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This paper deals with a diffusive logistic equation on one dimensional isotropically growing domain. The model equation on growing domains is first presented, and the comparison principle is then proved. The asymptotic behavior of temporal solutions to the reaction-diffusion problem is given by constructing upper and lower solutions. Our result shows that when the domain grows slowly, the species successfully spreads to the whole habitat and stabilizes at a positive steady state, while it dies out in the long run if the domain grows fast. Numerical simulations are also presented to illustrate the analytical result. [ABSTRACT FROM AUTHOR]

A mathematical model of mediated bioelectrocatalysis is restudied in this paper. Here He's Homotopy perturbation method is implemented to give approximate and analytical solutions of steady-state non-linear reaction diffusion equation containing a non-linear term related to Michaelis-Menten kinetics of the enzymatic reaction. Approximate analytical expressions for mediator concentration and current have been derived for all values of saturation parameter α and reaction diffusion parameter k for the various types of boundary conditions. The Homotopy perturbation method which produces the solutions in terms of convergent series, requiring no linearization or small perturbation. These analytical results are compared with numerical results (Matlab program) and are found to be in good agreement. [ABSTRACT FROM AUTHOR]

In this paper, we present a new quartic B-spline collocation method for linear and nonlinear fifth order boundary value problems. The method is fourth order convergent. It develops not only the quartic spline approximation solution but also the higher order approximation derivatives. Numerical examples are performed and comparisons with some other methods are also made. Numerical results show our method has higher accuracy and efficiency. [ABSTRACT FROM AUTHOR]

In this paper, a new method for analysis of the pseudoelastic response of shape memory alloy thick-walled cylinders subjected to internal pressure is proposed. Two cases of short and long cylinders are considered by assuming the plane stress and plane strain conditions. In each case, a three-dimensional phenomenological SMA constitutive model is simplified to obtain the corresponding two-dimensional constitutive relations. The cylinder is partitioned into a finite number of narrow annular regions, and appropriate assumptions are made in order to find a closed-form solution for the equilibrium equations in each annular region. The global solution is obtained by enforcing the stress continuity condition at the interface of the annular regions and imposing the boundary conditions. Several numerical examples are presented to demonstrate the efficiency of the proposed method, and the results are compared with three-dimensional finite element simulations. [ABSTRACT FROM AUTHOR]

In this paper, we consider a fractional impulsive boundary value problem on infinite intervals. We obtain the existence, uniqueness and computational method of unbounded positive solutions. [ABSTRACT FROM AUTHOR]

In this paper, the static and free vibration analysis of laminated shells is performed by radial basis functions (RBFs) collocation, according to a layerwise deformation theory (LW). The present LW theory accounts for through-the-thickness deformation, by considering an Mindlin-like evolution of all displacements in each layer. The equations of motion and the boundary conditions are obtained by Carrera's unified formulation, and further interpolated by collocation with RBFs. [ABSTRACT FROM AUTHOR]

In this paper, we consider the nonlinear complementarity problem with band structure. This problem occurs, for example, if certain classes of free boundary problems are discretized. We compute error bounds for the approximate solution of the discretized problems. The error bounds are improved by an iterative method and can be made arbitrarily small. The ideas are illustrated by numerical experiments. [ABSTRACT FROM AUTHOR]

Our motivation is a mathematical model describing the spatial propagation of an epidemic disease through a population. In this model, the pathogen diversity is structured into two clusters and then the population is divided into eight classes which permits to distinguish between the infected/uninfected population with respect to clusters. In this paper, we prove the weak and the global existence results of the solutions for the considered reaction-diffusion system with Neumann boundary. Next, mathematical Turing formulation and numerical simulations are introduced to show the pattern formation for such systems. [ABSTRACT FROM AUTHOR]

