Abstract: A physical-mathematical approach to anomalous diffusion is based on a generalized diffusion equation containing derivatives of fractional order. In this paper, an anomalous subdiffusion equation (ASub-DE) is considered. A new implicit numerical method (INM) and two solution techniques for improving the order of convergence of the INM for solving the ASub-DE are proposed. The stability and convergence of the INM are investigated by the energy method. Some numerical examples are given. The numerical results demonstrate the effectiveness of theoretical analysis. These methods and supporting theoretical results can also be applied to other fractional integro-differential equations and higher-dimensional problems.

Abstract: Extensions of finite-difference time-domain (FDTD) and finite-element time-domain (FETD) algorithms are reviewed for solving transient Maxwell equations in complex media. Also provided are a few representative examples to illustrate the modeling capabilities of FDTD and FETD for complex media. The term complex media refers here to media with dispersive, (bi)anisotropic, inhomogeneous, and/or nonlinear properties present in the constitutive tensors.

Abstract: An analysis is made for the steady two-dimensional magneto-hydrodynamic flow of an incompressible viscous and electrically conducting fluid over a stretching vertical sheet in its own plane. The stretching velocity, the surface temperature and the transverse magnetic field are assumed to vary in a power-law with the distance from the origin. The transformed boundary layer equations are solved numerically for some values of the involved parameters, namely the magnetic parameter M, the velocity exponent parameter m, the temperature exponent parameter n and the buoyancy parameter λ, while the Prandtl number Pr is fixed, namely Pr = 1, using a finite difference scheme known as the Keller-box method. Similarity solutions are obtained in the presence of the buoyancy force if n = 2m−1. The features of the flow and heat transfer characteristics for different values of the governing parameters are analyzed and discussed. It is found that both the skin friction coefficient and the local Nusselt number decrease as the magnetic parameter M increases for fixed λ and m. For m = 0.2 (i.e. n = −0.6), although the sheet and the fluid are at different temperatures, there is no local heat transfer at the surface of the sheet except at the singular point of the origin (fixed point).

Abstract: Recently, it is found that telegraph equation is more suitable than ordinary diffusion equation in modelling reaction diffusion for such branches of sciences. In this article, we propose a numerical scheme to solve the one-dimensional hyperbolic telegraph equation using collocation points and approximating the solution using thin plate splines radial basis function. The scheme works in a similar fashion as finite difference methods. The results of numerical experiments are presented, and are compared with analytical solutions to confirm the good accuracy of the presented scheme. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2008

Abstract: Frequency-domain waveform inversion is typically perfomed using frequency-domain finite-difference modelling techniques. In 3D, these methods face significant computational challenges that limit any application to full-scale seismic applications. An alternative approach is to use a 3D time-domain finite-difference method and extract the frequency-domain wavefield by computing the terms of a discrete Fourier transform at each time step. This method combines the computational efficiency of 3D time-domain modelling while permitting casting the inverse problem in the frequency domain.

Abstract: A three-dimensional unconditionally-stable locally-one-dimensional finite-difference time-domain (LOD-FDTD) method is proposed and is proved unconditionally stable analytically. In it, the number of equations to be computed is the same as that with the conventional three-dimensional alternating direction implicit FDTD (ADI-FDTD) but with reduced arithmetic operations. The reduction in arithmetic operations leads to approximately 20% less computational time in comparisons with the ADI-FDTD method.

Abstract: The solution of the elastohydrodynamic lubrication (EHL) problem involves the simultaneous resolution of the hydrodynamic (Reynolds equation) and elastic problems (elastic deformation of the contacting surfaces) Up to now, most of the numerical works dealing with the modeling of the isothermal EHL problem were based on a weak coupling resolution of the Reynolds and elasticity equations (semi-system approach) The latter were solved separately using iterative schemes and a finite difference discretization Very few authors attempted to solve the problem in a fully coupled way, thus solving both equations simultaneously (full-system approach) These attempts suffered from a major drawback which is the almost full Jacobian matrix of the nonlinear system of equations This work presents a new approach for solving the fully coupled isothermal elastohydrodynamic problem using a finite element discretization of the corresponding equations The use of the finite element method allows the use of variable unstructured meshing and different types of elements within the same model which leads to a reduced size of the problem The nonlinear system of equations is solved using a Newton procedure which provides faster convergence rates Suitable stabilization techniques are used to extend the solution to the case of highly loaded contacts The complexity is the same as for classical algorithms, but an improved convergence rate, a reduced size of the problem and a sparse Jacobian matrix are obtained Thus, the computational effort, time and memory usage are considerably reduced

Abstract: A new method for solving the time-domain integral equations of electromagnetic scattering from conductors is introduced. This method, called finite difference delay modeling, appears to be completely stable and accurate when applied to arbitrary structures. The temporal discretization used is based on finite differences. Specifically, based on a mapping from the Laplace domain to the z-transform domain, first- and second-order unconditionally stable methods are derived. Spatial convergence is achieved using the higher-order divergence-conforming vector bases of Graglia et al. Low frequency instability problems are avoided with the loop-tree decomposition approach. Numerical results will illustrate the accuracy and stability of the technique.

Abstract: Based on the Crank-Nicolson (CN) scheme, an improved alternating direction implicit finite-difference time-domain (ADI-FDTD) method is proposed. By introducing local correction, the proposed method reduces the splitting error while the unconditional stability and computational efficiency are maintained. Theoretical analyses are given.

