A Kansa-type MFS scheme for two-dimensional time fractional diffusion equations

In this paper, an efficient Kansa-type method of fundamental solutions (MFS-K) is extended to the solution of two-dimensional time fractional sub-diffusion equations. To solve initial boundary value problems for these equations, the time dependence is removed by time differencing, which converts the original problems into a sequence of boundary value problems for inhomogeneous Helmholtz-type equations. The solution of this type of elliptic boundary value problems can be approximated by fundamental solutions of the Helmholtz operator with different test frequencies. Numerical results are presented for several examples with regular and irregular geometries. The numerical verification shows that the proposed numerical scheme is accurate and computationally efficient for solving two-dimensional fractional sub-diffusion equations.     

Canonical duality for solving nonconvex and nonsmooth optimization problem

This paper presents an application of the canonical duality theory for solving a class of nonconvex and nonsmooth optimization problems. It is shown that by use of the canonical dual transformation, these difficult optimization problems in R ⁿ can be converted into a one-dimensional canonical dual problems, which can be solved to obtain all extremal points. Both global and local extremality conditions can be identified by the triality theory. Applications are illustrated.     

Performance evaluation of a multi-product system under CONWIP control

Analytical models of multi-product manufacturing systems operating under CONWIP control are composed of closed queuing networks with synchronization stations. Under general assumptions, these queuing networks are hard to analyze exactly and therefore approximation methods must be used for performance evaluation. This research proposes a new approach based on parametric decomposition. Two-moment approximations are used to estimate the performance measures at individual stations. Subsequently, the traffic process parameters at the different stations are linked using stochastic transformation equations. The resulting set of non-linear equations is solved using an iterative algorithm to obtain estimates of key performance measures such as throughput, and mean queue lengths. Numerical studies indicate that the proposed method is computationally efficient and yields fairly accurate results when compared to simulation.     

The horn-feed problem: sound waves in a tube joined to a cone,and related problems

A semi-infinite tube is joined to a semi-infinite cone. Waves propagating in the tube towards the join are partly reflected and partly radiated into the cone. The problem is to determine these wave fields. Two modal expansions are used, one in the tube and one in the cone. However, their regions of convergence do not overlap: there is a region D{\mathcal{D}} near the join where neither expansion converges. It is shown that the expansions can be connected by judicious applications of Green’s theorem in D{\mathcal{D}}. The resulting equations are solved asymptotically, for long waves or for narrow cones. Related two-dimensional problems are also solved. Applications to acoustics, electromagnetics and hydrodynamics are considered.     

QPSchur: A dual, active-set, Schur-complement method for large-scale and structured convex quadratic programming

We describe an active-set, dual-feasible Schur-complement method for quadratic programming (QP) with positive definite Hessians. The formulation of the QP being solved is general and flexible, and is appropriate for many different application areas. Moreover, the specialized structure of the QP is abstracted away behind a fixed KKT matrix called K_o and other problem matrices, which naturally leads to an object-oriented software implementation. Updates to the working set of active inequality constraints are facilitated using a dense Schur complement, which we expect to remain small. Here, the dual Schur complement method requires the projected Hessian to be positive definite for every working set considered by the algorithm. Therefore, this method is not appropriate for all QPs. While the Schur complement approach to linear algebra is very flexible with respect to allowing exploitation of problem structure, it is not as numerically stable as approaches using a QR factorization. However, we show that the use of fixed-precision iterative refinement helps to dramatically improve the numerical stability of this Schur complement algorithm. The use of the object-oriented QP solver implementation is demonstrated on two different application areas with specializations in each area; large-scale model predictive control (MPC) and reduced-space successive quadratic programming (with several different representations for the reduced Hessian). These results demonstrate that the QP solver can exploit application-specific structure in a computationally efficient and fairly robust manner as compared to other QP solver implementations.     

The annihilator method for computing particular solutions to partial differential equations

We generalize the well-known annihilator method, used to find particular solutions for ordinary differential equations, to partial differential equations. This method is then used to find particular solutions of Helmholtz-type equations when the right hand side is a linear combination of thin plate and higher order splines. These particular solutions are useful in numerical algorithms for solving boundary value problems for a variety of elliptic and parabolic partial differential equations.     

