Comparative analysis of variational iteration method and Haar wavelet method for the numerical solutions of Burgers–Huxley and Huxley equations

In this paper, variational iteration method is applied to compute the numerical solutions of non-linear partial differential equations like Huxley and Burgers–Huxley equation. The approximate solutions of the Huxley and Burgers–Huxley equations are compared with the Haar wavelet solution as well as with the exact solutions. The present method is very simple, effective and convenient analytical method with small computational overhead.     

The mu-derivative and its applications to finding exact solutions of the Cahn-Hilliard, Korteveg-de Vries, and Burgers equations

A new transformation termed the mu-derivative is introduced. Applying it to the Cahn-Hilliard equation yields dynamical exact solutions. It is shown that the mu-transformed Cahn-Hilliard equation can be presented in a separable form. This transformation also yields dynamical exact solutions and separable forms for other nonlinear models such as the modified Korteveg-de Vries and the Burgers equations. The general structure of a nonlinear partial differential equation that becomes separable upon applying the mu-derivative is described.     

The wavelet methods to linear and nonlinear reaction–diffusion model arising in mathematical chemistry

In this paper, we have applied an accurate and efficient wavelet scheme (due to Legendre polynomial) to find the numerical solutions for a set of coupled reaction–diffusion equations. This technique provides the solutions in rapid convergence series with computable terms for the problems with high degree of non linear terms appearing in the governing differential equations. The highest derivative in the differential equation is expanded into wavelet series, this approximation is then integrated while the boundary conditions are applied by using integration constants. With the help of operational matrices, the nonlinear reaction–diffusion equations are converted into a system of algebraic equations. Finally, some numerical examples to demonstrate the validity and applicability of the method have been furnished. The use of Legendre wavelets is found to be accurate, efficient, simple, and computationally attractive. This wavelet method can be used for obtaining quick solution in many chemical Engineering problems.     

A computational modeling of two dimensional reaction–diffusion Brusselator system arising in chemical processes

In this article, the authors proposed a modified cubic B-spline differential quadrature method (MCB-DQM) to show computational modeling of two-dimensional reaction–diffusion Brusselator system with Neumann boundary conditions arising in chemical processes. The system arises in the mathematical modeling of chemical systems such as in enzymatic reactions, and in plasma and laser physics in multiple coupling between modes. The MCB-DQM reduced the Brusselator system into a system of nonlinear ordinary differential equations. The obtained system of nonlinear ordinary differential equations is then solved by a four-stage RK4 scheme. Accuracy and efficiency of the proposed method successfully tested on four numerical examples and obtained results satisfy the well known result that for small values of diffusion coefficient, the steady state solution converges to equilibrium point $(B,A/B)$ if $1-A+B^{2}>0$ .     

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.     

A family of parametric schemes of arbitrary even order for solving nonlinear models: CMMSE2016

Many problems related to gas dynamics, heat transfer or chemical reactions are modeled by means of partial differential equations that usually are solved by using approximation techniques. When they are transformed in nonlinear systems of equations via a discretization process, this system is big-sized and high-order iterative methods are specially useful. In this paper, we construct a new family of parametric iterative methods with arbitrary even order, based on the extension of Ostrowski’ and Chun’s methods for solving nonlinear systems. Some elements of the proposed class are known methods meanwhile others are new schemes with good properties. Some numerical tests confirm the theoretical results and allow us to compare the numerical results obtained by applying new methods and known ones on academical examples. In addition, we apply one of our methods for approximating the solution of a heat conduction problem described by a parabolic partial differential equation.     

New algorithms for the computation of column dynamics of multicomponent liquid phase adsorption

New and efficient numerical algorithms were developed for simulating column dynamics of multicomponent liquid phase adsorption. Simple and realistic models are used for the simulation. Langmuir form of isotherm and linear driving force rate expressions are employed in the model equations. Algorithms were formulated for three different rate control mechanisms, namely, film diffusion control, particle diffusion control and combined film and particle diffusion control. The algorithms derived are explicit with the exception of the requirement of solving a nonlinear equation in one single variable which is the concentration of a reference species. Thus the tedious iterative calculation procedure for solving simultaneous nonlinear equations in a multicomponent fixed bed system is avoided. Example calculations indicated very good numerical accuracy as verified from an independent check by means of an overall mass balance.     

Numerical solution of a partial differential equation system describing chemical kinetics and diffusion in a cell with the aid of compartmentalization

To build a kinetic model of a cell with diffusion one has to solve a coupled nonlinear partial differential equation system consisting of several hundred equations. (Several hundred chemical components undergoing several hundred reactions.) To solve this formidable mathematical problem the division of the model cell into compartments (most biochemical reactions take place in a certain part of the cell) was suggested.¹ Solving the differential equation system in one compartment, the results can be used as input at other compartments until mutually consistent solutions are achieved. To test this suggestion 10 coupled chemical reactions with diffusion were investigated in a model that contains three compartments. The results in the case of pure diffusion are in excellent agreement with and without compartmentalization. After this the full problem was treated by compartmentalization using for the solution of the differential equation system a discretization of the concentrations as functions of space and time and the Newton–Raphson iterative procedure. The results obtained give reasonable space and time dependence for the concentrations of all 10 components.     

