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Green’s functions for Poisson’s equation in inhomogeneous media with material interfaces have many practical applications. In the present work, we focus on Green’s functions for Poisson’s equation in inhomogeneous media with diffuse material interfaces where a gradual and continuous transition in the material constant is assumed in a small region around the interfaces between different materials. We present a unified general framework for calculating Green’s functions for Poisson’s equation in such inhomogeneous media and the framework can apply to all eleven orthogonal coordinate systems in which the three-dimensional Laplace equation is separable. Within this framework, the idea on how to design the so-called quasi-harmonic diffuse interface is discussed, formulations for building Green’s function for Poisson’s equation in an inhomogeneous medium with such a diffuse interface is elaborated, and a robust numerical method for calculating Green’s functions for Poisson’s equation in inhomogeneous media with general diffuse interfaces is developed. Several practically relevant separable coordinate systems are briefly surveyed at the level of definition and basic facts relevant for implementing the unified framework in these coordinate systems.

Two-step algorithms for the stationary incompressible Navier–Stokes equations with friction boundary conditions are considered in this paper. Our algorithms consist of solving one Navier–Stokes variational inequality problem used the linear equal-order finite element pair (i.e., P 1 – P 1 ) and then solving a linearization variational inequality problem used the quadratic equal-order finite element pair (i.e., P 2 – P 2 ). Moreover, the stability and convergence of our two-step algorithms are derived. Finally, numerical tests are presented to check theoretical results.

Spheroidal harmonics and modified Bessel functions have wide applications in scientific and engineering computing. Recursive methods are developed to compute the logarithmic derivatives, ratios, and products of the prolate spheroidal harmonics ( $$P_n^m(x)$$ , $$Q_n^m(x)$$ , $$n\ge m\ge 0$$ , $$x>1$$ ), the oblate spheroidal harmonics ( $$P_n^m(ix)$$ , $$Q_n^m(ix)$$ , $$n\ge m\ge 0$$ , $$x>0$$ ), and the modified Bessel functions ( $$I_n(x)$$ , $$K_n(x)$$ , $$n\ge 0$$ , $$x>0$$ ) in order to avoid direct evaluation of these functions that may easily cause overflow/underflow for high degree/order and for extreme argument. Stability analysis shows the proposed recursive methods are stable for realistic degree/order and argument values. Physical examples in electrostatics are given to validate the recursive methods.

In this note, we point out two errors in the article “On the Neumann function and the method of images in spherical and ellipsoidal geometry” by Dassios and Sten. Two corrections are then proposed.

In a preceding paper [Comput. Phys. Commun. 184 (1): 51–59, 2013], we revisited the problem of calculating Coulomb Green’s function and image potential near a planar diffuse interface within which the dielectric permittivity of the inhomogeneous medium changes continuously along one Cartesian direction in a transition layer between two dissimilar dielectric materials. In the present paper, we consider a cylindrical diffuse interface within which the dielectric permittivity changes continuously along the radial direction instead. First we propose a specific cylindrical diffuse interface model, termed the quasi-harmonic diffuse interface model, that can admit analytical solution for the Green’s function in terms of the modified Bessel functions. Then and more importantly we develop a robust numerical method for building Green’s functions for any cylindrical diffuse interface models. The main idea of the numerical method is, after dividing a diffuse interface into multiple sublayers, to approximate the dielectric permittivity profile in each one of the sublayers by one of the quasi-harmonic functional form rather than simply by a constant value as one would normally do. Next we describe how to efficiently compute well-behaved ratios, products, and logarithmic derivatives of the modified Bessel functions so as to avoid direct evaluations of individual modified Bessel functions in our formulations. Finally we conduct numerical experiments to show the effectiveness of the quasi-harmonic diffuse interface model in overcoming the divergence of the image potential, to validate the numerical method in terms of its accuracy and convergence, and to demonstrate its capability for computing Green’s functions for any cylindrical diffuse interface models.

In this work, we revisit the problem of calculating the Coulomb Green’s function and the image potential near a planar diffuse interface in which the dielectric constant changes continuously in a transition layer between two dielectrics. In particular, we extend previous work in two ways. Firstly, a new diffuse interface model, termed the quasi-harmonic interface model, is constructed, for which analytical calculation of Green’s function and the image potential is easy to achieve and need not use any special function. Secondly and also more importantly, a robust semi-analytical numerical procedure to build Green’s functions for general diffuse interface models is developed in analogy to the analysis of transmission lines, thus opening the way to treat in principle any well-behaving and physically plausible dielectric permittivity profile for the transition layer. Numerical experiments are given to illustrate the quasi-harmonic diffuse interface model, and to validate the semi-analytical numerical method particularly by demonstrating its convergence as the number of the sublayers used to partition the transition layer goes to infinity.