This paper is concerned with a two-species predator-prey reaction-diffusion system with Beddington-DeAngelis functional response and subject to homogeneous Neumann boundary conditions. By linearizing the system at the positive constant steady-state solution and analyzing the associated characteristic equation in detail, the asymptotic stability of the positive constant steady-state solution and the existence of local Hopf bifurcations are investigated. Also, it is shown that the appearance of the diffusion and homogeneous Neumann boundary conditions can lead to the appearance of codimension two Bagdanov-Takens bifurcation. Moreover, by applying the normal form theory and the center manifold reduction for partial differential equations (PDEs), the explicit algorithm determining the direction of Hopf bifurcations and the stability of bifurcating periodic solutions is given. Finally, numerical simulations supporting the theoretical analysis are also included. [ABSTRACT FROM AUTHOR]

The paper addresses the problem of finding a full-strength contour in the problem of plate bending for a cycle-symmetric doubly connected domain. An isotropic elastic plate, bounded by a regular polygon, is weakened by a required full-strength hole whose symmetry axes are the regular polygon diagonals. Rigid bars are attached to each component of the broken line of the outer boundary of the plate. The plate bends under the action of concentrated moments applied to the middle points of the bars. An unknown part of the boundary is free from external forces. Using the methods of complex analysis, the analytical image of Kolosov-Muskhelishvili's complex potentials (characterising an elastic equilibrium of the body) and of an unknown full-strength contour are determined. A numerical analysis is performed and the corresponding plots are obtained by means of the Mathcad system. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the asymptotic behavior of solutions to the initial boundary value problem for the interaction between the temperature field and the porosity fields in a homogeneous and isotropic mixture from the linear theory of porous Kelvin-Voigt materials. Our main result is to establish conditions which insure the analyticity and the exponential stability of the corresponding semigroup. We show that under certain conditions for the coefficients we obtain a lack of exponential stability. A numerical scheme is given. [ABSTRACT FROM AUTHOR]

This paper concerns the study of the numerical approximation for the following initialboundary value problem where f( s) is a positive, increasing, C convex function for the nonnegative values of s, f(0) > 0, $$ \frac{{ds}} {{f\left( s \right)}} $$ < ∞, u ∈ C([0, 1]), u(0) = 0, u′(1) = 0. We find some conditions under which the solution of a semidiscrete form of the above problem blows up in a finite time and estimate its semidiscrete blow-up time. We also prove the convergence of the semidiscrete blow-up time to the theoretical one. A similar study has been also undertaken for a discrete form of the above problem. Finally, we give some numerical results to illustrate our analysis. [ABSTRACT FROM AUTHOR]

This paper deals with the problem of asymptotic stability for delayed genetic regulatory networks with reaction--diffusion terms. For three types of boundary conditions, delay-dependent criteria are established by the Lyapunov--Krasovskii functional approach, respectively. The obtained results are expressed in terms of linear matrix inequalities. Numerical examples illustrating the effectiveness of the proposed approach are provided. [ABSTRACT FROM AUTHOR]

This paper investigates the existence and uniqueness of C[0, 1] positive solutions for a second order integral boundary value problem. We mainly use the method of lower and upper solutions and the maximal principle. Our nonlinearity f(t, u) may be singular at u = 0, t = 0, 1. [ABSTRACT FROM AUTHOR]

This paper presents a parallel generalized finite element method ( GFEM) that uses customized enrichment functions for applications where limited a priori knowledge about the solution is available. The procedure involves the parallel solution of local boundary value problems using boundary conditions from a coarse global problem. The local solutions are in turn used to enrich the global solution space using the partition of unity methodology. The parallel computation of local solutions can be implemented using a single pair of scatter-gather communications. Several numerical experiments demonstrate the high parallel efficiency of these computations. For problems requiring non-uniform mesh refinement and enrichment, load unbalance is addressed by defining a larger number of small local problems than the number of parallel processors and by sorting and solving the local problems based on estimates of their workload. A simple and effective estimate of the largest number of processors where load balance among processors is maintained is also proposed. Several three-dimensional fracture mechanics problems aiming at investigating the accuracy and parallel performance of the proposed GFEM are analyzed. [ABSTRACT FROM AUTHOR]

