equations is impossible. Hence, in each iteration, using a collocation method based on multi-dimensional barycentric rational basis functions, the solution to the linear integral equation is approximated. The quadratic convergence of the quasilinearization approach and the error estimation of the combined method are investigated theoretically. In the end, the efficiency and the validity of this method are illustrated with some numerical examples and compared with those of the existing numerical methods.

Iterative compact finite difference method for the numerical study of fully wet porous fins with different profile shapes

This paper investigates the temperature distribution of porous fins with different profiles (rectangular, triangular, convex parabolic and concave parabolic profiles) under fully wet conditions. The porous fin material is Al and the tips are insulated. An iterative discrete method is employed to solve the nonlinear governing equations. For this purpose, the governing equations are converted into a sequence of linear differential equations by the quasi-linearization method. Then, the numerical solutions of these linear equations are estimated using the compact finite difference scheme. The main advantage of the method is that it does not require to solve the system of non-linear algebraic equations. Moreover, unlike the Padé standard finite difference method, there is no need to calculate the derivatives of the solution from the governing differential equation in order to obtain the finite difference formulas with higher accuracy. The convergence analysis of the method is studied in detail. The effect of geometric and thermophysical parameters on temperature distribution and fins' efficiency is discussed using the plotted graphs. Comparing our numerical solutions with those obtained by the least squares method and the Chebfun package in Matlab shows the accuracy and efficiency of the proposed method.

Statistical Analysis of Non-Newtonian Couple Stress Fluid Induced in Stretching Cylinder

The paper provides the impact of suction and injection on convection laminar incompressible couple stress fluid flow and magnetic field using spectral quasi linearization methods as the major novelty of our work. This work is to addresed heat transfer is an important process in many engineering, industrial, residential, and commercial buildings. Thus, this study aims to analyze the effect of MHD and non-Newtonian couple stress fluid runs over a permeable stretched cylinder. The leading formulation is transmuted into ordinary differential equations via similarity functions. The coupled equations with non-linearly terms are resolved numerically through utilization of MATLAB code for spectal quasi linearization methods (SQLM). Convergence regions for solutions are discussed. Graphical results illustrating the impacts of various emerging parameters are presented in discussion. The statistical declaration and probable error for skin friction and Nusselt number are numerically computed and discussed through Tables. From obtained outcomes it is concluded that magnitude of skin friction increases at the cylindrical surface for higher values of couple stress parameter and Reynolds number. Nusselt number or heat transfer rate also enhances at the surface of cylinder in the presence of Reynolds number.

Powell-Eyring Fluid Flow Over a Stretching Surface with Variable Properties

The objective of this study is to investigate the effect of temperature dependent viscosity and thermal conductivity on the stretching surface and heat transfer of a Powell-Eyring fluid with the effect of nonlinear thermal radiation. The Prandtl number is also considered to be varying within the boundary layer. The governing model which consists of a set of coupled non-linear partial differential equations is converted into a system of ordinary differential equations via suitable similarity transformations and then solved by a recent and reliable numerical method called the spectral quasi-linearization method is used in the computational analysis. The result shows that there is large variation in the value of the Nusselt number and skin friction co-efficient.

Generalized Shifted Airfoil Polynomials of the Second Kind to Solve a Class of Singular Electrohydrodynamic Fluid Model of Fractional Order

In this manuscript, we find the numerical solutions of a class of fractional-order differential equations with singularity and strong nonlinearity pertaining to electrohydrodynamic flow in a circular cylindrical conduit. The nonlinearity of the underlying model is removed by the quasilinearization method (QLM) and we obtain a family of linearized equations. By making use of the generalized shifted airfoil polynomials of the second kind (SAPSK) together with some appropriate collocation points as the roots of SAPSK, we arrive at an algebraic system of linear equations to be solved in an iterative manner. The error analysis and convergence properties of the SAPSK are established in the L2 and L∞ norms. Through numerical simulations, it is shown that the proposed hybrid QLM-SAPSK approach is not only capable of tackling the inherit singularity at the origin, but also produces effective numerical solutions to the model problem with different nonlinearity parameters and two fractional order derivatives. The accuracy of the present technique is checked via the technique of residual error functions. The QLM-SAPSK technique is simple and efficient for solving the underlying electrohydrodynamic flow model. The computational outcomes are accurate in comparison with those of numerical values reported in the literature.

