Some inverse problems of internal-diffusion kinetics of adsorption

Methods are proposed for determining the diffusion coefficients of adsorbed molecules and the adsorption isotherms from given concentrations of the solutes in the external solution. The methods are based on a comparison of experimental data with the numerical results produced by a mathematical model of internal-diffusion kinetics of adsorption from a constant bounded volume.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 66, pp. 41–46, 1988.     

Differential properties of the solution of a mathematical model of kinetic adsorption

Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 41, No. 5, pp. 698–702, May, 1989.     

Numerical solution of an inverse problem for a two-dimensional mathematical model of sorption dynamics

We consider a two-dimensional mathematical model of sorption that allows for inner-diffusion kinetics as well as longitudinal and transverse diffusion. The inverse problem of determining the sorption isotherm from an experimental dynamic output curve is investigated for this model and stable solution methods are proposed for the inverse and the direct problem. The efficiency of the solution methods is explored in computer experiments.     

Numerical analysis of a mathematical model of the myocardial blood-supply system

A hydraulic model of the hemodynamics of the arterial part of the myocardium is considered, and a numerical analysis of the model is conducted. Computer experiments are used to investigate the dependence of blood flows on parametric and structural changes in the system. Translated from Prikladnaya Matematika i Informatika, No. 2, pp. 56–62, 1999.     

Stability of the trivial solution of a one-dimensional mathematical model of thermoelasticity

Lyapunov stability is established for a one-dimensional physically linear mathematical model of thermoelasticity. For this purpose, the convergent iteration process is constructed; it consists of solving hyperbolic and parabolic problems successively by using new estimates for the solution of a mixed problem for the wave equation.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 9, pp. 1239–1252, September, 1993.     

Numerical solution of calcium-mediated dendritic branch model

The standard algorithms for spatial discretizations of calcium-mediated dendritic branch models via finite difference methods are quite accurate, but they are also extremely slow. To improve computational efficiency we apply spatial discretization using a spectral collocation method. Simulations using the spectral collocation method are compared to the finite difference approach using a model for calcium-mediated restructuring with spine pruning. We find that the spectral collocation method is about fifteen times more efficient to achieve similar accuracy than the finite difference approach even though spectral collocation requires more steps.     

Solvability of a mathematical model of dissociative adsorption and associative desorption type

A mathematical model of dissociative adsorption and associative desorption for diatomic molecules is generalized. The model is described by a coupled system of parabolic and ordinary differential equations. The existence and uniqueness theorem of the classical solution is proved.     

Numerical treatment of a mathematical model arising from a model of neuronal variability

In this paper, we describe a numerical approach based on finite difference method to solve a mathematical model arising from a model of neuronal variability. The mathematical modelling of the determination of the expected time for generation of action potentials in nerve cells by random synaptic inputs in dendrites includes a general boundary-value problem for singularly perturbed differential-difference equation with small shifts. In the numerical treatment for such type of boundary-value problems, first we use Taylor approximation to tackle the terms containing small shifts which converts it to a boundary-value problem for singularly perturbed differential equation. A rigorous analysis is carried out to obtain priori estimates on the solution of the problem and its derivatives up to third order. Then a parameter uniform difference scheme is constructed to solve the boundary-value problem so obtained. A parameter uniform error estimate for the numerical scheme so constructed is established. Though the convergence of the difference scheme is almost linear but its beauty is that it converges independently of the singular perturbation parameter, i.e., the numerical scheme converges for each value of the singular perturbation parameter (however small it may be but remains positive). Several test examples are solved to demonstrate the efficiency of the numerical scheme presented in the paper and to show the effect of the small shift on the solution behavior.     

A mathematical model and analytical solution for buckling of inclined beam-columns

The differential equation that governs the buckling behavior of an inclined beam-column is obtained using the energy method, and the use of a suitable change of variable reduces the various geometrical and physical parameters into a single dimensionless length parameter (X). An exact solution is presented via the use of some members of the family of generalized hypergeometric functions. One type of boundary condition (pinned-ends) is presented and analyzed, and others boundary conditions may be easily studied following the same process. The analysis shows the singular behavior of the inclined beam-column and demonstrates the procedure used to obtain the critical values of the axial force. An important application of this model is its usefulness in the analysis of buckling of drillstrings within curved boreholes.     