This paper introduces a new model of pedestrian flow, formulated within a measure-theoretic framework. It consists of a macroscopic representation of the system via a family of measures which, pushed forward by some flow maps, provide an estimate of the space occupancy by pedestrians at successive times. From the modeling point of view, this setting is particularly suitable for treating nonlocal interactions among pedestrians, obstacles, and wall boundary conditions. In addition, the analysis and numerical approximation of the resulting mathematical structures, which are the principal objectives of this work, follow more easily than for models based on standard hyperbolic conservation laws. [ABSTRACT FROM AUTHOR]

This paper presents a numerical analysis of the method of factorization for elliptic boundary value problems. A second-order elliptic boundary value problem is transformed into a decoupled system of first-order initial value problems. This elaborates the decoupled system's occupying operator, the solution of the problem, and the affine part of the operator. The solution of the problem is thus obtained by solving the factorized system using the finite-element method. A comparative study is made with the help of an example that illustrates many of its properties. [ABSTRACT FROM AUTHOR]

The stress and displacement fields for a viscoelastic axisymmetric plane problem involving time-dependent boundary regions under mixed boundary condition are presented in this paper. The viscoelastic fields are determined by solving two unknown functions; one is governed by a resulting second kind Volterra integral equation from the stress condition at the outer radius of an annular region, which is dependent on the other by the displacement condition at the inner radius. The integral equation has an analytical solution for the case of an infinite region with large enough outer radius while it can be solved using a numerical integral method for the case of a finite region. Numerical examples for the Boltzmann viscoelastic model are given, and the responses of the displacement and stresses are presented in detail to illustrate the effects of velocity of change in the inner radius and the magnitude of the outer radius. Meanwhile, the effect of the aspect ratio (void concentration parameter) on the viscoelastic fields is presented. When the ratio of the outer radius to the inner radius is large enough, the responses can be evaluated by directly adopting the resulting analytical solution to avoid a complex numerical procedure. The resulting solutions and computational results are helpful to a better understanding of mechanical behaviors for (large) excavation or finite void growth in viscoelastic media. [ABSTRACT FROM AUTHOR]

The paper establishes existence, uniqueness and regularity for the solution of a phase-field transition system with non-homogeneous Cauchy-Neumann boundary conditions. Moreover, some estimates for the solution are given. The non-linear parabolic problem considered here can be used to modeling the solidification (liquidification) process to a matter. The non-homogeneous boundary conditions (depending both on time and space variables) can be understood as a boundary control. [ABSTRACT FROM AUTHOR]

In this paper we prove the convergence of an iterative scheme of fractional steps type for a non-homogeneous Cauchy-Neumann boundary optimal control problem governed by non-linear phase-field system, when the boundary control is dependent both on time and spatial variables. Moreover, necessary optimality conditions are established for the approximating process. The advantage of such approach leads to a numerical algorithm in order to approximate the original optimal control problem. [ABSTRACT FROM AUTHOR]

We study the numerical solution of an initial-boundary value problem for parabolic integro-differential equation with a weakly singular kernel. The main purpose of this paper is to construct and analyze stable and high order scheme to efficiently solve the integro-differential equation. The equation is discretized in time by the finite central difference and in space by the finite element method. We prove that the full discretization is unconditionally stable and the numerical solution converges to the exact one with order O(Δ t + h). A numerical example demonstrates the theoretical results. [ABSTRACT FROM AUTHOR]

This paper presents a precise method for solving singularly perturbed boundary-value problems with the boundary layer at one end. The method divides the interval evenly and gives a set of algebraic equations in a matrix form by the precise integration relationship of each segment. Substituting the boundary conditions into the algebraic equations, the coefficient matrix can be transformed to the block tridiagonal matrix. Considering the nature of the problem, an efficient reduction method is given for solving singular perturbation problems. Since the precise integration relationship introduces no discrete error in the discrete process, the present method has high precision. Numerical examples show the validity of the present method. [ABSTRACT FROM AUTHOR]

