Stability analysis of the Fourier–Bessel method for the Cauchy problem of the Helmholtz equation

This paper is concerned with the Cauchy problem connected with the Helmholtz equation in a smooth-bounded domain. The Fourier–Bessel method with Tikhonov regularization is applied to achieve a regularized solution to the problem with noisy data. The convergence and stability are obtained with a suitable choice of the regularization parameter. Numerical experiments are also presented to show the effectiveness of the proposed method.     

A Dirichlet–Neumann cost functional approach for the Bernoulli problem

The Bernoulli problem is rephrased into a shape optimization problem. In particular, the cost function, which turns out to be a constitutive law gap functional, is borrowed from inverse problem formulations. The shape derivative of the cost functional is explicitly determined. The gradient information is combined with the level set method in a steepest descent algorithm to solve the shape optimization problem. The efficiency of this approach is illustrated by numerical results for both interior and exterior Bernoulli problems.     

Boundary element–minimal error method for the Cauchy problem associated with Helmholtz-type equations

An iterative procedure, namely the minimal error method, for solving stably the Cauchy problem associated with Helmholtz-type equations is introduced and investigated in this paper. This method is compared with another two iterative algorithms previously proposed by Marin et al. (Comput Mech 31:367–377, 2003; Eng Anal Bound Elem 28:1025–1034, 2004), i.e. the conjugate gradient and Landweber–Fridman methods, respectively. The inverse problem analysed in this study is regularized by providing an efficient stopping criterion that ceases the iterative process in order to retrieve stable numerical solutions. The numerical implementation of the aforementioned iterative algorithms is realized by employing the boundary element method for both two-dimensional Helmholtz and modified Helmholtz equations.     

More on ‘Models and algorithms for the dynamic-demand joint replenishment problem’

This paper presents tight mathematical programming formulations for the dynamic demand joint replenishment problem (DJRP). Experimental studies using general-purpose software document the improved computational efficiency of the new formulations versus the earlier models of Boctor et al. (Boctor, F.F., Laporte, G. and Renaud, J., Models and algorithms for the dynamic-demand joint replenishment problem. Int. J. Prod. Res., 2004, 42, 2667–2678.) and Robinson and Gao (Robinson, E.P. and Gao, L., A dual-ascent procedure for multi-product dynamic demand coordinated replenishment with backlogging. Manag. Sci., 1996, 42, 1556–1564.). The findings encourage the development of specialized algorithms for their solution. The paper also evaluates the performance of the perturbation and dual-ascent heuristics for solving DJRP finding the perturbation heuristic is superior at relatively low set-up cost ratios and high joint set-up cost levels, while the dual-ascent heuristic strongly dominates at relatively high set-up cost ratios and low joint set-up costs. Considering that both heuristics are computationally efficient, the best application strategy is to solve the problem with both heuristics and implement the best found solution.     

On blow-up phenomena for the weakly dissipative Camassa–Holm equation

We study the Cauchy problem for the weakly dissipative Camassa?CHolm equation: $$u_{t}-u_{txx}+3uu_{x}+\lambda \left( u-u_{xx} \right)=2u_{x}u_{xx}+uu_{xxx}, \quad t >0 ,x\in \mathbb{R}$$ . A new blow-up result for positive strong solutions of the equation with certain profiles is presented. In particular, we use a condition where the initial data u ₀ and its derivative u _0x are not simultaneously involved and the parameter ?? is not bounded from above.     

Cascadic meshfree method for the elliptic Monge–Ampère equation

The elliptic Monge–Ampère equation is a fully nonlinear partial differential equation, which originated in geometric surface theory and has been widely applied in dynamic meteorology, elasticity, geometric optics, image processing and others. The numerical solution of the elliptic Monge–Ampère equation has been a subject of increasing interest recently. In this paper, we design a cascadic algorithm which is meshfree. We first generate hierarchical scattered data sets. Then on each successive refinement levels, the Monge–Ampère equation can be solved by Kansa's method. We call this method as cascadic meshfree method (CMF). Different from cascadic multigrid method, CMF avoids tedious interpolation and is more easy for implementation and coding. Finally, numerical experiments confirm the efficiency and robustness of CMF method.     