Spectral Quasilinearization Method for the Numerical Solution of the Non-standard Volterra Integral Equations

In this paper, an iterative scheme is proposed through quasilinearization and the Legendre spectral collocation method to approximate the solution of the non-standard Volterra integral equations (NVIEs). At first, the process of quasilinearization serves to turn a NVIE into a sequence of linear Volterra integral equations. Then, in each iteration, the linear problem is solved by the Legendre spectral collocation method. After that, a convergence analysis is performed. The efficiency and accuracy of the proposed scheme are also verified through several illustrative examples.

On numerical solution of boundary layer flow of viscous incompressible fluid past an inclined stretching sheet in porous medium and heat transfer using spline technique

In this paper, the boundary layer flow of viscous incompressible fluid over an inclined stretching plate in porous media with body force and heat transfer has been studied. To solve this problem, we develop a suitable spline method which is used to calculate the velocity function of the flow problem. We proceed as follows:•With a suitable stream function, the concerned boundary layer equation is converted into non-linear third order ordinary differential equation together with appropriate boundary conditions in an infinite domain which has been further linearized by using quasi-linearization method.•Then, we develop a non polynomial quintic spline technique which has been used to find the numerical values of the velocity function of the flow problem. The convergence analysis of the developed spline technique has been discussed.•Later, the method developed so far has been applied to solve nonlinear boundary value problem for different angles of inclination and Froude number. The values obtained so far have been used to study heat flow problem. Finally, skin friction has been discussed.

Identification of low and medium intensity gas leakage location in medium and ultra-high pressure gas pipelines

An efficient method for calculating the location of stationary gas leakage in wells and gas pipelines with medium and ultra-high pressures is presented. An algorithm for constructing the dependence of the compressibility factor of the gas mixture with a predominance of methane in the region of ultra-high pressures based on experimental data of pressure and temperature is proposed. The algorithm for calculating the leakage location is based on the Bellman method of quasi-linearization of nonlinear problems, with an additional hypothesis of the linear dependence of pressure and temperature on the leakage location coordinates used at each iteration. The issue of selecting the initial approximation is considered, and the advantage of calculating the initial approximation using a simplified non-isothermal flow model is justified. Examples of solving practical problems of high interest are given, and the high accuracy of calculating the gas leakage location using the proposed algorithm in gas pipelines with medium pressures (around 9 MPa) and ultra-high pressures (around 20 MPa) for small and medium gas leaks is demonstrated.

Entropy Generation and Regression Analysis of Magnetohydrodynamic Stagnation Point Flow of a Casson Fluid with Radiative and Dissipative Heat Transfer and Hall Effects

The phenomenon of heat transfer is prevalent in industries and has an extensive range of applications. However, mostly the discussion of heat transfer problems is limited to the study of the first law of thermodynamics, which deals with energy conservation. It is just restricted to the quantity of energy, not to its quality; i.e., there is no difference between the work (high‐grade energy) and the heat (low‐grade energy). A measurement of the degree of randomness of energy in a system is known as entropy. It is unavailable for doing useful work because work takes place only from ordered molecular motion. Even though many boundary layer models exist in the literature to investigate the flow and heat transfer of various fluids along a stretching surface, they have not yet been used at their maximum ability. The main motive of the current research is to discuss entropy generation or its minimization during heat transfer. This work presents an entropy generation analysis for the transient three‐dimensional stagnation point flow of a hydromagnetic Casson fluid flowing over a stretching surface in the existence of Hall current, viscous dissipation, and nonlinear radiation. The physical configuration of the present work is described in terms of partial differential equations (PDEs) of nonlinear nature. Furthermore, these PDEs are converted into ordinary differential equations by using some relevant similarity transformations. An efficient numerical method named as the spectral quasilinearization method (SQLM) is used to solve this model. The expression of the Bejan number and volumetric entropy generation rate is also computed. A parametric analysis, including the essential physical parameters, is performed to examine the influences of distinct flow parameters on the velocity profile, temperature profile, Bejan number, entropy generation number, and the coefficients of skin friction and the Nusselt number. In order to further insight into the emerging physical quantities of engineering interest, multiple quadratic regression models are used to estimate the coefficients of skin friction and heat transfer.

