A hybrid solver of size modified Poisson–Boltzmann equation by domain decomposition,finite element,and finite difference

The size-modified Poisson–Boltzmann equation (SMPBE) is one important variant of the popular dielectric model, the Poisson–Boltzmann equation (PBE), to reflect ionic size effects in the prediction of electrostatics for a biomolecule in an ionic solvent. In this paper, a new SMPBE hybrid solver is developed using a solution decomposition, Schwartz’s overlapped domain decomposition, finite element, and finite difference. It is then programmed as a software package in C, Fortran, and Python based on the state-of-the-art finite element library DOLFIN from the FEniCS project. This software package is well validated on a Born ball model with analytical solution and a dipole model with known physical properties. Numerical results on six proteins with different net charges demonstrate its high performance. Finally, this new SMPBE hybrid solver is shown to be numerically stable and convergent in the calculation of electrostatic solvation free energy for 216 biomolecules and binding free energy for a DNA-drug complex.     

A new analysis of electrostatic free energy minimization and Poisson–Boltzmann equation for protein in ionic solvent

In this paper, a novel solution decomposition of the Poisson dielectric model is proposed to modify a traditional electrostatic free energy minimization problem into one that is well defined for the case of protein in ionic solvent. The target function of this modified problem is shown to be strictly convex, weak sequentially lower semicontinuous, and twice continuously Fréchet differentiable. Its first and second Gâteaux derivatives are then found. Moreover, it is proved that this modified electrostatic free energy minimization problem has a unique solution, and its solution existence and uniqueness is equivalent to that of the Poisson–Boltzmann equation, a widely-used implicit solvent model for computing the electrostatics of biomolecules.     

A simple Taylor-series expansion method for a class of second kind integral equations

A simple yet effective Taylor-series expansion method is presented for a class of Fredholm integral equations of the second kind with smooth and weakly singular kernels. The equations studied in this paper arise in a number of applications, e.g., potential theory, radiative equilibrium, radiative heat transfer, and electrostatics. The approach leads to an approximate solution of the integral equation which can be expressed explicitly in a simple, closed form. The approximate solution is of sufficient accuracy as illustrated by the numerical examples arising from radiative heat transfer and electrostatics.     

A phase field model in electrowetting and related phenomena

The term electrowetting is commonly used for some techniques to change the shape and wetting behaviour of liquid droplets by the application of electric fields and charges. We developand analyze a model for electrowetting that combines the Navier-Stokes system for fluid flow, a phase-field model of Cahn-Hilliard type for the movement of the interface, a charge transport equation, and the potential equation of electrostatics. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Identification of some source densities of the distribution type

In this paper we investigate the solvability of an ill-posed two-dimensional Fredholm integral equation of the first kind which allows the solutions of distribution type. The problem is first transformed into a well-posed differential–integral equation using output least-squares approach with a regularization of bounded variations. A globally convergent iterative method is proposed and some numerical results are presented. The methodology discussed may be applied for the identification of the boundary shape of the defects of a dielectric material or the interface between different materials.     

The Integral Form of the Equation of Radiative Transfer for an Inhomogeneous, Anisotropically Scattering, Solid Sphere

The integral form of the equation of radiative transfer is developedfor an absorbing, emitting, inhomogeneous, anisotropically scattering,solid sphere having internal energy sources, externally incidentradiation and a specularly or diffusively reflecting boundarysurface. The resulting integral form is useful for developingsolutions to radiation problems in a solid sphere by the applicationof projection techniques.     

Integral-equation Perturbation Theory for Electrostatic Torus Problems with Applied Fields

A perturbation method has been given for solving the Fredholmintegral equations of electrostatics as applied to tori of revolution,but under the restriction that there be in no case an appliedfield other than perhaps one that is uniform and parallel tothe axis of the torus. In the present paper this restrictionis removed, both for the conductor situation (Robin's equation)and the dielectric one (Durand's equation), and the generalizedmethod is illustrated by three applications. Two of these areto new problems which prove to be of considerable interest inthemselves, and one of the two leads to an example of significantfailure of the perturbation method.     

Quadruple Trigonometrical Series Equations and Their Application to an Inclusion Problem in the Theory of Elasticity

Closed form solution of quadruple series equations involving cosine kernels has been obtained by reducing the series equations into triple Abel's type integral equations which in turn are reduced to a single integral equation. Making use of finite Hilbert transforms the solution of the single integral equation is obtained in closed form. This solution is used to solve an electrostatic problem. The results of this paper have also been used in a two-dimensional elastostatic problem under anti-plane shear and the effect of rigid line inclusions with thickness on the Griffith cracks has been examined. The expressions for shear stress and stress intensity factor at the tip of the crack are obtained. Finally, some numerical results for the stress intensity factor and shear stress distribution are obtained.     