The paper is devoted to the upscaling method appropriate for single-phase flow in media with discontinuous permeability distribution. The suggested algorithm is a modification of the iterative adaptive local–global upscaling developed by Chen and coauthors. The key feature of this method is a consistency between local and coarse global calculated characteristics. In this work, we apply a modified procedure to determine the boundary conditions used in the local fine-scale computation. To increase the accuracy of these boundary conditions on each iteration, we involve an additional preliminary step based on the results of coarse scale calculations from the previous iteration. Numerical tests show an essential improvement of the accuracy of upscaled flow rates for most of the realizations of statistical permeability distribution. Although the developed method is universal, its efficiency increases with increasing of permeability contrast. [ABSTRACT FROM AUTHOR]

Consider the over-determined system Fx = b where $${{\bf F}\in\mathcal{R}^{m \times n}, m \geq n}$$ and rank ( F) = r ≤ n, the effective condition number is defined by $${{\rm Cond_{-}eff }= \frac {\|{\bf b}\|}{\sigma_r\|{\bf x}\|}}$$, where the singular values of F are given as σmax = σ1 ≥ σ2 ≥ . . . ≥ σ r > 0 and σ r+1 = . . . = σ n = 0. For the general perturbed system ( A+Δ A)( x+Δ x) = b+Δ b involving both ΔA and Δb, the new error bounds pertinent to Cond_eff are derived. Next, we apply the effective condition number to the solutions of Motz’s problem by the collocation Trefftz methods (CTM). Motz’s problem is the benchmark of singularity problems. We choose the general particular solutions $${v_L = \sum\nolimits_{k=0}^L d_k (\frac {r}{R_p})^{k+\frac 12}}$$ $${{\rm cos}(k +\frac 12)\theta}$$ with a radius parameter R p. The CTM is used to seek the coefficients d i by satisfying the boundary conditions only. Based on the new effective condition number, the optimal parameter R p = 1 is found. which is completely in accordance with the numerical results. However, if based on the traditional condition number Cond, the optimal choice of R p is misleading. Under the optimal choice R p = 1, the Cond grows exponentially as L increases, but Cond_eff is only linear. The smaller effective condition number explains well the very accurate solutions obtained. The error analysis in [14,15] and the stability analysis in this paper grant the CTM to become the most efficient and competent boundary method. [ABSTRACT FROM AUTHOR]

In Xu et al. (Comput Struct 87:1416–1426, 2009) a novel Green-function-based multiscale stochastic finite element method (MSFEM) was proposed to model boundary value problems involving random heterogeneous materials. In this paper, we describe in detail computational aspects of the MSFEM explicitly across macro–meso–micro scales. Different numerical algorithms are introduced and compared in terms of numerical accuracy and convergence. [ABSTRACT FROM AUTHOR]

A class of periodic initial value problems for two-dimensional Newton-Boussinesq equations are investigated in this paper. The Newton-Boussinesq equations are turned into the equivalent integral equations. With iteration methods, the local existence of the solutions is obtained. Using the method of a priori estimates, the global existence of the solution is proved. [ABSTRACT FROM AUTHOR]

An enriched collocation method with the modified equilibrium on line method (ELM) for imposition of Neumann boundary conditions is presented for solving the two-dimensional elastic fracture problems. In the modified ELM, equilibrium over the lines on the Neumann boundary is satisfied as Neumann boundary condition equations. In this paper, two different enriched methods are used to construct meshless shape functions: enriched moving least squares approximation and enriched radial basis point interpolation. A number of numerical examples are presented to demonstrate the stability, accuracy, and convergence of the proposed method for solving two-dimensional elastic fracture problems. [ABSTRACT FROM AUTHOR]

Abstract  This main issue of this paper is a conjunction of the structural design sensitivity analysis using the Boundary Element Method with the polynomial response function determination. The procedure is so general that it enables sensitivity analysis for potential and elasticity problems within both homogeneous and heterogeneous plane and 3D problems. The essential difference with respect to the previous approaches like the Direct Differentiation Method or the Adjoint Variable Method is in discrete evaluation of the structural response using the response polynomials of some state parameters and design variable as the independent parameter. Such a determination is carried out via the several solutions of the given boundary value problem, where design parameter mean value is regularly perturbed in each of the solutions to cover the closest neighborhood of this mean value. Those few solutions make it possible to recover the polynomial response function from node-to node within the boundary elements, so that further symbolic differentiation using MAPLE returns the sensitivity gradients particular values. The entire procedure is tested here twice—first example deals with the homogeneous cantilever beam, where comparison against pure analytical differentiation is done and, separately, for two-component composite cantilever, where such a comparison is made against the central difference method linked with the same BEM solution. [ABSTRACT FROM AUTHOR]

