A free energy based mathematical study for molecular motors

We present a Parrondo’s paradox for free energy in a classical flashing ratchet model and use it as an alternative way to interpret the working mechanism of molecular motors. We also study the efficiency of molecular motors measured by their free energies. Our example demonstrates that a molecular motor can gain up to 20% in its free energy during the process. In addition, we report a noise induced free energy increasing phenomenon, which is similar to the stochastic resonance, in flashing ratchet models.     

One-step Taylor–Galerkin methods for convection–diffusion problems

Third and fourth order Taylor–Galerkin schemes have shown to be efficient finite element schemes for the numerical simulation of time-dependent convective transport problems. By contrast, the application of higher-order Taylor–Galerkin schemes to mixed problems describing transient transport by both convection and diffusion appears to be much more difficult. In this paper we develop two new Taylor–Galerkin schemes maintaining the accuracy properties and improving the stability restrictions in convection–diffusion. We also present an efficient algorithm for solving the resulting system of the finite element method. Finally we present two numerical simulations that confirm the properties of the methods.     

Intermittency and deterministic diffusion in chaotic ratchets

We consider the problem of deterministic transport of particles in an asymmetric periodic ratchet potential of the rocking type. When the inertial term is taken into account, the dynamics can be chaotic and modify the transport properties. We calculate the bifurcation diagram as a function of the amplitude of forcing and analyze in detail the crisis bifurcation that leads to current reversals. Near this bifurcation we obtain intermittency and anomalous deterministic diffusion.     

Numerical solution of a PDE model for a ratchet-cap pricing with BGM interest rate dynamics

In this paper we present a new numerical method to price an interest rate derivative. The financial product consists of a particular ratchet cap contract which contains a set of ratchet caplets. For this purpose, we first pose the PDE pricing model for each ratchet caplet by means of Feynman-Kac theorem. The underlying interest rates are the forward LIBOR rates, the dynamics of which are assumed to follow the recently introduced BGM (LMM) market model. For the set of PDEs associated to the ratchet caplets pricing problems, we propose a second order Crank-Nicolson characteristics time discretization scheme combined with a finite element discretization in the interest rate variables. In order to illustrate the performance of the numerical methods, we present an academic test and a real example of a particular ratchet cap pricing. In the second case, a comparison between the results obtained by Monte Carlo simulation and the proposed method is presented.     

Comparative study on order-reduced methods for linear third-order ordinary differential equations

The linear third-order ordinary differential equation (ODE) can be transformed into a system of two second-order ODEs by introducing a variable replacement, which is different from the common order-reduced approach. We choose the functions p(x) and q(x) in the variable replacement to get different cases of the special order-reduced system for the linear third-order ODE. We analyze the numerical behavior and algebraic properties of the systems of linear equations resulting from the sinc discretizations of these special second-order ODE systems. Then the block-diagonal preconditioner is used to accelerate the convergence of the Krylov subspace iteration methods for solving the discretized system of linear equation. Numerical results show that these order-reduced methods are effective for solving the linear third-order ODEs.     

Calculation of effective moduli of composite materials

Effective properties of composite and porous materials are determined by using an approach based on two-scale asymptotic expansions. Explicit approximate formulas are derived for the effective moduli of composite and porous materials of elongated structures. A numerical method is proposed for finding solutions to cell problems, which are used to determine “exact” effective moduli. Examples are computed for a two-dimensional porous medium with variously shaped pores and various degrees of “elongation.” The effective moduli produced by the explicit approximate formulas prove to be similar to those found by numerically solving cell problems.     

An alternative to the Bathe algorithm

This paper presents a new composite sub-steps algorithm for solving reliable numerical responses in structural dynamics. The newly developed algorithm is a two sub-steps, second-order accurate and unconditionally stable implicit algorithm with the same numerical properties as the Bathe algorithm. The detailed analysis of the stability and numerical accuracy is presented for the new algorithm, which shows that its numerical characteristics are identical to those of the Bathe algorithm. Hence, the new sub-steps scheme could be considered as an alternative to the Bathe algorithm. Meanwhile, the new algorithm possesses the following properties: (a) it produces the same accurate solutions as the Bathe algorithm for solving linear and nonlinear problems; (b) it does not involve any artificial parameters and additional variables, such as the Lagrange multipliers; (c) The identical effective stiffness matrices can be obtained inside two sub-steps; (d) it is a self-starting algorithm. Some numerical experiments are given to show the superiority of the new algorithm and the Bathe algorithm over the dissipative CH-α algorithm and the non-dissipative trapezoidal rule.     