The simulation of electrified liquid jets

The simulation of a two-dimensional electrified liquid jet is described. A finite difference technique is coupled with a computational fluid dynamic code to solve Poisson's equation and the Navier-Stokes equations at the electrostatic fluid-flow interface. The dynamics of free-surface electrohydrodynamic fluid flow are simulated for a dielectric fluid in an electrostatics nozzle and a conducting two-dimensional jet. The fluid flow in a nozzle is compared with and without an applied electric field, and the effects of adding a grounded conducting cylinder to the configuration is demonstrated. A set of time sequence graphs are used to illustrate the breakup of a charged jet into droplets, and the influence of viscosity on jet formation and breakup is depicted.     

A new coupled model for alloy solidification

A new coupled model in the binary alloy solidification has been developed. The model is based on the cellular automaton (CA) technique to calculate the evolution of the interface governed by temperature, solute diffusion and Gibbs-Thomson effect. The diffusion equation of temperature with the release of latent heat on the solid/liquid (S/L) interface is valid in the entire domain. The temperature diffusion without the release of latent heat and solute diffusion are solved in the entire domain. In the interface cells, the     

A new coupled model for alloy solidification

A new coupled model in the binary alloy solidification has been developed. The model is based on the cellular automaton (CA) technique to calculate the evolution of the interface governed by temperature, solute diffusion and Gibbs-Thomson effect. The diffusion equation of temperature with the release of latent heat on the solid/liquid (S/L) interface is valid in the entire domain. The temperature diffusion without the release of latent heat and solute diffusion are solved in the entire domain. In the interface cells, the energy and solute conservation, thermodynamic and chemical potential equilibrium are adopted to calculate the temperature, solid concentration, liquid concentration and the increment of solid fraction. Compared with other models where the release of latent heat is solved in implicit or explicit form according to the solid/liquid (S/L) interface velocity, the energy diffusion and the release of latent heat in this model are solved at different scales, i.e. the macro-scale and micro-scale. The variation of solid fraction in this model is solved using several algebraic relations coming from the chemical potential equilibrium and thermodynamic equilibrium which can be cheaply solved instead of the calculation of S/L interface velocity. With the assumption of the solute conservation and energy conservation, the solid fraction can be directly obtained according to the thermodynamic data. This model is natural to be applied to multiple (< 2) spatial dimension case and multiple (< 2) component alloy. The morphologies of equiaxed dendrite are obtained in numerical experiments.     

Nonlinear transmission eigenvalue problem describing TE wave propagation in two-layered cylindrical dielectric waveguides

The paper is concerned with propagation of surface TE waves in a circular nonhomogeneous two-layered dielectric waveguide filled with a Kerr nonlinear medium. The problem is reduced to the analysis of a nonlinear integral equation with a kernel in the form of a Green’s function. The existence of propagating TE waves is proved using the contraction mapping method. For the numerical solution of the problem, two methods are proposed: an iterative algorithm (whose convergence is proved) and a method based on solving an auxiliary Cauchy problem (the shooting method). The existence of roots of the dispersion equation (propagation constants of the waveguide) is proved. Conditions under which k waves can propagate are obtained, and regions of localization of the corresponding propagation constants are found.     

Diffraction by a Dielectric Wedge

Approximate expressions are obtained for the field producedwhen an electromagnetic source field is diffracted by a dielectricwedge. The boundary value problem, of the diffraction of an E-or H-pokarizedelectromagnetic line source by an arbitrary angled dielectricwedgeis formulated, and its solution is given in the form ofa Fredholm integral equation. The solution of the integral equationis obtained by a standard perturbation technique. The perturbationparameter is dependent on the refractive index of the dielectricwedge. The right angle dielectric wedge, which is illuminatedby an E-polarized plane wate, and whose refractive index (=n)is such that 1<n < 2, is considered in detail.     

An all-frequency weakly-singular surface integral equation for electromagnetism in dielectric media: Reformulation and well-posedness analysis

We develop and analyze a surface integral equation (SIE) whose solution pertains to numerical simulations of propagating time-harmonic electromagnetic waves in three-dimensional dielectric media. The formulae to evaluate the far-field pattern and propagation of the electric and magnetic fields in the interior and exterior of a dielectric body, through surface integrals, require the solution of a 2×2$2\times 2$ system of weakly-singular SIEs for the two unknown electric and magnetic fields at the interface surface of the dielectric body. The SIE is governed by an operator that is of the classical identity plus compact form. The tangential surface currents and normal surface charges of the dielectric model can be easily computed from the surface electric and magnetic fields.     