On the anisotropic piezoelastic contact problem for an elliptical punch

Summary This paper firstly conducts a systematic investigation of the problem of a rigid punch indenting an anisotropic piezoelectric half-space. The Fourier transform method is employed to the mixed boundary value problem. Using the principle of linear superposition, the resulting transformed (algebraic) equations, whose right-hand sides contain both pressure and electric displacement terms, can be solved by superposing the solutions of two sets of algebraic equations, one containing pressure and another containing electric displacement. For an arbitrarily shaped punch, two governing equations are derived, which can be solved numerically. In the case of transversely isotropic piezoelectric media, the two governing equations are corresponding with that given by others using potential theory. Particularly, when the punch has elliptic cross-section, and the pressure and electric displacement are given by some certain forms of polynomial functions, then the displacement and electric potential are prescribed by polynomial functions in the contact area. The parameters contained in it satisfy a set of linear algebraic equations, whose coefficients involve contour integrals. The problem of indentation by a smooth flat punch is examined for special orthotropic piezoelectric media, and some results obtained can be degenerated to the case of transversely isotropic piezoelectric media.     

Iterative computational identification of a space-wise dependent source in parabolic equation

Coefficient inverse problems related to identifying the right-hand side of an equation with use of additional information is of interest among inverse problems for partial differential equations. When considering non-stationary problems, tasks of recovering the dependence of the right-hand side on time and spatial variables can be treated as independent. These tasks relate to a class of linear inverse problems, which sufficiently simplifies their study. This work is devoted to finding the dependence of right-hand side of multidimensional parabolic equation on spatial variables using additional observations of the solution at the final point of time – the final overdetermination. More general problems are associated with some integral observation of the solution in time – the integral overdetermination. The first method of numerical solution of inverse problems is based on iterative solution of boundary value problem for time derivative with non-local acceleration. The second method is based on the known approach with iterative refinement of desired dependence of the right-hand side on spatial variables. Capabilities of proposed methods are illustrated by numerical examples for a two-dimensional problem of identifying the right-hand side of a parabolic equation. The standard finite-element approximation in space is used, whilst the time discretization is based on fully implicit two-level schemes.     

Acoustic and effective material parameters of heterogeneous viscoelastic bodies

Summary Acoustic waves propagating through a special class of random viscoelastic bodies are considered. These media are assumed to be composed ofM (M=1, 2, 3, ...) distinct and linearly viscoelastic materials (phases). All these phases are randomly dispersed in the volume of the body in the form of grains, perfectly bonded together. Assuming the mean displacement field to be in the form of attenuated plane waves, the equations for the fluctuating part of the displacement were solved formally for the case of small, moderate and large inhomogeneities. Subsequent to this, equations and formulae determining the effective complex moduli tensor of the heterogeneous body as well as the propagation velocity and attenuation coefficient are derived for the mean displacement field propagating through such media in the form of acoustic plane waves.     

Limiting equilibrium of a spherical elastoplastic shell with two colinear cracks

The problem of the stress-strain state of an elastoplastic spherical shell with two colinear cracks is reduced to a system of singular integral equations with unknown integration limits and discontinuous right-hand sides. The system obtained is solved simultaneously for conditions of stress finiteness and plasticity. The algorithm of the numerical solution of such a system, which is based on the method of mechanical quadratures, is constructed. The dependence of the opening of the crack tips on the applied load and the distance between the crack centers is analyzed numerically.Translated from Problemy Prochnosti, No. 8, pp. 24–28, August, 1994.     

A closed formula for local sensitivity analysis in mathematical programming

This article introduces a method for local sensitivity analysis of practical interest. A theorem is given that provides a general and neat manner to obtain all sensitivities of a general nonlinear programming problem (around a local minimum) with respect to any parameter irrespective of it being a right-hand side, objective function or constraint constant. The method is based on the well-known duality property of mathematical programming, which states that the partial derivatives of the primal objective function with respect to the constraints' right-hand side parameters are the optimal values of the dual problem variables. For the parameters or data for which sensitivities are sought to appear on the right-hand side, they are converted into artificial variables and set to their actual values, thus obtaining the desired constraints. If the problem is degenerated and partial derivatives do not exist, the method also permits obtaining the right, left, and also directional derivatives, if they exist. In addition to its general applicability, the method is also computationally inexpensive because the necessary information becomes available without extra calculations. Moreover, analytical relations among sensitivities, locally valid, are straightforwardly obtained. It is also shown how the roles of the objective function and any of the active constraints (equality or inequality) can be exchanged leading to equivalent optimization problems. This permits obtaining the sensitivities of any constraint with respect to the parameters without the need of repeating the calculations. The method is illustrated by its application to two examples, one degenerated and the other one of a competitive market.     