Numerical solution of nonequilibrium expansion of viscous gas using the Navier-Stokes equation with electron-chemical kinetics

An algorithm is constructed for numerical solution of the problem of nonequilibrium supersonic expansion of hot viscous gas in short plane nozzles allowing for the excitation of the molecular electron levels. The suitability of chlorine as an active medium for gas-dynamic lasers in the visible and ultraviolet regions is demonstrated.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 56, pp. 68–75, 1985     

Numerical techniques for simulating a fractional mathematical model of epidermal wound healing

A number of mathematical models investigating certain aspects of the complicated process of wound healing are reported in the literature in recent years. However, effective numerical methods and supporting error analysis for the fractional equations which describe the process of wound healing are still limited. In this paper, we consider the numerical simulation of a fractional mathematical model of epidermal wound healing (FMM-EWH), which is based on the coupled advection-diffusion equations for cell and chemical concentration in a polar coordinate system. The space fractional derivatives are defined in the Left and Right Riemann-Liouville sense. Fractional orders in the advection and diffusion terms belong to the intervals (0,1) or (1,2], respectively. Some numerical techniques will be used. Firstly, the coupled advection-diffusion equations are decoupled to a single space fractional advection-diffusion equation in a polar coordinate system. Secondly, we propose a new implicit difference method for simulating this equation by using the equivalent of Riemann-Liouville and Grünwald-Letnikov fractional derivative definitions. Thirdly, its stability and convergence are discussed, respectively. Finally, some numerical results are given to demonstrate the theoretical analysis.     

Numerical solution for the model of vibrating elastic membrane with moving boundary

In this work, we are interested in obtaining an approximated numerical solution for the model of vibrating elastic membranes with moving boundary. The model is an extension of Kirchhoff’s model, which takes into account the change of size during the vibration. We apply the finite element method with a finite difference method in time to obtain an approximated numerical solution. Some numerical experiments are presented to show the effect of moving boundary effects in vibrating elastic membranes.     

Numerical solution of a thermomechanical coupled model governing glacier evolution

In this paper the numerical solution of a highly nonlinear model for the thermomechanical behavior of polythermal glaciers is presented. The modeling follows the shallow ice approximation (SIA) for glaciers introduced in Fowler (1997) [13]. The model has been extended to incorporate additional moving boundaries and other nonlinear features. Moreover, a fixed domain formulation is proposed to avoid the computational drawbacks of a time-dependent domain in the numerical simulation with front tracking methods. In this setting, the coupled problem is decomposed into different nonlinear problems which allow one to obtain sequentially the profile evolution, the velocity field, the glacier surface and atmospheric temperatures, basal magnitudes and the temperature distribution inside the ice mass. A fixed point iteration algorithm converges to the solution of the nonlinear coupled problem. Among different numerical methods involved in the solution of the subproblems, characteristic schemes for time discretization, finite elements for spatial discretization, duality methods for the nonlinearities associated to maximal monotone operators and a Newton scheme for the nonlinear viscous term are proposed. Several numerical simulation examples illustrate the performance of the numerical methods and the behavior of the involved physical magnitudes.     

Numerical solution and optimal control for fractional tumor immune model

In this article, the numerical model of fractional tumor immunity has been described. We have proved and analyzed the model does have a stable solution. In addition to this, the optimal control of their form as well as the numerical approach for the simulation of the control problem, are both brought up and examined. We have presented evidence that demonstrates the existence of the solution. We use an algorithm modeled after the generalized Adams-Bashforth-Moulton style (GABMS) to solve the fractional tumor immune model. This amendment is predicated on changing the form to a memristive one for the first time because such a notion is being utilized for the first time to control this ailment. The dissection results have been interpreted using numerical simulations we created. To calculate the results, we relied on the Maple 15 programming language.     

Partitioning mathematical programs for parallel solution

This paper describes heuristics for partitioning a generalM × N matrix into arrowhead form. Such heuristics are useful for decomposing large, constrained, optimization problems into forms that are amenable to parallel processing. The heuristics presented can be easily implemented using publicly available graph partitioning algorithms. The application of such techniques for solving large linear programs is described. Extensive computational results on the effectiveness of our partitioning procedures and their usefulness for parallel optimization are presented. © 1998 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V.This material is based on research supported by National Science Foundation Grants CCR-9157632 and CDA-9024618, the Air Force Office of Scientific Research Grant F49620-94-1-0036 and the AT&T Foundation.     

Related solution operators in mathematical physics

Transmutation operators are derived relating many of the frequently encountered linear partial differential equations in mathematical physics. The setting for this study is vector-valued distributions. Examples are given showing how fundamental solutions are derived for both homogeneous and nonhomogeneous partial differential equations.     

Nonlinear mathematical model for intensive steel quenching and its analytical solution in closed form

In the present paper the nonlinear mathematical model for intensive steel quenching processes is considered. This model involves the hyperbolic heat conduction equation, the initial conditions and the Newton type nonlinear boundary conditions characterizing the boiling process on quenchant surface. The reduction of the original problem to the Volterra type nonlinear integral equation is the essence of offered approach. Besides, in the present work some analytical procedure for solving of obtained nonlinear equation is proposed. It is proved that this analytical procedure permits finding the unique solution of the original nonlinear model. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)