Entropy generation analysis of MHD hybrid nanofluid flow due to radiation with non-erratic slot-wise mass transfer over a rotating sphere

Flow over rotating spheres has serious applications in fiber coating, rotating machinery design and parts, and projectile missions. Therefore, the current study investigates the flow and entropy generation of a radiative magneto-hybrid nanofluid flow over rotating sphere. Further, the temperature-sensorial water properties in studying water-based nanofluids are ignored, though this erratic behavior of water characteristics influences the physical characteristics of the corresponding hybrid nanofluid. Hence the current framework is one of the foremost projects to introduce the variable nature of water properties in the water-hosted hybrid nanofluid flow analysis. The investigation is accomplished in the account of the modified Buongiorno model (MBM). The flow separation in this kind of flow geometry is controlled by using non-erratic slot-mass transfer. The mathematical representations of the physical principles of the flow are solved using (i) congenial transformation (ii) quasilinearization (iii) methods of finite differences to form a block matrix system, and (iv) Varga’s iterative algorithm. Some of the major outcomes are: The flow separation can be significantly delayed using slot suctions, precisely appointing them downstream with higher strengths; heat transport performance drastically subsidies for high viscous dissipation (Ec), entropy produces at higher rate for viscous heating (Br) and high angular speed (Ω).

Investigation of second grade viscoelastic non-Newtonian nanofluid flow on the curve stretching surface in presence of MHD

MHD extended flow of second-grade viscoelastic nanofluid is studied in two-dimensional mode on a curve stretching surface. Items like Joule heating and curvature parameter have been applied to inspect the heat transfer and mass transfer rates. To produce the nonlinear ordinary differential system, appropriate transformations are used. The quasi-linearization method is utilized to derive the solutions numerically. The impact of specific variables on the properties of fluid has been studied. Conclusions indicate that the increase in Schmidt number halted the fluid concentration, whereas temperature grew because of an increment in radiation parameter. Viscous fluid velocity and concentration are reduced faster in comparison to viscoelastic fluid. The surface drag force is an increasing function of the non-Newtonian fluid. When values of Nt and Nb are increased, the heat transfer rate of second-grade fluid increases compared to Newtonian fluid. Schmidt number and chemical reaction parameters highly affect the mass transfer of second-grade fluid compared to Newtonian fluid.

MHD Powell–Eyring nanofluid motion with convective surface condition and Dufour–Soret impact past a vertical plate: Lie group analysis

The formerly neglected impact of the convective surface condition and the Soret–Dufour effect on a Powell–Eyring fluid over a vertical plate is brought to light in the present study. As such, this study focuses on a Powell–Eyring hydromagnetic nanofluid through a vertical plate with the convective surface condition and the Dufour–Soret effect. The governing flow equations containing all the thermo-physical properties are modelled mathematically through the momentum, energy and species conservation equations. We convert the governing partial differential equation into ordinary differential equations using the Lie symmetry group of scaling. The technique is theorized on the self-transformation mappings and symmetric properties of differential equations. The spectral-based quasi-linearization method was introduced to account for all the resulting flow parameters. The Biot number has a significant influence on the flow profile. Graphical analyses of some parameters are shown for the skin friction, the Nusselt and Sherwood numbers. The results herein give some insight into the use of the Powell–Eyring nanofluid for industrial applications.

Analysis of Reciprocal Thermal Conductivity on Free Convection Flow along a Wavy Vertical Surface

The effects of thermal conductivity which depend on temperature are conversely proportional with the linear function of temperature on free convective flow where the fluid is viscous and incompressible along a heated uniform and the vertical wavy surface has been examined in this study. The boundary layer equations with the associated boundary conditions that govern the flow are converted into a nondimensional form by using an appropriate transformation. In the domain of a vertical plate that is flat, the resulting method of nonlinear PDEs is mapped and then worked out numerically by applying the implicit central finite difference technique with Newton’s quasilinearization method, and the block Thomas algorithm is well known as the Keller-box method. The outputs are obtained in the terms of the heat transferring rate, the frictional coefficient of skin, the isotherms, and streamlines. The outcomes showed that the local heat transferring rate, the local skin friction coefficient, the temperature, and the velocity all are decreasing, and both the thermal layer of boundary and velocity become narrower with the rising values of reciprocal variation of temperature-dependent thermal conductivity. On the other hand, the friction coefficient of skin, the velocity, and the temperature decrease where the friction coefficient of skin and velocity decrease by 43% and 64%, respectively, but the heat transfer rate increases by 61% approximately, and both the boundary layer thermal and velocity become thinner when the Prandtl number increases.