An Eulerian–Lagrangian method for optimization problems governed by multidimensional nonlinear hyperbolic PDEs

We present a numerical method for solving tracking-type optimal control problems subject to scalar nonlinear hyperbolic balance laws in one and two space dimensions. Our approach is based on the formal optimality system and requires numerical solutions of the hyperbolic balance law forward in time and its nonconservative adjoint equation backward in time. To this end, we develop a hybrid method, which utilizes advantages of both the Eulerian finite-volume central-upwind scheme (for solving the balance law) and the Lagrangian discrete characteristics method (for solving the adjoint transport equation). Experimental convergence rates as well as numerical results for optimization problems with both linear and nonlinear constraints and a duct design problem are presented.     

A characteristic finite element method with local mesh refinements for the Lamm equation in analytical ultracentrifugation

The Lamm equation is a fundamental differential equation in analytical ultracentrifugation, for describing the transport of solutes in an ultracentrifuge cell. In this article, we present a characteristic finite element method with local mesh refinements for solving the Lamm equation. The numerical method is mass‐conservative by design and allows relatively large time steps to be used. Numerical experiments indicate that the numerical solutions are oscillation‐free in the region near the cell bottom, where mass build up and large concentration gradients occur. Positivity of solutions is also well kept. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009     

Convergence of an operator splitting method on a bounded domain for a convection-diffusion-reaction system

We solve a convection-diffusion-sorption (reaction) system on a bounded domain with dominant convection using an operator splitting method. The model arises in contaminant transport in groundwater induced by a dual-well, or in controlled laboratory experiments. The operator splitting transforms the original problem to three subproblems: nonlinear convection, nonlinear diffusion, and a reaction problem, each with its own boundary conditions. The transport equation is solved by a Riemann solver, the diffusion one by a finite volume method, and the reaction equation by an approximation of an integral equation. This approach has proved to be very successful in solving the problem, but the convergence properties where not fully known. We show how the boundary conditions must be taken into account, and prove convergence in L1,loc of the fully discrete splitting procedure to the very weak solution of the original system based on compactness arguments via total variation estimates. Generally, this is the best convergence obtained for this type of approximation. The derivation indicates limitations of the approach, being able to consider only some types of boundary conditions. A sample numerical experiment of a problem with an analytical solution is given, showing the stated efficiency of the method.     

A mixed multiscale finite element method for elliptic problems with oscillating coefficients

The recently introduced multiscale finite element method for solving elliptic equations with oscillating coefficients is designed to capture the large-scale structure of the solutions without resolving all the fine-scale structures. Motivated by the numerical simulation of flow transport in highly heterogeneous porous media, we propose a mixed multiscale finite element method with an over-sampling technique for solving second order elliptic equations with rapidly oscillating coefficients. The multiscale finite element bases are constructed by locally solving Neumann boundary value problems. We provide a detailed convergence analysis of the method under the assumption that the oscillating coefficients are locally periodic. While such a simplifying assumption is not required by our method, it allows us to use homogenization theory to obtain the asymptotic structure of the solutions. Numerical experiments are carried out for flow transport in a porous medium with a random log-normal relative permeability to demonstrate the efficiency and accuracy of the proposed method.     

An analytical solution for diffusion and nonlinear uptake of oxygen in a spherical cell

We develop a new analytical solution for a reactive transport model that describes the steady-state distribution of oxygen subject to diffusive transport and nonlinear uptake in a sphere. This model was originally reported by Lin [S.H. Lin, Oxygen diffusion in a spherical cell with nonlinear oxygen uptake kinetics, J. Theor. Biol. 60 (1976) 449–457] to represent the distribution of oxygen inside a cell and has since been studied extensively by both the numerical analysis and formal analysis communities. Here we extend these previous studies by deriving an analytical solution to a generalized reaction–diffusion equation that encompasses Lin’s model as a particular case. We evaluate the solution for the parameter combinations presented by Lin and show that the new solutions are identical to a grid-independent numerical approximation.     