Abstract: Numerical solutions of the Benjamin-Bona-Mahony-Burgers equation in one space dimension are considered using Crank-Nicolson-type finite difference method. Existence of solutions is shown by using the Brower's fixed point theorem. The stability and uniqueness of the corresponding methods are proved by the means of the discrete energy method. The convergence in L∞-norm of the difference solution is obtained. A conservative difference scheme is presented for the Benjamin-Bona-Mahony equation. Some numerical experiments have been conducted in order to validate the theoretical results.© 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007

Abstract: In order to interpret the logging data from triaxial induction tools, inversion technology has been adopted. To make the inversion process reasonably fast, a fast forward method must be developed. Numerical methods, such as the finite element method and finite difference method, are flexible but slow for inversion purposes. In this paper, an analytic method is developed to simulate induction tools in deviated wells drilled in anisotropic formations. This method can be applied to the triaxial induction tools with transmitting and receiving coils oriented in three mutually perpendicular directions. The axis of the tool may intercept a formation with dip, azimuthal, and orientation angles. Formulations of the electromagnetic fields generated by these three transmitting coils are derived. The derivation uses the coefficient propagator method with the assumption of negligible borehole effect and invasion zones. This method overcomes the problem of numerical overflow without compromising the accuracy of the solution. Because the new induction tool has three transmitting and three receiving coils, a total of nine logs are obtained at each logging depth compared with one or two logs in regular induction or logging while drilling tools.

Abstract: We adapt a finite difference method of solution of the two-dimensional massless Dirac equation, developed in the context of lattice gauge theory, to the calculation of electrical conduction in a graphene sheet or on the surface of a topological insulator The discretized Dirac equation retains a single Dirac point (no ``fermion doubling''), avoids intervalley scattering as well as trigonal warping, and preserves the single-valley time-reversal symmetry $(=\text{symplectic}\text{ }\text{symmetry})$ at all length scales and energies\char22{}at the expense of a nonlocal finite difference approximation of the differential operator We demonstrate the symplectic symmetry by calculating the scaling of the conductivity with sample size, obtaining the logarithmic increase due to antilocalization We also calculate the sample-to-sample conductance fluctuations as well as the shot-noise power and compare with analytical predictions

Abstract: The direct numerical simulation (DNS) of the Taylor–Couette flow in the fully turbulent regime is described. The numerical method extends the work by Quadrio and Luchini [M. Quadrio, P. Luchini, Eur. J. Mech. B/Fluids 21 (2002) 413–427], and is based on a parallel computer code which uses mixed spatial discretization (spectral schemes in the homogeneous directions, and fourth-order, compact explicit finite-difference schemes in the radial direction). A DNS is carried out to simulate for the first time the turbulent Taylor–Couette flow in the turbulent regime. Statistical quantities are computed to complement the existing experimental information, with a view to compare it to planar, pressure-driven turbulent flow at the same value of the Reynolds number. The main source for differences in flow statistics between plane and curved-wall flows is attributed to the presence of large-scale rotating structures generated by curvature effects.

Abstract: A Chebyshev finite difference method has been proposed in order to solve linear and nonlinear second-order Fredholm integro-differential equations. The approach consists of reducing the problem to a set of algebraic equations. This method can be regarded as a nonuniform finite difference scheme. Some numerical results are also given to demonstrate the validity and applicability of the presented technique.

Abstract: The focus of this article is the numerical study of particle deposition profiles on a solid substrate during the evaporation of a sessile drop of a colloidal particle suspension. The evaporation flux along the drop interface, the induced fluid dynamics inside the drop, and the particle deposition profile on the solid substrate are solved simultaneously. The governing equations are solved numerically using the Galerkin/finite element method (G/FEM) for discretization of the spatial domain and an adaptive finite difference method for discretization in the time domain. Several particle deposition profiles, such as a ring-shaped deposit and uniform particle distribution, are obtained from the numerical simulations. The particle deposition profile is found to be influenced by the mass transfer (both convective and diffusive mass transfer) of the particles in the bulk liquid and by the deposition rate along the substrate. © 2008 American Institute of Chemical Engineers AIChE J, 2008

Abstract: Boundary value problems for loaded ordinary and partial differential equations are considered. A priori bounds are obtained for solutions to differential and difference equations. These bounds imply the stability and convergence of difference schemes for the equations under consideration.

Abstract: Mathematical Foundations Algorithmic Complexity Ordinary Differential Equations Partial Differential Equations Roots of Nonlinear Equations Numerical Integration Computational Linear Algebra Interpolation Finite Difference Methods for ODEs Finite Difference Methods for PDEs Finite Volume Method Finite Element Method Mathematical Optimization Mathematical Programming Stochastic Models Data Modeling Metaheuristic Methods Bee Algorithms Swarm Optimization.

Abstract: Impulse responses in a hall were calculated by the finite-difference time-domain (FDTD) method, and typical room acoustic parameters were obtained from the responses. The calculated parameters were compared with those actually measured in the hall. In the FDTD calculation, the impedance boundary condition was modeled by an equivalent mechanical system comprising masses, springs, and dampers. To calculate the impulse responses, the normal acoustic impedance of the interior finishing materials of the various surfaces in the hall were measured by applying the impedance-tube method, and the model of the room boundary condition was determined for the respective parts. A comparison between the calculated and measured values showed that the values of reverberation time RT, definition D50, clarity C80, and center time Ts were in good agreement in the middle-frequency bands. However, in low-frequency bands, large discrepancies were observed because of the difficulties in determining and modeling the boundary conditions.