For a second-order symmetric uniformly elliptic differential operator with rapidly oscillating coefficients, we study the asymptotic behavior of solutions of a mixed inhomogeneous boundary-value problem and a spectral Neumann problem in a thin perforated domain with rapidly varying thickness. We obtain asymptotic estimates for the differences between solutions of the original problems and the corresponding homogenized problems. These results were announced in Dopovidi Akademii Nauk Ukrainy, No. 10, 15-19 (1991). The new results obtained in the present paper are related to the construction of an asymptotic expansion of a solution of a mixed homogeneous boundary-value problem under additional assumptions of symmetry for the coefficients of the operator and for the thin perforated domain. [ABSTRACT FROM AUTHOR]

This paper is concerned with the variationally consistent incorporation of time dependent boundary conditions. The proposed methodology avoids ad hoc procedures and is applicable to both linear as well as nonlinear problems. An integral formulation of the dynamic problem serves as a basis for the imposition of the corresponding constraints, which are enforced via the consistent form of the penalty method, e.g. a form that complies with the norm and inner product of the functional space where the weak formulation is posed. Also, it is shown that well known and broadly implemented modelling techniques such as “large mass” and “large spring” methods, arise as limiting cases of the penalty formulation. Further extension of the proposed methodology is provided in the case where a generalized weak formulation of the dynamic problem with an independent velocity field is assumed. Finally, a few examples serve to illustrate the above concepts. [ABSTRACT FROM AUTHOR]

In this paper we investigate two-dimensional (2D) Rayleigh-Bénard convection using direct numerical simulation in Boussinesq fluids with Prandtl number P = 6:8 confined between thermally conducting plates. We show through the simulation that in a small range of reduced Rayleigh number r (770 < r < 890) the 2D rolls move chaotically in a direction normal to the roll axis. The lateral shift of the rolls may lead to a global flow reversal of the convective motion. The chaotic travelling rolls are observed in simulations with free-slip as well as no-slip boundary conditions on the velocity field. We show that the travelling rolls and the flow reversal are due to an interplay between the real and imaginary parts of the critical modes. [ABSTRACT FROM AUTHOR]

In this paper we present a multiscale framework suited for geometrically nonlinear computations of foam-like materials applying high-order finite elements ( p-FEM). This framework is based on a nested finite element analysis (FEA) on two scales, one nonlinear boundary value problem on the macroscale and k independent nonlinear boundary value problems on the microscale allowing for distributed computing. The two scales are coupled by a numerical projection and homogenization procedure. On the microscale the foam-like structures are discretized by high-order continuum-based finite elements, which are known to be very efficient and robust with respect to locking effects. In our numerical examples we will discuss in detail three characteristic test cases (simple shear, tension and bending). Special emphasis is placed on the material’s deformation-induced anisotropy and the macroscopic load-displacement behavior. [ABSTRACT FROM AUTHOR]

This paper develops and analyzes two techniques to extend the use of generalized finite element method techniques to structural shell problems. The first one is a procedure to define local domains for enrichment functions based on the use of pseudo-tangent planes. The second one is a procedure for imposing homogeneous essential boundary conditions and treatment of boundary layer problems by utilizing special functions. The main idea supporting the pseudo-tangent proposition is the separation of the geometric description, with its intrinsical distortions with respect to the physical domain, from the approximation space, which is defined in a locally undistorted domain. The treatment of essential boundary conditions allows an adequate enrichment in the boundary vicinity, preserving the completeness of the polynomials defining the basis functions. A set of numerical cases are tested in order to show the behavior of the proposed strategies, and a number of observations are drawn from the results, as follows. First, the technique of constructing the enrichment functions on a pseudo-tangent plane shows good results, even with strongly curved shell surfaces. With respect to the locking problem, the method behaves in a similar way as the classical hierarchical finite element methods, avoiding locking for appropriate levels of p-refinements. The procedure considered to impose essential boundary conditions in strong form appears to be more accurate than with the penalty or Lagrange multiplier methods. The inclusion of exponential modes for the treatment of boundary layers in shells provided extremely good results, even with integration elements much larger than the shell thickness. [ABSTRACT FROM AUTHOR]