New stochastic model for dispersion in heterogeneous porous media: 1. Application to unbounded domains

A new model of solute dispersion in porous media that avoids Fickian assumptions and that can be applied to variable drift velocities as in non-homogeneous or geometrically constricted aquifers, is presented. A key feature is the recognition that because drift velocity acts as a driving coefficient in the kinematical equation that describes random fluid displacements at the pore scale, the use of Ito calculus and related tools from stochastic differential equation theory (SPDE) is required to properly model interaction between pore scale randomness and macroscopic change of the drift velocity. Solute transport is described by formulating an integral version of the solute mass conservation equations, using a probability density. By appropriate linking of this to the related but distinct probability density arising from the kinematical SPDE, it is shown that the evolution of a Gaussian solute plume can be calculated, and in particular its time-dependent variance and hence dispersivity. Exact analytical solutions of the differential and integral equations that this procedure involves, are presented for the case of a constant drift velocity, as well as for a constant velocity gradient. In the former case, diffusive dispersion as familiar from the advection–dispersion equation is recovered. However, in the latter case, it is shown that there are not only reversible kinematical dispersion effects, but also irreversible, intrinsically stochastic contributions in excess of that predicted by diffusive dispersion. Moreover, this intrinsic contribution has a non-linear time dependence and hence opens up the way for an explanation of the strong observed scale dependence of dispersivity.     

Water waves diffraction by a submerged sphere and dual integral transforms

A solution of the diffraction problem for a submerged sphere in finite water depth based on the linearized potential theory is presented. The sphere can take different positions relative to the bottom. A new method is suggested to solve this problem. This method is a generalization of the integral transforms. Two systems of the curvilinear coordinates are used, two spectral systems are constructed and two spectral functions are introduced to obtain the solution. For the first spectral function an integral representation is obtained, for the second spectral function an integro-operator equation is derived. Different asymptotic approximations are considered.     

A complete steady state model of solute and water transport in the kidney

The purpose of this paper is to incorporate a detailed model, along with an optimized set of parameters for the proximal tubule, into J. L. Stephenson's current central core model of the nephron. In this model a set of equations for the proximal tubule are combined with Stephenson's equations for the remaining four tubules and interstitium, to form a complete nonlinear system of 34 ordinary differential and algebraic equations governing fluid and solute flow in the kidney. These equations are then discretized by the Crank-Nicholson scheme to form an algebraic system of nonlinear equations for the unknown concentrations, flows, hydrostatic pressure, and potentials. The resulting system is solved via factored secant update with a finite-difference approximation to the Jacobian. Finally, numerical simulations performed on the model showed that the modeled behavior approximates, in a general way, the physiological mechanisms of solvent and solute flow in the kidney.     

A COLLOCATION METHOD FOR THE CONDUCTIVITY PROBLEM WITH DISCONTINUOUS COEFFICIENT

In this paper, a new collocation BEM for the Robin boundary value problem of the conductivity equation ▽(γ▽u) = 0 is discussed, where the 7 is a piecewise constant function. By the integral representation formula of the solution of the conductivity equation on the boundary and interface, the boundary integral equations are obtained. We discuss the properties of these integral equations and propose a collocation method for solving these boundary integral equations. Both the theoretical analysis and the error analysis are presented and a numerical example is given.     

An integral formulation procedure for the solutions to Helmholtz's equation in spherically symmetric media

Starting from Helmholtz's equation in inhomogeneous media, the associated radial second‐order equation is investigated through a Volterra integral equation. First the integral equation is considered in a sphere. Boundedness, uniqueness and existence of the (regular) solution are established and the series form of the solution is provided. An estimate is determined for the error arising when the series is truncated. Next the analogous problem is considered for a spherical layer. Again, boundedness, uniqueness and existence of two base solutions are established and error estimates are determined. The procedure proves more effective in the sphere. Copyright © 2009 John Wiley & Sons, Ltd.     

Eine Integralgleichungsmethode für akustische Reflexionsprobleme an lokal gestörten Ebenen

In this paper we consider the reflection of acoustic waves at an unbounded surface which coincides with a plane outside a sufficiently large sphere. We prove uniqueness and existence theorems for the corresponding boundary value problems for the reduced wave equation with Dirichlet and Neumann data by employing integral equation methods.