A study of one dimensional nonlinear diffusion equations by Bernstein polynomial based differential quadrature method

The purpose of this work is to study numerical solutions of nonlinear diffusion equations such as Fisher’s equation, Burgers’ equation and modified Burgers’ equation by applying Bernstein differential quadrature method (BDQM). These nonlinear diffusion equations occur in many applications of theoretical, engineering and environmental sciences. Therefore, finding numerical solutions of these equations is very important. In BDQM, Bernstein polynomials are used as base functions to find weighting coefficients of differential quadrature method. We have applied BDQM on eleven different test problems taken from the literature and the computed results confirm that BDQM is an efficient method for finding solution of nonlinear partial differential equations. It is found that BDQM produces very good results even at small number of grid points.     

An optimal scheme for multiple roots of nonlinear equations with eighth-order convergence

We have a good number of eighth-order iterative methods for simple zeros of nonlinear equations in the available literature. But, unfortunately, we don’t have a single iterative method of eighth-order for multiple zeros with known or unknown multiplicity. Some scholars from the worldwide have tried to present optimal or non-optimal multipoint eighth-order iteration functions. But, unfortunately, none of them get success in this direction and attained maximum sixth-order convergence in the case of multiple zeros with known multiplicity m. Motivated and inspired by this fact, we propose an optimal scheme with eighth-order convergence based on weight function approach. In addition, an extensive convergence study is discussed in order to demonstrate the eighth-order convergence of the present scheme. Moreover, we also show the applicability of our scheme on some real life and academic problems. These problems illustrate that our methods are more efficient among the available multiple root finding techniques.     

Discretization of unknown functions: A new method to solve partial differential equations with mixed boundary conditions

In this paper, we develop a new numerical technique to obtain an approximate solution of partial differential equations subject to mixed boundary conditions (MBCs). The approach has been applied to a class of differential equations which frequently arise in a large variety of problems such as heat conduction, potential theory, and diffusion‐controlled chemical reactions. In our approach, based on the discretization of unknown functions (DF), the solution is expressed as a series expansion and the determination of the series coefficients is reduced to the solution of a system of algebraic equations. The main advantages of the DF procedure are: (a) the smoothness of the function and of its first derivative in the different domains, whereas the other numerical methods generally show a highly oscillating behavior; (b) the fast convergence of the series expansion. This method has been applied to solve diffusion problems in different coordinate systems (trigonometric, cylindrical and spherical). The obtained results have been compared with the analytical solution (when available) as well as with other numerical methods commonly used to solve MBCs problems. This revised version was published online in July 2006 with corrections to the Cover Date.     

Multidimensional stability analysis of a family of biparametric iterative methods: CMMSE2016

In this paper, we present a multidimensional real dynamical study of the Ostrowsky–Chun family of iterative methods to solve systems of nonlinear equations. This family was defined initially for solving scalar equations but, in general, scalar methods can be transferred to make them suitable to solve nonlinear systems. The complex dynamical behavior of the rational operator associated to a scalar method applied to low-degree polynomials has shown to be an efficient tool for analyzing the stability and reliability of the methods. However, a good scalar dynamical behavior does not guarantee a good one in multidimensional case. We found different real intervals where both parameters can be defined assuring a completely stable performance and also other regions where it is dangerous to select any of the parameters, as undesirable behavior as attracting elements that are not solution of the problem to be solved appear. This performance is checked on a problem of chemical wave propagation, Fisher’s equation, where the difference in numerical results provided by those elements of the class with good stability properties and those showed to be unstable, is clear.     

Nuclear reactor application on Jeffrey fluid flow with Falkner-skan factor,Brownian and thermophoresis,non linear thermal radiation impacts past a wedge

In this paper, an impact of non-linear thermal radiation, Brownian and thermophoresis on an MHD through a wedge with dissipative impacts for Jeffrey fluid is investigated. In addition, heat transport analysis is carried out. This work's originality is attributable to the Jeffrey fluid formulation, nonlinear thermal radiation, Brownian and Thermophoresis. The boundary layer approximations are examined, to transform the governing equations into partial differential equations. Utilizing appropriate similarity transformations, the boundary value issue is expressed in ordinary differential form. BVP4C, a nonlinear numerical method, was utilized to determine the outcomes of velocity, concentration and temperature fields at multiple points of the measured quantities. The skin friction term, Sherwood and Nusselt numbers were analyzed in depth, and the findings are achieved graphically and tabularly. A comparison via the previously published data reveals a good degree of concordance. This research focuses mostly on the modelling of flow in a nuclear reactor. The boundary layer flow caused by a wedge surface play s a crucial role the aspects of geothermal and heat exchangers systems.     

Analytic solution of nonlinear singularly perturbed initial value problems through iteration

This paper is concerned with singularly perturbed initial value problems for systems of ordinary differential equations. Here our emphasis will be on nonlinear phenomena and properties, particularly those with physical relevance. Since very few nonlinear systems can be solved explicitly, one must typically rely on a numerical scheme to accurately approximate the solution. However, numerical schemes do not always give accurate results, and we discuss the class of stiff differential equations, which present a more serious challenge to numerical analysts. In this paper, we derive in closed from, analytic solution of stiff nonlinear initial value problems, through iteration. The obtained sequence of iterates is based on the use of Lagrange multipliers. Moreover, the illustrative examples shows the efficiency of the method.     