In this work, for the first time in an article, we present in a comprehensive way the explicit advanced step-point (EAS) methods. The EAS methods is a family of methods designed for the numerical solution of non-stiff and mildly stiff initial value problems (IVPs) and comprises three distinct schemes: EAS1, EAS2 and EAS3. A thorough theoretical analysis of the EAS family of predictor–corrector methods is presented in terms of their accuracy and stability characteristics and requirements, as well as the rationale for creating the three distinct schemes mentioned above. In this paper we also examine in detail one of the three schemes, the EAS1 methods. EAS1 are assessed for the very first time, are meticulously studied and their superior regions of absolute stability are presented. Furthermore the computational efficiency of EAS1 is examined and comparative numerical results are presented with the use of a variable step, variable order EAS1 code. The numerical results provide good evidence that EAS1 could be seen as superior to the well established Adams methods for the numerical solution of mildly stiff initial value problems. [ABSTRACT FROM AUTHOR]

Abstract  In this paper the problem of finding the shape of distributed piezoelectric modal sensors/actuators for plates with arbitrary boundary conditions is treated by an optimization approach. A binary function is used to model the design variable: the polarization profile of the piezoelectric layers. Contrary to what it could be expected, it is analytically proved that it is possible to find manufacturable polarization profiles taking on only two values, i.e. either positive or negative polarization. Several numerical examples are shown to corroborate that such topologies isolate particular vibration modes in the frequency domain. [ABSTRACT FROM AUTHOR]

The paper investigates model reduction techniques that are based on a nonlocal quasi-continuum-like approach. These techniques reduce a large optimization problem to either a system of nonlinear equations or another optimization problem that are expressed in a smaller number of degrees of freedom. The reduction is based on the observation that many of the components of the solution of the original optimization problem are well approximated by certain interpolation operators with respect to a restricted set of representative components. Under certain assumptions, the “optimize and interpolate” and the “interpolate and optimize” approaches result in a regular nonlinear equation and an optimization problem whose solutions are close to the solution of the original problem, respectively. The validity of these assumptions is investigated by using examples from potential-based and electronic structure-based calculations in Materials Science models. A methodology is presented for using quasi-continuum-like model reduction for real-space DFT computations in the absence of periodic boundary conditions. The methodology is illustrated using a basic Thomas–Fermi–Dirac case study. [ABSTRACT FROM AUTHOR]

In this paper a method is developed for solving hyperbolic initial boundary value problems in one space dimension using domain decomposition, which can be extended to problems in several space dimensions. We minimize a functional which is the sum of squares of the L² norms of the residuals and a term which is the sum of the squares of the L² norms of the jumps in the function across interdomain boundaries. To make the problem well posed the interdomain boundaries are made to move back and forth at alternate time steps with sufficiently high speed. We construct parallel preconditioners and obtain error estimates for the method. The Schwarz waveform relaxation method is often employed to solve hyperbolic problems using domain decomposition but this technique faces difficulties if the system becomes characteristic at the inter-element boundaries. By making the inter-element boundaries move faster than the fastestwave speed associated with the hyperbolic system we are able to overcome this problem. [ABSTRACT FROM AUTHOR]

This paper introduces a new spontaneous potential log model for the case in which formation resistivity is not piecewise constant. The spontaneous potential satisfies an elliptic boundary value problem with jump conditions on the interfaces. It has been shown that the elliptic interface problem has a unique weak solution. Furthermore, a jump condition capturing finite difference scheme is proposed and applied to solve such elliptic problems. Numerical results show validity and effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