The inverse problem of Lagrangian mechanics for Meshchersky’s equation

Variable-mass systems are not included in the conventional domain of the analytical and variational methods of classical mechanics. This is due to the fact that the fundamental principles of mechanics were primarily conceived for constant-mass systems. In the present article, an analytical and variational formulation for variable-mass systems will be proposed. This will be done from the solution of the here called ‘inverse problem of Lagrangian mechanics for Meshchersky’s equation’. The first problem of this nature was posed in 1887, by Helmholtz (J. reine angew. Math. 100:137–166, 1887). Investigations on the matter are far from being exhausted. Within mechanics, it means the construction of a Lagrangian from a given equation of motion. To the authors’ best knowledge, aiming at general results, the inverse problem of Lagrangian mechanics has not been properly connected to Meshchersky’s equation yet. This is the main goal of this article. We will address the issue by assuming that mass depends on generalized coordinate, generalized velocity and on time. After the construction of a Lagrangian from Meshchersky’s equation, a general and unifying mathematical formulation will emerge in accordance. Therefore, variable-mass systems will be accommodated at the level of analytical mechanics. A variational formulation, which will be written via a principle of stationary action, and a Hamiltonian formulation will be both stated. The latter could be read as the ‘Hamiltonization’ of variable-mass systems from the solution of the inverse problem of Lagrangian mechanics. An energy-like conservation law will naturally appear from the simplification of the general theory to the case of a system with mass solely dependent on a generalized coordinate.     

Conservation of energy of non-Newtonian media for the Rayleigh–Stokes problem

The energetic balance of the Rayleigh–Stokes problem for Newtonian-, second grade- and Maxwell fluids is studied for different initial and boundary conditions. We get the solutions of the differential equations by Fourier sine transform or by series expansion. The result for the kinetic energy E _kin, the dissipation Φ and the power of the shear stresses at the wall L are important for nature and technology.     

On the ‘interior Helmholtz integral equation formulation’ in sound radiation problems

This paper discusses the accuracy of the sound radiation (Helmholtz equation) BEM analysis when the source points are located inside the vibrating surface. It is shown that when these points are arranged in such a way that they compose a fictitious internal boundary, geometrically similar to the vibrating surface, then fictitious eigenfrequencies appear at larger values than those corresponding to the eigenfrequencies of the internal boundary. In this way, the direct-BEM analysis becomes capable of treating the non-uniqueness problem in a simple and efficient practical manner, which makes the method applicable for industrial purposes. Results are presented for the spherical monopole and dipole, while discussion is extended to a similarly vibrating cube.     

A Petrov–Galerkin finite element scheme for the regularized long wave equation

A Petrov-Galerkin finite element method (FEM) for the regularized long wave (RLW) equation is proposed. Finite elements are used in both the space and the time domains. Dispersion correction and a highly selective dissipation mechanism are introduced through additional streamline upwind terms in the weight functions. An implicit, conditionally stable, one-step predictor–corrector time integration scheme results. The accuracy and stability are investigated by means of local expansion by Taylor series and the resulting equivalent differential equation. An analysis based on a linear Fourier series solution and the Von Neumanns stability criterion is also performed. Based on the order of the analytical approximations and of the domain discretization it is concluded that the scheme is of third order in the nonlinear version and of fourth order in the linear version. Three numerical experiments of wave propagation are presented and their results compared with similar ones in the literature: solitary wave propagation, undular bore propagation, and cnoidal wave propagation. It is concluded that the present scheme possesses superior conservation and accuracy properties.This work has been partially supported by the Fundação para a Ciência e Tecnologia, under project POCTI/ECM/41800/2001.     

Efficient parallel algorithms for elastic–plastic finite element analysis

This paper presents our new development of parallel finite element algorithms for elastic–plastic problems. The proposed method is based on dividing the original structure under consideration into a number of substructures which are treated as isolated finite element models via the interface conditions. Throughout the analysis, each processor stores only the information relevant to its substructure and generates the local stiffness matrix. A parallel substructure oriented preconditioned conjugate gradient method, which is combined with MR smoothing and diagonal storage scheme are employed to solve linear systems of equations. After having obtained the displacements of the problem under consideration, a substepping scheme is used to integrate elastic–plastic stress–strain relations. The procedure outlined controls the error of the computed stress by choosing each substep size automatically according to a prescribed tolerance. The combination of these algorithms shows a good speedup when increasing the number of processors and the effective solution of 3D elastic–plastic problems whose size is much too large for a single workstation becomes possible.     

A numerical method based on the boundary integral equation and dual reciprocity methods for one-dimensional Cahn–Hilliard equation

This paper describes a numerical method based on the boundary integral equation and dual reciprocity methods for solving the one-dimensional Cahn–Hilliard (C–H) equation. The idea behind this approach comes from the dual reciprocity boundary element method that introduced for higher order dimensional problems. A time-stepping method and a predictor–corrector scheme are employed to deal with the time derivative and the nonlinearity respectively. Numerical results are presented for some examples to demonstrate the usefulness and accuracy of this approach. For these problems the energy functional dissipation and the mass conservation properties are investigated.     