Two reliable wavelet methods to Fitzhugh–Nagumo (FN) and fractional FN equations

Fractional reaction–diffusion equations serve as more relevant models for studying complex patterns in several fields of nonlinear sciences. In this paper, we have developed the wavelet methods to find the approximate solutions for the Fitzhugh–Nagumo (FN) and fractional FN equations. The proposed method techniques provide the solutions in rapid convergence series with computable terms. To the best of our knowledge, until now there is no rigorous wavelet solutions have been reported for the FN and fractional FN equations arising in gene propagation and model. With the help of Laplace operator and Legendre wavelets operational matrices, the FN equation is converted into an algebraic system. Finally, we have given some numerical examples to demonstrate the validity and applicability of the wavelet methods. The power of the manageable method is confirmed. Moreover, the use of the wavelet methods is found to be accurate, efficient, simple, low computation costs and computationally attractive.     

A structure-preserving computational method in the simulation of the dynamics of cancer growth with radiotherapy

In this work, we consider a two-dimensional mathematical model that describes the growth dynamics of cancer when radiotherapy is considered. The mathematical model for the local density of the tumor is described by a parabolic partial differential equation with variable diffusion coefficient. The nonlinear reaction term considers both the logistic law of proliferation of tumor cells and the effect of a treatment against cancer. Suitable initial-boundary conditions are imposed on a bounded spatial domain, and a theorem on the existence and the uniqueness of solutions for the initial-boundary-value problem is proved. Motivated by this result, we design a finite-difference methodology to approximate the solutions of our mathematical model. The scheme is a linear method that is capable of preserving the positivity and the boundedness of the approximations. Some simulations are presented in order to illustrate the performance of the method. Among other conclusions, the numerical results show that the method is able to preserve the analytical features of the relevant solutions of the model.     

Theoretical analysis of the enzyme reaction processes within the multiscale porous biocatalytic electrodes

Mathematical model describing the oxidation of glucose in a multiscale porous biocatalytic electrode is discussed. The model considers herein is composed of two nonlinear differential equations accounting for reaction and diffusion within the hydrogel film. In this letter, approximate analytical expressions for the concentration of mediator, substrate and current have been obtained using the Adomian decomposition method (ADM). Furthermore, a comparison confirmed that our analytical result fitted very well with the numerical solution (Matlab). Sensitivity analysis of the parameters is also reported.     

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.     

Approximate series solution of fourth-order boundary value problems using decomposition method with Green’s function

This paper proposes a new efficient approach for obtaining approximate series solutions to fourth-order two-point boundary value problems. The proposed approach depends on constructing Green’s function and Adomian decomposition method (ADM). Unlike existing methods like ADM or modified ADM, the proposed approach avoids solving a sequence of nonlinear equations for the undetermined coefficients. In fact, the proposed method gives a direct recursive scheme for obtaining approximations of the solution with easily computable components. We also discuss the convergence and error analysis of the proposed scheme. Moreover, several 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.     

Soliton solutions for time fractional coupled modified KdV equations using new coupled fractional reduced differential transform method

In this paper, new Coupled Fractional Reduced Differential Transform has been implemented to obtain the soliton solutions of coupled time fractional modified KdV equations. This new method has been revealed by the author. The fractional derivatives are described in the Caputo sense. By using the present method, we can solve many linear and nonlinear coupled fractional differential equations. The results reveal that the proposed method is very effective and simple for obtaining approximate solutions of fractional coupled modified KdV equations. Numerical solutions are presented graphically to show the reliability and efficiency of the method. Solutions obtained by this new method have been also compared with Adomian decomposition method (ADM).     

INTERFACIAL RHEOLOGY AND KINETICS OF ADSORPTION FROM SURFACTANT SOLUTION

A theoretical approach to the diffusion controlled kinetics of adsorption on the expanding interface of surfactant solutions is developed and compared with the experiment. This approach being an analogue of von Karman's approach to the hydrodynamic boundary layer is applicable to both submicellar and micellar surfactant solutions under large deviations from equilibrium. The partial differential equations of the convective diffusion are reduced to a set of ordinary differential equations of first order and algebraic equations. This simplifies the numerical computations and enhances the interpretation of the experimental data. Dynamic surface tension data for solutions of sodium dodecyl sulfate obtained by the maximum bubble pressure method are interpreted. Reasonable results for the diffusivity of monomers and the rate constant of micellar disintegration have been obtained.

A local approach to interfacial rheology is briefly considered. The applicability of this approach to studies of visco-elastic dilational properties of adsorption layers from low molecular surfactants and proteins is demonstrated.     

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.     

On the Pace–Datyner theory of diffusion of small molecules through polymers

The Pace–Datyner theory for diffusion of penetrant molecules in polymers has been analyzed. It has been found that the correct solution of the problem they pose is possible only at 0 K, since then the separation of two chains at x = ∞ is equal to the minimum of the DiBenedetto potential for their interaction. Otherwise the energy of symmetrical separation is infinite. By using the linearization method to solve the differential equation, Pace and Datyner neglected the problem of unnatural boundary conditions at x = ∞ for temperatures above 0 K. The exact numerical solutions of differential equations at temperature 0 K were therefore compared with the results of the Pace–Datyner linear approximation. For temperatures different from 0 K the solution of the problem is possible only when the proper cutoff is imposed. The analytical expression for the coeffients in the DiBenedetto potential has been found, and the potential can be written as