A multi-domain spectral method for non-Darcian mixed convection flow in a power-law fluid with viscous dissipation

The purpose of this study is to investigate non-Darcian mixed convection flow, heat and mass transfer in a non-Newtonian power-law fluid over a flat plate embedded in porous medium with suction and viscous dissipation and also is to demonstrate the application and utility of a recently developed multi-domain bivariate spectral quasi-linearisation method (MD-BSQLM) in finding the solutions of highly nonlinear differential equations. The flow is subject to, among other source terms, internal heat generation, thermal radiation and partial velocity slip. The coupled system of nonlinear partial differential equations are solved using a MD-BSQLM to find the fluid properties, the skin friction, as well as the heat and mass coefficients. We have presented selected results that give the significance of some system parameters on the fluid properties. This MD-BSQLM has not been used before in the literature to find the nature of the solutions of power-law fluids. Indeed, validation of this numerical method for general fluid flows, heat and mass transfer problems has not yet been done. This study presents the first opportunity to evaluate the accuracy and robustness of the MD-BSQLM in finding solutions of non-Newtonian fluids.     

On a partial differential equation method for determining the free energies and coexisting phase compositions of ternary mixtures from light scattering data

In this paper we present a method for determining the free energies of ternary mixtures from light scattering data. We use an approximation that is appropriate for liquid mixtures, which we formulate as a second-order nonlinear partial differential equation. This partial differential equation (PDE) relates the Hessian of the intensive free energy to the efficiency of light scattering in the forward direction. This basic equation applies in regions of the phase diagram in which the mixtures are thermodynamically stable. In regions in which the mixtures are unstable or metastable, the appropriate PDE is the nonlinear equation for the convex hull. We formulate this equation along with continuity conditions for the transition between the two equations at cloud point loci. We show how to discretize this problem to obtain a finite-difference approximation to it, and we present an iterative method for solving the discretized problem. We present the results of calculations that were done with a computer program that implements our method. These calculations show that our method is capable of reconstructing test free energy functions from simulated light scattering data. If the cloud point loci are known, the method also finds the tie lines and tie triangles that describe thermodynamic equilibrium between two or among three liquid phases. A robust method for solving this PDE problem, such as the one presented here, can be a basis for optical, noninvasive means of characterizing the thermodynamics of multicomponent mixtures.     

Numerical approximation of a nonisothermal two-dimensional general rate model of reactive liquid chromatography

A nonlinear and nonisothermal two-dimensional general rate model is formulated and approximated numerically to allow quantitatively analyzing the effects of temperature variations on the separations and reactions in liquid chromatographic reactors of cylindrical geometry. The model equations form a nonlinear system of convection-diffusion-reaction partial differential equations coupled with algebraic equations for isotherms and reactions. A semidiscrete high-resolution finite volume method is modified to approximate the system of partial differential equations. The coupling between the thermal waves and concentration fronts is demonstrated through numerical simulations, and important parameters are pointed out that influence the reactor performance. To evaluate the precision of the model predictions, consistency checks are successfully carried out proving the accuracy of the predictions. The results allow to quantify the influence of thermal effects on the performance of the fixed beds for different typical values of enthalpies of adsorption and reaction and axial and radial Peclet numbers for mass and heat transfer. Furthermore, they provide useful insight into the sensitivity of nonisothermal chromatographic reactor operation.     

SULFONATION IN A FALLING FILM REACTOR: COMPARATION OF EXPERIMENTAL RESULTS WITH MATHEMATICAL MODEL PREDICTIONS

A mathematical model is developed to simulate a falling film reactor for sulfonation/sulfation. In the model, the reaction rate is considered to be controlled by the mass transfer in the gas phase or in the liquid phase. The gas phase mass and heat transfers are calculated by empiric equations; in the liquid phase, they are calculated by solving with numerical methods the partial differential equations which describe the system. In these equations, and eddy diffusion is considered, following the Levich's theories

The model results are compared with the experimental results obtained by the authors in a pilot plant, for the dodecylbenzene sulfonation.     

Numerical solution of nonlinear and non-isothermal general rate model of reactive liquid chromatography

Abstract

A nonlinear general rate model (GRM) of liquid chromatography is formulated to analyze the influence of temperature variations on the dynamics of multi-component mixtures in a thermally insulated liquid chromatographic reactor. The mathematical model is formed by a system of nonlinear convection–diffusion reaction partial differential equations (PDEs) coupled with nonlinear algebraic equations for reactions and isotherms. The model equations are solved numerically by applying a semi-discrete high-resolution finite volume scheme (HR-FVS). Several numerical case studies are conducted for two different types of reactions to demonstrate the influence of heat transfer on the retention time, separation, and reaction. It was found that the enthalpies of adsorption and reaction significantly influence the reactor performance. The ratio of density time heat capacity of solid and liquid phases significantly influences the magnitude and velocity of concentration and thermal waves. The results obtained could be very helpful for further developments in non-isothermal reactive chromatography and provide a deeper insight into the sensitivity of chromatographic reactor operating under non-isothermal conditions.     