求解无约束优化问题的一个秩一适定方法

在本文中,我们给出一个求解无约束优化问题的秩一适定方法,该方法具有下述较好性质:校正矩阵是对称正定的;在适当条件下,对非凸函数拥有全局收敛性.我们还给出数值检验结果.     

二维土壤溶质输运方程的有限体积元格式

本文导出二维的土壤溶质输运方程的有限体积元格式, 并分析其误差.通过数值例子说明, 有限体积元格式比有限元格式稳定.     

On the homotopy analysis method for solving a particle transport equation

The purpose of this paper consists in the finding of the solution for a stationary neutron transport equation that is accompanied by the homogeneous boundary conditions, using the techniques of homotopy analysis method (HAM) and a numerical integration formula. Also, algorithm presented can be used for solving the integral–differential equations in which the unknown function depends on two variables, such as a radiative transfer equation. Results of a numerical example illustrate the accuracy and computational efficiency of the new proposed method.     

The hydrodynamic equations for chemically equilibrium flows of a multielement plasma with exact transport coefficients

Explicit expressions for all of the effective transport coefficients are derived for thermochemically equilibrium flows using the exact mass and heat transfer equations, which are resolved with respect to the “forces” (the gradients of the hydrodynamic variables) via the flukes. It is shown that, in a mixture where the components have different diffusion properties, separation (diffusion) of the chemical elements occurs which leads to a state of affairs where the equilibrium concentrations, and together with them, the effective transport coefficients will be functions not only of the pressure and temperature but will also depend on the concentrations of the elements, determined when solving the problem (self-consistent concentrations of the elements). It is shown that the existence of an electric current and lack of quasineutrality (flow around electrically conducting walls—electrodes) does not change the structure of the expressions for the effective transport coefficients and does not add anything new. The approximate and incomplete treatment of thermochemically equilibrium flows of multicomponent gas mixtures and a plasma in previously published papers are especially noted. Numerical estimates of the effective transport coefficients are presented for an air plasma and the domains in the pressure-temperature plane with the required number of approximations in order to obtain results with an error of no worse than 5% are indicated.     

The role of microvascular tortuosity in tumor transport phenomena and future perspective for drug delivery

We tackle the fluid transport problem in vascularized tumors by solving a double Darcy model obtained via multiscale homogenization. The hydraulic conductivities of the capillary and interstitial compartments are computed solving classical problems on the representative periodic cell, which encodes details of the microvasculature. Microvascular tortuosity leads to a dramatic decrease of the capillary hydraulic conductivity, and the corresponding lowering in the pressure drop impairs tumor blood flow and consequently advection of molecules. Further perspectives for anti-cancer agents delivery are illustrated. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On bounded solutions of transport equation solved with a spectral method

In this article, a spectral method accompanied by finite difference method has been proposed for solving a boundary value problem that accompanies a stationary transport equation. We also prove that the solution is bounded by a value that depends of the source function. The accuracy and computational efficiency of the proposed method are verified with the help of a numerical example. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2012     

Chaotic transport of a matter-wave soliton in a biperiodically driven optical superlattice

Under the effective particle approximation, we study the temporal ratchet effect for chaotic transport of a matter-wave soliton consisting of an attractive Bose–Einstein condensate held in a quasi-one-dimensional symmetric optical superlattice with biperiodic driving. It is known that chaos can substitute for disorder in Anderson’s scenario [Wimberger S, Krug A, Buchleitner A. Phys Rev Lett 2002;89:263601] and only a higher level of disorder can induce Anderson localization for some special systems [Schwartz T, Bartal G, Fishman S, Segev M. Nature 2007;46:52], and a matter-wave soliton can transit to chaos with high or low probability in a high- or low-chaoticity region [Zhu Q, Hai W, Rong S. Phys Rev E 2009;80:016203]. Here we demonstrate that varying the driving phase to break the time reversal symmetry of the system can increase the size of the high-chaoticity region for low- and moderate-frequency regions. Consequently, the parameter region of the exponential spatial localization increases to the same size, and the low-chaoticity and delocalization region, which includes subregions of the ratchet effect and its inverse effect, correspondingly decreases. The positive dependence of the localization on the driving frequency is also revealed. The results indicate that a high-chaoticity region could replace higher disorder and assists in Anderson localization. From the results we suggest a method for controlling directed motion of a matter-wave soliton by adjusting the driving frequency and amplitude to strengthen or suppress, or even reverse, the temporal ratchet effect.