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Abstract: In this note, we point out an error in the recently published article [R.K. Mohanty, M.K. Jain, D. Dhall, A cubic spline approximation and application of TAGE iterative method for the solution of two-point boundary value problems with forcing function in integral form, Appl. Math. Model. 35 (2011) 3036–3047] and then correct it. [Copyright &y& Elsevier]

We present a simple and effective Chebyshev polynomial scheme (CPS) combined with the method of fundamental solutions (MFS) and the equilibrated collocation Trefftz method for the numerical solutions of inhomogeneous elliptic partial differential equations (PDEs). In this paper, CPS is applied in a two-step approach. First, Chebyshev polynomials are used to approximate a particular solution of a PDE. Chebyshev nodes which are the roots of Chebyshev polynomials are used in the polynomial interpolation due to its spectral convergence. Then the resulting homogeneous equation is solved by boundary type methods including the MFS and the equilibrated collocation Trefftz method. Numerical results for problems on various irregular domains show that our proposed scheme is highly accurate and efficient. [ABSTRACT FROM AUTHOR]

The aim of this paper is to present an efficient and reliable treatment of the homotopy perturbation method (HPM) for two dimensional time-fractional wave equation (TFWE) with the boundary conditions. The fractional derivative is described in the Caputo sense. The initial approximation can be determined by imposing the boundary conditions. The method provides approximate solutions in the form of convergent series with easily computable components. The obtained results shown that the technique introduced here is efficient and easy to implement. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we develop numerical approximations of the wave equation in mixed form supplemented with non-reflecting boundary conditions (NRBCs) of Sommerfeld-type on artificial boundaries for truncated domains. We consider three different variational forms for this problem, depending on the functional space for the solution, in particular, in what refers to the regularity required on artificial boundaries. Then, stabilized finite element methods that can mimic these three functional settings are described. Stability and convergence analyses of these stabilized formulations including the NRBC are presented. Additionally, numerical convergence test are evaluated for various polynomial interpolations, stabilization methods and variational forms. Finally, several benchmark problems are solved to determine the accuracy of these methods in 2D and 3D. [Copyright &y& Elsevier]

Abstract: This paper deals with the construction and characterization of discrete variational PDE splines. To formulate the problem, we need an evolutionary PDE equation, certain boundary conditions and a set of points to approximate. We show the existence and the uniqueness of the solution of such a problem and we establish a convergence result of a discrete variational evolutionary PDE spline to a given function. We present several numerical and graphical examples of construction and approximation of surfaces in order to prove the validity of the presented method. [Copyright &y& Elsevier]

Abstract: This paper presents a numerical method for solving a class of fractional variational problems (FVPs) with multiple dependent variables, multi order fractional derivatives and a group of boundary conditions. The fractional derivative in the problem is in the Caputo sense. In the presented method, the given optimization problem reduces to a system of algebraic equations using polynomial basis functions. An approximate solution for the FVP is achieved by solving the system. The choice of polynomial basis functions provides the method with such a flexibility that initial and boundary conditions can be easily imposed. We extensively discuss the convergence of the method and finally present illustrative examples to demonstrate validity and applicability of the new technique. [Copyright &y& Elsevier]

Abstract: In this paper, a least-squares finite element method for second-order two-point boundary value problems is considered. The problem is recast as a first-order system. Standard and improved optimal error estimates in maximum-norms are established. Superconvergence estimates at interelement, Lobatto, and Gauss points are developed. Numerical experiments are given to illustrate theoretical results. [Copyright &y& Elsevier]

Abstract: The Wave Based Method is an efficient alternative prediction technique based on a Trefftz approach which, as compared to element-based methods, can be applied in a wider frequency range at a reasonable cost. This paper discusses the use of this method for the particular case of 2D acoustic problems where singularities are present in the corners of the acoustic domain. The conventional set of expansion functions used to approximate the dynamic pressure field is extended with special purpose functions that incorporate the singular behaviour in the region around the singular corners. This enrichment comes with a beneficial convergence rate while only a small increase of computational effort is incurred. This is illustrated for various problem settings, including different combinations of both homogeneous and non-homogeneous boundary conditions and bounded as well as unbounded problems are considered. [Copyright &y& Elsevier]

Abstract: In this paper, we study the initial-boundary value problem of the usual Rosenau-RLW equation by finite difference method. We design a conservative numerical scheme which preserves the original conservative properties for the equation. The scheme is three-level and linear-implicit. The unique solvability of numerical solutions has been shown. Priori estimate and second order convergence of the finite difference approximate solutions are discussed by discrete energy method. Numerical results demonstrate that the scheme is efficient and accurate. [Copyright &y& Elsevier]

Abstract: A Galerkin boundary node method (GBNM), for boundary only analysis of partial differential equations, is discussed in this paper. The GBNM combines an equivalent variational form of a boundary integral equation with the moving least-squares (MLS) approximations for generating the trial and test functions of the variational formulation. In this approach, only a nodal data structure on the boundary of a domain is required, and boundary conditions can be implemented directly and easily despite of the fact that the MLS shape functions lack the delta function property. Formulations of the GBNM using boundary singular integral equations of the second kind for potential problems are developed. The theoretical analysis and numerical results indicate that it is an efficient and accurate numerical method. [Copyright &y& Elsevier]