Superfluid Mass-Energy Densities of Nonlocal Particle and Gravitational Field

Dynamical gravitational and geodesic equations are derived for superfluid densities of nonlocal self-coherent particles. The geometrized gravitational particle is the r ⁻⁴ distribution of inertial mass that balances Ricci curvatures in the Einstein equation without the right-hand side. The spatial energy integral of such an infinite radial particle is finite and determines its nonlocal gravimechanical charge for energy-to-energy interactions with other nonlocal particles. Non-empty space of the flat material world is filled continuously by overlapping energy-flows of all nonlocal particles and their fields.     

Elastic interaction between screw dislocations and the internal crack near a free surface

The elastic interaction between screw dislocation and the internal crack near a free surface has been investigated. The stress intensity factor at the crack tip, crack extension force, the image force on the dislocation are affected by the free surface. The number and nature of dislocations, m, inside the crack also play an important role in fracture. In order to understand the plastic zone, the zero-force points of dislocation along the x-axis are involved. The dislocation emitted from the right-hand crack tip is enhanced by positive m and reduced by negative m. On the other hand, if the internal crack is closer to the free surface, a dislocation generated from the right-hand crack tip is easier for negative m and more difficult for positive m. However, the role of m on the dislocation emission for the left-hand crack tip is opposite to that for the right-hand crack tip. Finally, three special cases can be obtained from our results. (1) The interaction between a dislocation and a surface crack; (2) the interaction between a dislocation and an internal crack; (3) the interaction between two dislocations.     

AN ITERATIVE SOLUTION SCHEME FOR SYSTEMS OF WEAKLY CONNECTED BOUNDARY ELEMENT EQUATIONS

In this paper an iterative scheme of first degree is developed for solving linear systems of equations. The systems investigated are those which are derived from boundary integral equations and are of the form ∑^N_j=1H_ijx_j=c_i, i=1, 2,…,N, where H_ij are matrices, x_j and c_i are column vectors. In addition, N denotes the number of domains and for i≠j, H_ij is considered to be small in some sense. These systems, denoted as weakly connected, are solved using first-order iterative techniques initially developed by the authors for solving single-domain problems. The techniques are extended to solve multi-domain problems. Novel solution strategies are investigated and procedures are developed which are computationally efficient. Computation times are determined for the iterative procedures and for elimination techniques indicating the benefits of iterative techniques over direct methods for problems of this nature.     

An application of modified reductive perturbation method to long water waves

In this work, we extended the application of “the modified reductive perturbation method” to long water waves and obtained the governing equations as the KdV hierarchy. Seeking a localized travelling wave solutions to these evolution equations we determined the scale parameter c₁so as to remove the possible secularities that might occur. The present method is seen to be fairly simple as compared to the renormalization method [Kodama, Y., & Taniuti, T. (1977). Higher order approximation in reductive perturbation method 1. Weakly dispersive system. Journal of Physics Society of Japan, 45, 298–310] and the multiple scale expansion method [Kraenkel, R. A., Manna, M. A., & Pereira, J. G. (1995). The Korteweg–deVries hierarchy and long water waves. Journal of Mathematics Physics, 36, 307–320].     

Computer programs for the solution of systems of linear algebraic equations

FORTRAN subprograms for the solution of systems of linear algebraic equations are evaluated and compared on the basis of execution speed and accuracy. A symmetric, positive definite, banded test matrix is used in each case. The procedures considered are direct solution, iteration and matrix inversion. Both in-core schemes and those requiring the use of auxiliary data storage devices are included. Some of the techniques used require the full coefficient matrix, whereas others account for symmetry, banding, or sparseness of the system. Matrix inversion is found to be an inefficient technique, even if multiple right-hand side constant vectors are to be solved. In such cases, either Gauss elimination with multiple constant vectors treated simultaneously, or decomposition with the retention of the upper and lower triangular matrices, is recommended. Double precision arithmetic is suggested as a means of reducing round-off errors, and should always be employed when permitted by the computer's core capacity. On the basis of the results obtained with the single test system of equations, specific subprograms are recommended for each category of problem.     