The Rayleigh–Stokes problem for a generalized second grade fluid in a porous half-space with a heated flat plate is investigated. For the description of such a viscoelastic fluid, a fractional calculus approach in the constitutive relationship model is used. By using the Fourier sine transform and the fractional Laplace transform, exact solutions of the velocity and temperature fields are obtained, from which some classical results can be recovered.

Modified Darcy’s law for fractional generalized Burgers’ fluid in a porous medium is introduced. The flow near a wall suddenly set in motion for a fractional generalized Burgers’ fluid in a porous half-space is investigated. The velocity of the flow is described by fractional partial differential equations. By using the Fourier sine transform and the fractional Laplace transform, an exact solution of the velocity distribution is obtained. Some previous and classical results can be recovered from our results, such as the velocity solutions of the Stokes’ first problem for viscous Newtonian, second grade, Maxwell, Oldroyd-B or Burgers’ fluids.

A Hopf bundle, whose base manifold is the ring surface T 2 and whose fiber is the group U ( 1 ) , is established in this paper. On this Hopf bundle, the lifting of the Laplace operator on the base manifold is proved to be the Laplace operator on the Hopf bundle. The solutions of covariant derivative equations of cross section in associated bundles and the index theorem on the ring surface are also discussed.

By using ghost points on either side of the interfaces, a global second-order accurate upwinding boundary condition capturing method for time-domain Maxwell's equations in media with material interfaces is proposed. The equations are discretized on a uniform Cartesian grid and the interfaces are allowed to intersect the grid in an arbitrary fashion. The method is then obtained by combining central finite difference schemes with applicable nodes being replaced by the ghost points and upwinding technique with jump conditions across the interfaces being captured in a manner that the upwind property is always satisfied. The resulting discretization has the desirable property that the allowed time step size is independent of the locations and the shapes of the interfaces. Numerical examples are then given to demonstrate the second-order accuracy as well as the stability of the method, where it is used to study wave equations with various types of material interfaces, including electromagnetic scattering of a plane incident wave by a dielectric circular cylinder.

A recent article by Deng and Cai [Extending the fast multipole method for charges inside a dielectric sphere in an ionic solvent: High-order image approximations for reaction fields, J. Comput. Phys. (2007), doi: 10.1016/j.jcp.2007.09.001] introduced two fourth-order image approximations to the reaction field for a charge inside a dielectric sphere immersed in a solvent of low ionic strength. To represent such a reaction field, the image approximations employ a point charge at the classical Kelvin image point and two line charges that extend from this Kelvin image point along the radial direction to infinity, with one decaying to zero and the other growing to infinity. In this paper, alternative versions of the fourth-order image approximations are presented, using the same point charge but three different line charges, all decaying to zero along the radial direction. Similar discussions on how to approximate the line charges by discrete image charges and how to apply the resulting multiple discrete image approximations together with the fast multipole method are also included.

The Rayleigh-Stokes problem for a generalized Maxwell fluid in a porous half-space with a heated flat plate is investigated. For the description of such a viscoelastic fluid, a fractional calculus approach in the constitutive relationship model is used. By using the Fourier sine transform and the fractional Laplace transform, exact solutions of the velocity and the temperature are obtained. Some classical results can be regarded as particular cases of our results, such as the classical solutions of the first problem of Stokes for Newtonian viscous fluids, Maxwell fluids, and Maxwell fluids in a porous half-space.

We investigate the stochastic bifurcation of a damped sine-Gordon equation with Dirichlet boundary conditions under the influence of multiplicative Gaussian white noise. Introducing a slow time scale, we derive the amplitude equations near the trivial solution by multiscale analysis. And the stationary probability density functions are formulated analytically using the stochastic averaging of energy envelope. The numerical calculations show that the system undergoes a stochastic D-bifurcation of energy envelope from a delta measure to new stationary measures when the control parameter crosses a critical point.

This article is concerned with the existence of mild solutions for fractional differential inclusions with nonlocal conditions in Banach spaces. The results are obtained by using fractional calculus, Hausdorff measure of noncompactness, and the multivalued fixed point theorem. The results obtained in the present paper extend some related results on this topic.