The Effects of Electrical Conductivity on Fluid Flow between Two Parallel Plates in a Porous Medioum

This paper deals with a mathematical model of a fluid flowing between two parallel plates in a porous medium under the influence of electromagnetic forces (EMF). The continuity, momentum, and energy equations were utilized to describe the flow. These equations were stated in their nondimensional forms and then processed numerically using the method of lines. Dimensionless velocity and temperature profiles were also investigated due to the impacts of assumed parameters in the relevant problem. Moreover, we investigated the effects of Reynolds number , Hartmann number M, magnetic Reynolds number , Prandtl number , Brinkman number , and Bouger number , beside those of new physical quantities (N , ). We solved this system by creating a computer program using MATLAB.

Approximate Models of Microbiological Processes in a Biofilm Formed on Fine Spherical Particles

This paper concerns the dynamical modeling of the microbiological processes that occur in the biofilms that are formed on fine inert particles. Such biofilm forms e.g. in fluidized-bed bio-reactors, expanded bed biofilm reactors and biofilm air-lift suspension reactors. An approximate model that is based on the Laplace–Carson transform and a family of approximate models that are based on the concept of the pseudo-stationary substrate concentration profile in the biofilm were proposed. The applicability of the models to the microbiological processes was evaluated following Monod or Haldane kinetics in the conditions of dynamical biofilm growth. The use of approximate models significantly simplifies the computations compared to the exact one. Moreover, the stiffness that was present in the exact model, which was solved numerically by the method of lines, was eliminated. Good accuracy was obtained even for large internal mass transfer resistances in the biofilm. It was shown that significantly higher accuracy was obtained using one of the proposed models than that which was obtained using the previously published approximate model that was derived using the homotopy analysis method.

Numerical algorithms for solving an elliptic optimal control problem with a power-law nonlinearity

The paper is devoted to the development and analysis of approximation-iteration algorithms based on the method of grids and the method of lines for solving an elliptic optimal control problem with a power-law nonlinearity. For the numerical solution of the main boundary value problem and the adjoint one, the second order of accuracy difference schemes are applied using the implicit method of simple iteration. Computational schemes of the method of lines for solving the above-mentioned elliptic boundary value problems are implemented in combination with the shooting method for the approximate solution of boundary value problems for the corresponding ordinary differential equations systems arising in the considered domain after lattice approximation. To minimize the objective functional, well-known gradient-type methods (gradient projection and conditional gradient methods) of constrained optimization are used. The essence of the proposed approximation-iteration approach consists in replacing the original extremal problem with a sequence of grid problems that approximate it on a set of refining grids, and applying an iterative gradient-type method to each of the "approximate" extremal problems. In this case, we propose to construct only a few approximations to the solution for each of the "approximate" problems and to take the last of these approximations, using piecewise linear interpolation, as the initial approximation in the iterative process for the next "approximate" problem. The sequence of the corresponding piecewise linear interpolants is considered as a sequence of approximations to the solution of the original extremal problem. The paper discusses the theoretical foundations of this combined approach, as well as its advantages over traditional methods using the example of solving a model optimal control problem

Method of lines for solving linear equations of mathematical physics with the third and first types boundary conditions

The use of the method of lines in solving multidimensional problems of mathematical physics makes it possible to eliminate the discrepancies caused by the use of the sweep method in certain coordinates. As a result, the solution of the Poisson equation, for example, is obtained without using the relaxation method. In the article, the problem on the eigenvalues and vectors of the transition matrix is solved for boundary conditions of the third and first types, used to solve a one-dimensional equation of parabolic type by the method of lines. Due to the features of boundary conditions of the third type for determining the eigenvalues, a mixed method was proposed based on the Vieta theorem and the representation of the characteristic equation in trigonometric form typical for the method of lines. To solve the eigenvector problem, a simple sweep method was used with the algebraic compliments to the transition matrix. Discontinuous solutions of a one-dimensional parabolic equation were presented for various values of complex 1 -αl; the method for solving the characteristic equation was selected based on these values. The calculation results are in good agreement with the analytical solution.

Black hole initial data by numerical integration of the parabolic–hyperbolic form of the constraints

The parabolic–hyperbolic form of the constraints is integrated numerically. The applied numerical stencil is fourth-order accurate (in the spatial directions) while “time”-integration is made by using the method of lines with a fourth-order order accurate Runge–Kutta scheme. The proper implementation of the applied numerical method is verified by convergence tests and monitoring the relative and absolute errors is determined by comparing numerically and analytically known solutions of the constraints involving boosted and spinning vacuum single black hole configurations. The main part of our investigations is, however, centered on the construction of initial data for distorted black holes which, in certain cases, have non-negligible gravitational wave content. Remarkably, the applied new method is unprecedented in that it allows to construct initial data for highly boosted and spinning black holes, essentially for the full physical allowed ranges of these parameters. In addition, the use of the evolutionary form of the constraints is free from applying any sort of boundary conditions in the strong field regime.

Nonlinear dynamics of the predator – prey system in a heterogeneous habitat and scenarios of local interaction of species

The purpose of this work is to study the influence of various local models in the equations of diffusion–advection– reaction on the spatial processes of coexistence of predators and prey under conditions of a nonuniform distribution of the carrying capacity. We consider a system of nonlinear parabolic equations to describe diffusion, taxis, and local interaction of a predator and prey in a one-dimensional habitat. Methods. We carried out the study of the system using the dynamical systems approach and a computational experiment based on the method of lines and a scheme of staggered grids. Results. The behavior of the predator – prey system has been studied for various scenarios of local interaction, taking into account the hyperbolic law of prey growth and the Holling effect with nonuniform carrying capacity. We have established paradoxical scenarios of interaction between prey and predator for several modifications of the trophic function. Stationary and nonstationary solutions are analyzed considering diffusion and directed migration of species. Conclusion. The trophic function that considers the heterogeneity of the resource is proposed, which does not lead to paradoxical dynamics.

Nonlinear Longitudinal Vibrations in Accreting One-dimensional Nanostructures

A governing partial differential equation that describes an accreting nanochain containing an attachment of infinite atoms was considered in this paper. We transformed the space variable u(t,r) → v(τ,x) (for a governing PDE formulated in previous research studies) and introduced a function of linear growth. The boundary conditions were also transformed into the new variables, the left end of the accreting chain was free u(t, r=0)=0 while the right end was fixed. The method of lines was also employed to numerically analyze the governing partial differential equation. We detailed the differential transformation for the change of variables used in obtaining the transformed partial differential equation. We also considered what happens with the introduction of the viscous damping term, (δ). The governing partial differential equation was formulated. Numerical simulations for both cases, was then carried out.

Application of SPD-RBF method of lines for solving nonlinear advection–diffusion–reaction equation with variable coefficients

Purpose The purpose of this study is to use the method of lines to solve the two-dimensional nonlinear advection–diffusion–reaction equation with variable coefficients. Design/methodology/approach The strictly positive definite radial basis functions collocation method together with the decomposition of the interpolation matrix is used to turn the problem into a system of nonlinear first-order differential equations. Then a numerical solution of this system is computed by changing in the classical fourth-order Runge–Kutta method as well. Findings Several test problems are provided to confirm the validity and efficiently of the proposed method. Originality/value For the first time, some famous examples are solved by using the proposed high-order technique.