Particular solutions of splines and monomials for polyharmonic and products of Helmholtz operators

This paper presents the particular solutions for the polyharmonic and the products of Helmholtz partial differential operators with polyharmonic splines and monomials right-hand side. By the application of the Hörmander linear partial differential operator theory, many of the systems can be reduced to a single equation involving the polyharmonic or the product of Helmholtz differential operators. If the inhomogeneous right-hand side of these operators can be removed by the method of particular solutions, then boundary-type numerical methods, such as the boundary element method, the method of fundamental solutions, and the Trefftz method, can be applied to solve these differential equations.     

A variational principle for magneto-elastic buckling

A variational principle that can serve as the basis for a magneto-elastic stability (or buckling) problem is constructed. For the two cases of soft ferromagnetic media and superconductors, respectively, it is shown how the variational principle directly yields an explicit expression for the buckling value. The formulation starts from a specific choice for a magneto-elastic Lagrangian L (associated with the so-called Maxwell-Minkowski model for magneto-elastic interactions). For the evaluation of the principle the first and second variations of L are calculated both inside and outside the solid magneto-elastic body. Thus, a general buckling criterion, consisting of an expression for the critical field value, together with a set of constraints for the field variables occurring in the right-hand side of this expression, is constructed. Finally, more detailed formulations are given for, successively, soft ferromagnetic bodies and superconductors. Applications to specific structures, yielding explicit numerical values for the magneto-elastic buckling fields, will be given in a forthcoming paper.     

Improvement of PUFEM for the numerical solution of high‐frequency elastic wave scattering on unstructured triangular mesh grids

The purpose of this paper, which builds on previous work (Int. J. Numer. Meth. Engng 2009; 77 :1646–1669), is to improve a numerical scheme based on the partition of unity finite element method (PUFEM) for the solution of the time harmonic elastic wave equations. The approach consists to approximate the displacement field by the standard finite element shape functions, enriched locally by superimposing pressure (P) and shear (S) plane waves. The aim is to accurately model two‐dimensional elastic wave problems on relatively coarse mesh grids, capable of containing many wavelengths per nodal spacing, for wide ranges of frequencies. This allows us to relax the traditional requirement of about 10 nodal points per S wavelength. In this work, an exact integration scheme for the linear triangular finite element is developed to evaluate the oscillatory integrals arising from the use of the PUFEM. The main contribution here consists in developing an explicit closed‐form solution for two‐dimensional wave‐based integrals, when the phase variation is linear in the local coordinate element system. The evaluation of the element mass matrix is performed from appropriate edge integrals. All other element matrices, obtained by adequate splitting of the element stress tensor matrix, are simply deduced from the element mass matrix entries. The results show clearly that the proposed integration scheme evaluates accurately the entries of the global matrix with drastic reduction of the computational time. Numerical tests dealing with the scattering of S elastic plane waves by a circular rigid body show that, for the same discretization level, it is possible to improve the accuracy by using large elements associated with high numbers of approximating plane waves rather than using small elements with less plane waves. However, this increases the conditioning and the fill‐in of the global matrix. At high frequency, it is even possible to push the number of degrees of freedom per S wavelength under 2 and still achieve good accuracy. Finally, some remarks on the choice of the numbers of P and S plane waves leading to better accuracy and conditioning are discussed. Copyright © 2010 John Wiley & Sons, Ltd.     

Rogue waves for a coupled nonlinear Schrödinger system in a multi-mode fibre

In this paper, we investigate the rogue waves for an integrable coupled nonlinear Schrödinger (CNLS) system with the self-phase modulation, cross-phase modulation and four-wave mixing term, which can describe the propagation of optical waves in a multi-mode fibre. We construct a generalized Darboux transformation (GDT) for the CNLS system and find a gauge transformation which converts the Lax pair into the constant-coefficient differential equations. Solving those equations, we can obtain the vector solutions of the Lax pair. Using the GDT, we derive an iterative formula for the nth-order rogue-wave solutions for the CNLS system. We derive the first- and second-order rogue-wave solutions for the CNLS system and analyse the profiles for the rogue waves with respect to the self-phase modulation term a, cross-phase modulation term c and four-wave mixing term b, respectively. The rogue waves become thinner with the increase in the value for the real part of b and that the effect of a or c on the rogue waves is the same as the one of the real part of b.