Three Green element models for the diffusion–advection equation and their stability characteristics

Solutions of the transient 1-D diffusion–advection equation by three models of the Green element method (GEM) and their stability characteristics are presented. GEM is a novel approach of implementing the singular boundary integral theory so that computational efficiency is enhanced, and the theory is made more versatile. The first model, denoted as the quasi-steady Green element (QSGE) model, employs the Green’s function of the Laplacian operator in deriving its integral representation, while the second, denoted the TGE model, uses the Green’s function of the transient diffusion differential operator, and the third, denoted the ADGE model, uses the Green’s function of the diffusion–advection differential operator. The first model, which had earlier been presented, is herein compared to the other two models. Three numerical examples are used to compare the accuracies of the three models. It is observed that incorporating the Crank–Nicholson scheme into the first model not only gives optimal results of the three models, but it more readily accommodates transport with nonuniform flow velocity field and first-order rate of decay of the pollutant. Further, the mathematical simplicity of the Green’s function of the first model is an added advantage which enhances computational efficiency. The numerical stability characteristics of these models are evaluated by examining their propagation of the amplitudes and speeds of Fourier wave components in relation to their corresponding theoretical values. The results from the stability analysis confirms the superiority of the QSGE model with the Crank–Nicholson scheme.     

On symmetries of the Fokker–Planck equation

This paper is devoted to Lie point symmetries of the Fokker–Planck (FP) equation. It describes the relation between symmetries and first integrals of stochastic differential equations (SDEs) and symmetries of the associated FP equations. This relation is illustrated on symmetries of (1 + 1)-dimensional FP equations specified by Lie group classification of the scalar SDE. Further, it is used to find symmetries of (1 + 2)-dimensional FP equations specified by Lie group classification of the system of two SDEs.     

An analysis for the elasto-plastic problem of the moderately thick plate using the meshless local Petrov–Galerkin method

A meshless local Petrov–Galerkin method for the analysis of the elasto-plastic problem of the moderately thick plate is presented. The discretized system equations of the moderately thick plate are obtained using a locally weighted residual method. It uses a radial basis function (RBF) coupled with a polynomial basis function as a trial function, and uses the quartic spline function as a test function of the weighted residual method. The shape functions have the Kronecker delta function properties, and no additional treatment to impose essential boundary conditions. The present method is a true meshless method as it does not need any grids, and all integrals can be easily evaluated over regularly shaped domains and their boundaries. An incremental Newton–Raphson iterative algorithm is employed to solve the nonlinear discretized system equation. Numerical results show that the present method possesses not only feasibility and validity but also rapid convergence for the elasto-plastic problem of the moderately thick plate.     

Piecewise oblique boundary treatment for the elastic–plastic wave equation on a cartesian grid

Numerical schemes for hyperbolic conservation laws in 2-D on a Cartesian grid usually have the advantage of being easy to implement and showing good computational performances, without allowing the simulation of “real-world” problems on arbitrarily shaped domains. In this paper a numerical treatment of boundary conditions for the elastic–plastic wave equation is developed, which allows the simulation of problems on an arbitrarily shaped physical domain surrounded by a piece-wise smooth boundary curve, but using a PDE solver on a rectangular Cartesian grid with the afore-mentioned advantages.     

Neighborhood search algorithms with restricted infeasible solution sets–application to a transportation project evaluation problem

Abstract

Using a transportation project evaluation problem as an example, in this paper we employ the local search method, the threshold accepting method, together with the combination of feasible and restricted infeasible solution sets in neighborhood searches, to develop four solution algorithms. The test results indicate that the threshold accepting algorithm and the local search algorithm, that combine feasible and restricted infeasible solution sets, can improve the conventional threshold accepting algorithm and local search algorithm, which are confined to feasible solution sets.     

A gradient free integral equation for diffusion–convection equation with variable coefficient and velocity

In this paper a boundary-domain integral diffusion–convection equation has been developed for problems of spatially variable velocity field and spatially variable coefficient. The developed equation does not require a calculation of the gradient of the unknown field function, which gives it an advantage over the other known approaches, where the gradient of the unknown field function is needed and needs to be calculated by means of numerical differentiation. The proposed equation has been discretized by two approaches—a standard boundary element method, which features fully populated system matrix and matrices of integrals and a domain decomposition approach, which yields sparse matrices. Both approaches have been tested on several numerical examples, proving the validity of the proposed integral equation and showing good grid convergence properties. Comparison of both approaches shows similar solution accuracy. Due to nature of sparse matrices, CPU time and storage requirements of the domain decomposition are smaller than those of the standard BEM approach.