Quasilinearization-based Legendre collocation method for solving a class of functional Volterra integral equations

In this paper, a combination of the quasilinearization and the Legendre spectral collocation methods is introduced to approximate the solution of the nonlinear functional Volterra integral equations. Throughout this process, the quasilinearization method converts the nonlinear functional Volterra integral equation to a sequence of linear integral equations. Then, in each iteration, the obtained linear integral equation is solved using the Legendre spectral collocation method. After that, a convergence analysis is discussed in detail. Finally, several numerical examples are included to demonstrate the capability and validity of the proposed method.

On the suitability of differential transform method for solving the self‐similar channel flow problems

AbstractThis article presents, for the first time, a comparative analysis of two different solution methodologies for the numerical investigation of flow and thermal behavior of Casson fluid through a channel having linearly extending lower wall while the upper wall is kept at rest. MHD Casson fluid boundary layer flow is supposed to be steady, laminar, viscous, and incompressible. We have taken into account the mixed convection effects, whereas the usual no slip condition has been imposed on both walls. The Navier–Stokes equations for the proposed model have been converted into nonlinear and coupled system of ordinary differential equations (ODEs) by employing similarity variables, which are then solved by using the differential transform method (DTM) and the quasi‐linearization method (QLM). We note that the quasi‐linearization approach is better than the DTM, as the latter becomes inappropriate for the increasing values of the parameters involved. Moreover, the variation of the different parameters for velocity and temperature profiles has been analyzed graphically.

Analysis of entropy generation and activation energy on double diffusive magnetohydrodynamic Williamson nanofluid flow featuring nonlinear thermal radiation

In this article, we study the effects of the generation of entropy on the transfer of heat and mass by considering the activation energy in the Williamson nanofluid. The importance of Brownian motion and thermophoresis are analysed in various fields over a vertical stretching sheet. The mathematical governing basic equations are solved efficiently by using the spectral quasilinearisation (SQL) method. Numerous physical parameters are considered to plot profiles of nanofluid velocity, temperature, concentration and entropy generation fields. It is exhibited that the velocity profiles tend to reduce due to enhancement in magnetic and Williamson parameters. Furthermore, the entropy generation profile diminishes for the higher Williamson parameters, whereas it enhances due to higher values of the Brinkman numbers. Also, it is noticed that the temperature profile increases by enhancing the values of the activation parameter, whereas the entropy generation diminishes due to increasing the values of the activation parameter.

An adaptive space-step simulation approach for steam heating network considering condensate loss

Steam heating plays an irreplaceable role in industrial productions and occupies a specific proportion of the total heating load. High-precision and efficient simulation is of substantial significance to the economical operation, safety and reliability and energy utilization efficiency of the steam heating network. Widely applicable and adequately accurate simulation approach for the steam heating network is in necessity. A novel adaptive space-step simulation approach for steam heating network considering the condensate loss is developed in this paper. The model equations are precisely derived from the primitive differential equations based on the trapezoidal formula and reorganized into the matrix form for computation acceleration. Meanwhile, a quasi-linear fitting method is proposed to simplify the calculation of steam enthalpy. To solve the model, the computation process is decoupled as hydraulic calculation and thermal calculation, and the converge is achieved through alternating iterative algorithm. The condensate check is introduced after the thermal calculation, in which the condensate loss can be calculated. To acquire the expected simulation precision, an adaptive space-step determination method is developed, in which minimum number of nodes are automatically added to divide long pipes into shorter segments. Finally, case study is conducted on single pipeline cases and a 20-node network. Results show that the proposed approach is applicable to networks containing superheated and saturated steam simultaneously. The error can be controlled to the initially set precision of 0.1 kPa and 0.1 °C through space-step adaption, and nodes are mostly added near the transition point from superheated steam to saturated steam for precision need.

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.