The Balancing Domain Decomposition by Constraints (BDDC) method is the most advanced method from the Balancing family of iterative substructuring methods for the solution of large systems of linear algebraic equations arising from discretization of elliptic boundary value problems. In the case of many substructures, solving the coarse problem exactly becomes a bottleneck. Since the coarse problem in BDDC has the same structure as the original problem, it is straightforward to apply the BDDC method recursively to solve the coarse problem only approximately. In this paper, we formulate a new family of abstract Multispace BDDC methods and give condition number bounds from the abstract additive Schwarz preconditioning theory. The Multilevel BDDC is then treated as a special case of the Multispace BDDC and abstract multilevel condition number bounds are given. The abstract bounds yield polylogarithmic condition number bounds for an arbitrary fixed number of levels and scalar elliptic problems discretized by finite elements in two and three spatial dimensions. Numerical experiments confirm the theory. [ABSTRACT FROM AUTHOR]

In this paper, a new alternating group explicit-implicit (nAGEI) scheme for dispersive equations with a periodic boundary condition is derived. This new unconditionally stable scheme has a fourth-order truncation error in space and a convergence ratio faster than some known alternating methods such as ASEI and AGE. Comparison in accuracy with the AGEI and AGE methods is presented in the numerical experiment. [ABSTRACT FROM AUTHOR]

The paper deals with the numerical solution of a generalized spectral boundary value problem for an elliptic operator with degenerating coefficients. We suggest an approximate method based on the multiplicative separation of the singularity, whereby the eigenfunctions are approximated by piecewise linear functions multiplied by a weight specially chosen depending on the order of degeneration of the coefficients. For this method, we obtain error estimates justifying its optimality. [ABSTRACT FROM AUTHOR]

The paper deals with the numerical implementation of local integral equations for solution of boundary value problems and interior computations of displacements and their gradients in functionally graded elastic solids. Two kinds of meshless approximations and one element based approximation are employed in various formulations. The numerical stability, accuracy, convergence of accuracy and cost efficiency are investigated in numerous test examples with exact benchmark solutions. [ABSTRACT FROM AUTHOR]

This paper presents a numerical method for solving the two-dimensional problem of a polygonal linear viscoelastic domain containing an arbitrary number of non-overlapping circular holes of arbitrary sizes. The solution of the problem is based on the use of the correspondence principle. The governing equation for the problem in the Laplace domain is a complex hypersingular boundary integral equation written in terms of the unknown transformed displacements on the boundaries of the holes and the exterior boundaries of the finite body. No specific physical model is involved in the governing equation, which means that the method is capable of handling a variety of viscoelastic models. A truncated complex Fourier series with coefficients dependent on the transform parameter is used to approximate the unknown transformed displacements on the boundaries of the holes. A truncated complex series of Chebyshev polynomials with coefficients dependent on the transform parameter is used to approximate the unknown transformed displacements on the straight boundaries of the finite body. A system of linear algebraic equations is formed using the overspecification method. The viscoelastic stresses and displacements are calculated through the viscoelastic analogs of the Kolosov–Muskhelishvili potentials, and an analytical inverse Laplace transform is used to provide the time domain solution. Using the concept of representative volume, the effective viscoelastic properties of an equivalent homogeneous material are then found directly from the corresponding constitutive equations for the average field values. Several examples are given to demonstrate the accuracy of the method. The results for the stresses and displacements are compared with the numerical solutions obtained by commercial finite element software (ANSYS). The results for the effective properties are compared with those obtained with the self-consistent and Mori–Tanaka schemes. [ABSTRACT FROM AUTHOR]

In the paper, we study the problem on the number of zeros of eigenfunctions of the fourth-order boundary-value problem with spectral and physical parameters in the boundary conditions. We show that the number of zeros of the eigenfunctions corresponding to eigenvalues of positive type behaves in a usual way (it is equal to the serial number of an eigenvalue increased by 1), but, however, the number of zeros of the eigenfunction corresponding to an eigenvalue of negative type can be arbitrary. In the case of a sufficient smoothness of coefficients of the differential expression, we write the asymptotics in the physical parameter for such a number. [ABSTRACT FROM AUTHOR]