Abstract: Generating finite element discretizations with direct interface parameterizations constitutes a considerable computational expense in case of complex interface geometries. The paper at hand introduces a B-spline finite element method, which circumvents parameterization of interfaces and offers fast and easy meshing irrespective of the geometric complexity involved. Its core idea is the adaptive approximation of discontinuities by hierarchical grid refinement, which adds several levels of local basis functions in the close vicinity of interfaces, but unfitted to their exact location, so that a simple regular grid of knot span elements can be maintained. Numerical experiments show that an hp-refinement strategy, which simultaneously increases the polynomial degree of the B-spline basis and the levels of refinement around interfaces, achieves exponential rates of convergence despite the presence of discontinuities. It is also demonstrated that the hierarchical B-spline FEM can be used to transfer the recently introduced Finite Cell concept to geometrically nonlinear problems. Its computational performance, imposition of unfitted boundary conditions and fast hierarchical grid generation are illustrated for a set of benchmark problems in one, two and three dimensions, and the advantages of the regular grid approach for complex geometries are demonstrated by the geometrically nonlinear simulation of a voxel based foam composite. [Copyright &y& Elsevier]

Abstract: In this paper, we present the homotopy analysis method (shortly HAM) for obtaining the numerical solutions of higher-order fractional integro-differential equations with boundary conditions. The series solution is developed and the recurrence relations are given explicitly. The initial approximation can be freely chosen with possible unknown constants which can be determined by imposing the boundary conditions. The comparison of the results obtained by the HAM with the exact solutions is made, the results reveal that the HAM is very effective and simple. The HAM contains the auxiliary parameter , which provides us with a simple way to adjust and control the convergence region of series solution. [Copyright &y& Elsevier]

Abstract: A novel approach is presented in this paper for approximate solution of parameterized unperturbed and singularly perturbed two-point boundary value problems. The problem is first separated into a simultaneous system regarding the unknown function and the parameter, and then a methodology based on the powerful homotopy analysis technique is proposed for the approximate analytic series solutions, whose convergence is guaranteed by optimally chosen convergence control parameters via square residual error. A convergence theorem is also provided. Several nonlinear problems are treated to validate the applicability, efficiency and accuracy of the method. Vicinity of the boundary layer is shown to be adequately treated and satisfactorily resolved by the method. Advantages of the method over the recently proposed conventional finite-difference or Runga–Kutta methods are also discussed. [Copyright &y& Elsevier]

Abstract: In this paper, we construct the exact solution for fluid motion caused by the uniform expansion of a cylindrical or spherical piston into still air. Following Lighthill , we introduce velocity potential into the analysis and seek a similarity form of the solution. We find both numerical and analytic solutions of the second order nonlinear differential equation, with the boundary conditions at the shock and at the piston. The results obtained from the analytic solutions justify numerical solution and the approximate solution of Lighthill . We find that although the approximate solution of Lighthill gives remarkably good numerical results, the analytic form of that solution is not mathematically satisfactory. We also find that in case of spherical piston motion Lighthill’s solution differs significantly from that of our analytic and numerical solutions. We use Pade′ approximation to extend the radius of convergence of the series solution. We also carry out some local analysis at the boundary to obtain some singular solutions. [Copyright &y& Elsevier]

Abstract: In this paper, using the quasilinearization method coupled with the method of upper and lower solutions, we study a class of second-order nonlinear boundary value problems with nonlocal boundary conditions. We establish some sufficient conditions under which corresponding monotone sequences converge uniformly and quadratically to the unique solution of the problem. An example is also included to illustrate the main result. [Copyright &y& Elsevier]

Abstract: In this paper, we introduce a new one-step iterative process to approximate common fixed points of two multivalued nonexpansive mappings in a real uniformly convex Banach space. We establish weak and strong convergence theorems for the proposed process under some basic boundary conditions. [ABSTRACT FROM AUTHOR]

Abstract: This paper addresses the finite element method for the two-dimensional time-dependent Schrödinger equation on an infinite strip by using artificial boundary conditions. We first reduce the original problem into an initial-boundary value problem in a bounded domain by introducing a transparent boundary condition, then fully discretize this reduced problem by applying the Crank–Nicolson scheme in time and a bilinear or quadratic finite element approximation in space. This scheme, by a rigorous analysis, has been proved to be unconditionally stable and convergent, and its convergence order has also been obtained. Finally, two numerical examples are given to verify the accuracy of the scheme. [Copyright &y& Elsevier]

Abstract: In this paper, we investigate the superconvergence properties of the - version of the finite element method (FEM) for two-point boundary value problems. A postprocessing technique for the - finite element approximation is analyzed. The analysis shows that the postprocess improves the order of convergence. Furthermore, we obtain asymptotically exact a posteriori error estimators based on the postprocessing results. Numerical examples are included to illustrate the theoretical analysis. [Copyright &y& Elsevier]

Abstract: In the present paper, static and dynamical problems for linearly elastic shells in curvilinear coordinates are considered. Hierarchies of two-dimensional models for corresponding boundary and initial boundary value problems are constructed within the variational settings. The existence and uniqueness of solutions of the reduced problems are investigated in suitable spaces. Under the conditions of solvability of the original static or dynamical problem, convergence of the sequence of vector functions of three variables restored from the solutions of the constructed two-dimensional problems to the solution of the three-dimensional problem is proved and approximation error is estimated. [Copyright &y& Elsevier]