Application of binomial coefficients in representing central difference solution to a class of PDE arising in chemistry

The use of differential equations for modeling chemical systems and solving by numerical approaches (e.g. finite difference methods) are prevalent in chemistry-related problems. As an extension to the direct use of Pascal’s Triangle to obtain the forward and backward difference equations to partial differentials by Lim [Mathematical Medley 31 (2004) 2], this paper proposes the use of binomial coefficient to generate central difference equations to odd-ordered partial differentials in a single-step operation. All finite difference equations to partial differentials shown herein display finite series of palindromic coefficients with alternating signs     

A new efficient technique for solving two-point boundary value problems for integro-differential equations

In this paper, we propose a new efficient method based on a combination of Adomian decomposition method (ADM) and Green’s function for solving second-order boundary value problems (BVPs) for integro-differential equations (IDEs). The proposed method depends on constructing Green’s function before establishing the recursive scheme for the solution components. Unlike the ADM or modified ADM , the proposed method avoids solving a sequence of difficult nonlinear equations (transcendental equations) for the unknown parameters. The proposed method provides a direct recursive scheme for obtaining the series solution with easily calculable components. We also provide a sufficient condition that guarantees a unique solution to the second-order BVPs for IDEs. Convergence and error analysis of the proposed method are also discussed. Convergence analysis is reliable enough to estimate the error bound of the series solution. Some numerical examples are included to demonstrate the accuracy, applicability, and generality of the proposed approach. The numerical results reveal that the proposed method is very effective and simple.     

A novel thermodynamic state recursion method for description of nonideal nonlinear chromatographic process of frontal analysis

A novel thermodynamic state recursion (TSR) method, which is based on nonequilibrium thermodynamic path described by the Lagrangian-Eulerian representation, is presented to simulate the whole chromatographic process of frontal analysis using the spatial distribution of solute bands in time series like as a series of images. TSR differs from the current numerical methods using the partial differential equations in Eulerian representation. The novel method is used to simulate the nonideal, nonlinear hydrophobic interaction chromatography (HIC) processes of lysozyme and myoglobin under the discrete complex boundary conditions. The results show that the simulated breakthrough curves agree well with the experimental ones. The apparent diffusion coefficient and the Langmuir isotherm parameters of the two proteins in HIC are obtained by the state recursion inverse method. Due to its the time domain and Markov characteristics, TSR is applicable to the design and online control of the nonlinear multicolumn chromatographic systems.     

Wavelet-based collocation method for stiff systems in process engineering

Abrupt phenomena in modelling real-world systems such as chemical processes indicate the importance of investigating stiff systems. However, it is difficult to get the solution of a stiff system analytically or numerically. Two such types of stiff systems describing chemical reactions were modelled in this paper. A numerical method was proposed for solving these stiff systems, which have general nonlinear terms such as exponential function. The technique of dealing with the nonlinearity was based on the Wavelet-Collocation method, which converts differential equations into a set of algebraic equations. Accurate and convergent numerical solutions to the stiff systems were obtained. We also compared the new results to those obtained by the Euler method and 4th order Runge–Kutta method.     

A generator of families of two-step numerical methods with free parameters and minimal phase-lag

This research work is oriented in the behaviour of oscillating systems. In order to study such problems, we deal with the solution of ordinary second order differential equations. A generator of families of numerical methods is developed in our effort to solve equations of that type. The families created have constant coefficients and free parameters. We calculate the free parameters taking into consideration the condition of minimal phase-lag. The new methods are applied to the problem of the time independent Schrödinger equation and their results are presented. We also examine the properties of stability and minimum local truncation error.     

A simplified model of oscillating electrolytes

Unstable electrophoretic transport leading to oscillations in concentration profiles occur in certain electrolyte systems known as oscillating electrolytes whose eigenmobilities are complex valued. The study of the nonlinear behavior of such systems is of great interest but is constrained due to a high degree of complexity in the governing equations. Here we present a simplified model of unstable electrophoretic transport in a binary system that reduces the governing equations to two partial differential equations only and does away with other equations that characterize acid–base dissociation reactions and electroneutrality. We present analytical expressions for electromigration fluxes and validate the model with full nonlinear simulations. The model exhibits similar nonlinear behavior as the actual unstable electrophoretic system under various initial disturbances. For comparison, we also show that similar modeling for a stable system predicts concentration profiles that quantitatively agree with its nonoscillating dynamics. Moreover, the unique feature of electromigration flux in oscillating electrolytes that unfolds from the modeling led us to find an elegant explanation of the instability mechanism. Our theory gives a qualitative understanding of the existence and growth of large oscillation patterns in oscillating electrolytes.