Abstract  The fixed gravimetric boundary-value problem (FGBVP) represents an exterior oblique derivative problem for the Laplace equation. Terrestrial gravimetric measurements located by precise satellite positioning yield oblique derivative boundary conditions in the form of surface gravity disturbances. In this paper, we discuss the boundary element method (BEM) applied to the linearized FGBVP. In spite of previous BEM approaches in geodesy, we use the so-called direct BEM formulation, where a weak formulation is derived through the method of weighted residuals. The collocation technique with linear basis functions is applied for deriving the linear system of equations from the arising boundary integral equations. The nonstationary iterative biconjugate gradient stabilized method is used to solve the large-scale linear system of equations. The standard MPI (message passing interface) subroutines are implemented in order to perform parallel computations. The proposed approach gives a numerical solution at collocation points directly on the Earth’s surface (on a fixed boundary). Numerical experiments deal with (i) global gravity field modelling using synthetic data (surface gravity disturbances generated from a global geopotential model (GGM)) (ii) local gravity field modelling in Slovakia using observed gravity data. In order to extend computations, the memory requirements are reduced using elimination of the far-zone effects by incorporating GGM or a coarse global numerical solution obtained by BEM. Statistical characteristics of residuals between numerical solutions and GGM confirm the reliability of the approach and indicate accuracy of numerical solutions for the global models. A local refinement in Slovakia results in a local (national) quasigeoid model, which when compared with GPS-levelling data, does not make a large improvement on existing remove-restore-based models. [ABSTRACT FROM AUTHOR]

Solving an inverse Sturm-Liouville problem requires a mathematical process to determine unknown function in the Sturm-Liouville operator from given data in addition to the boundary values. In this paper, we identify a Sturm-Liouville potential function by using the data of one eigenfunction and its corresponding eigenvalue, and identify a spatial-dependent unknown function of a Sturm-Liouville differential operator. The method we employ is to transform the inverse Sturm-Liouville problem into a parameter identification problem of a heat conduction equation. Then a Lie-group estimation method is developed to estimate the coefficients in a system of ordinary differential equations discretized from the heat conduction equation. Numerical tests confirm the accuracy and efficiency of present approach. Definite and random disturbances are also considered when comparing the present method with that by using a technique of numerical differentiation. [ABSTRACT FROM AUTHOR]

This paper deals with the existence of solutions of the periodic boundary value problem of the impulsive Duffing equations: x″(t) + αx′(t) + βx(t) = f(t, x(t), x(α1(t)), …, x(αn(t))), a.e. t ϵ [0, T], Δx(tk) = Iκ(x(tκ), x′(tκ)), k = 1,…, m, Δx′(tκ) = Jκ(x(tκ), x′(tκ)), k = 1,…, m, x(i)(0) = x(i)(T), i = 0, 1. Sufficient conditions are established for the existence of at least one solution of above-mentioned boundary value problem. Our method is based upon Schaeffer's fixed-point theorem. Examples are presented to illustrate the efficiency of the obtained results. [ABSTRACT FROM AUTHOR]

In this paper, We study the existence and uniqueness of solutions for boundary value problems to the singular one-dimension p-Laplacian by using mixed monotone method. Our results improve several recent results established in the literature. [ABSTRACT FROM AUTHOR]

The study in this paper is a numerical integration of second-order three-point boundary value problems under two imposed nonlocal boundary conditions at t = t0, t = ξ, and t = t1 in a general setting, where t0 < ξ < t1. We construct a two-stage Lie-group shooting method for finding unknown initial conditions, which are obtained through an iterative solution of derived algebraic equations in terms of a weighting factor r ϵ (0, 1). The best r is selected by matching the target with a minimal discrepancy. Numerical examples are examined to confirm that the new approach has high efficiency and accuracy with a fast speed of convergence. Even for multiple solutions, the present method is also effective to find them. [ABSTRACT FROM AUTHOR]