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This paper reports a multiscale asymptotic analysis and computation for predicting heat transfer performance of periodic porous materials with radiation boundary condition. In these porous materials thermal radiation effect at micro-scale have an important impact on the macroscopic temperature field, which is our particular interest in this study. The multiscale asymptotic expansions for computing temperature field of the problem are constructed, and associated explicit convergence rates are obtained on some regularity hypothesis. Finally, the corresponding finite element algorithms based on the multiscale method are brought forward and some numerical results are given in details. The numerical tests indicate that the developed method is feasible and valid for predicting the heat transfer performance of periodic porous materials, and support the approximate convergence results proposed in this paper. [ABSTRACT FROM AUTHOR]

This paper presents a semigroup approach for inverse source problems for the abstract heat equation, when the measured output data is given in subject to the integral overspecification over the spatial domain. The existence of a solution to the inverse source problem is shown in appropriate function spaces and a representation formula for the solution is proposed. Such representation permits the derivation of sufficient conditions for the uniqueness of the solution. Also an approximation method based on the optimal homotopy analysis method (OHAM) is designed, and the error estimates are discussed using graphical analysis. Moreover, we conjecture that our approach can be applied for the determination of a control parameter in an inverse problem with integral overspecialization data. The proposed algorithm is examined through various numerical examples for the reconstruction of continuous sources and the determination of a control parameter in parabolic equations. The accuracy and stability of the method are discussed and compared with several finite-difference techniques. Computational results show efficiency and high accuracy of the proposed algorithm. [ABSTRACT FROM AUTHOR]

This paper proposes a full grid interval collocation method (FGICM) and a sparse grid interval collocation method (SGICM) to solve the uncertain heat convection-diffusion problem with interval input parameters in material properties, applied loads and boundary conditions. The Legendre polynomial series is adopted to approximate the functional dependency of temperature response with respect to the interval parameters. In the process of calculating the expansion coefficients, FGICM evaluates the deterministic solutions directly on the full tensor product grids, while the Smolyak sparse grids are reconstructed in SGICM to avoid the curse of dimensionality. The eventual lower and upper bounds of temperature responses are easily predicted based on the continuously-differentiable property of the approximate function. Comparing results with traditional Monte Carlo simulations and perturbation method, the numerical example evidences the remarkable accuracy and effectiveness of the proposed methods for interval temperature field prediction in engineering. [ABSTRACT FROM AUTHOR]

A soft computing approach based on artificial neural network (ANN) and optimization is presented for the numerical solution of the unsteady one-dimensional advection–dispersion equation (ADE) arising in contaminant transport through porous media. A length factor ANN method, based on automatic satisfaction of arbitrary boundary conditions (BCs) was chosen for the numerical solution of ADE. The strength of ANN is exploited to construct a trial approximate solution (TAS) for ADE in a way that it satisfies the initial or BCs exactly. An unsupervised error is constructed in approximating the solution of ADE which is minimized by training ANN using gradient descent algorithm (GDA). Two challenging test problems of ADE are considered in this paper, in which, the first problem has steep boundary layers near x = 1 and many numerical methods create non-physical oscillation near steep boundaries. Also for the second problem many numerical schemes suffer from computational noise and instability issues. The proposed method is advantageous as it does not require temporal discretization for the solution of the ADEs as well as it does not suffer from numerical instability. The reliability and effectiveness of the presented algorithm is investigated by sufficient large number of independent runs and comparison of results with other existing numerical methods. The results show that the present method removes the difficulties arising in the solution of the ADEs and provides solution with good accuracy. [ABSTRACT FROM AUTHOR]

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]

Abstract: There has been a recent increase in interest in the use of distributed order differential equations (particularly in the case where the derivatives are given in the Caputo sense) to model various phenomena. Recent papers have provided insights into the numerical approximation of the solution, and some results on existence and uniqueness have been proved. In each case, the representation of the solution depends, among other parameters, on Caputo-type initial conditions. In this paper we discuss the existence and uniqueness of solutions and we propose a numerical method for their approximation in the case where the initial conditions are not known and, instead, some Caputo-type conditions are given away from the origin. [Copyright &y& Elsevier]

Abstract: The objective of this paper is to present a numerical study of a class of boundary value problems of singularly perturbed differential difference equations (SPDDE) which arise in computational neuroscience in particular in the modeling of neuronal variability. The mathematical modeling of the determination of the expected time for the generation of action potential in the nerve cells by random synaptic inputs in dendrites includes a general boundary-value problem for singularly perturbed differential difference equation with shifts. The problem considered in this paper exhibit turning point behavior which add to the complexity in the construction of numerical approximation to the solution of the problem as well as in obtaining theoretical estimates on the solution. Exponentially fitted finite difference scheme based on Il’in-Allen-Southwell fitting is used on a specially designed mesh. Some numerical examples are given to validate convergence and computational efficiency of the proposed numerical scheme. Effect of the shifts on the layer structure is illuminated for the considered examples. [Copyright &y& Elsevier]

Abstract: In this paper, we consider a backward heat problem that appears in many applications. This problem is ill-posed. The solution of the problem as the solution exhibits unstable dependence on the given data functions. Using a new regularization method, we regularize the problem and get some new error estimates. Some numerical tests illustrate that the proposed method is feasible and effective. This work is a generalization of many recent papers, including the earlier paper [A new regularized method for two dimensional nonhomogeneous backward heat problem, Appl. Math. Comput. 215(3) (2009) 873–880] and some other authors such as Chu-Li Fu et al. , Campbell et al. . [Copyright &y& Elsevier]

In thermal analysis of concrete with cooling pipes by conventional FEM (finite element method), extremely refined conforming meshes have to be generated around the pipe axis to obtain precise temperature field. It makes the meshing process demanding and cumbersome, especially in 3D cases. To avoid the difficulty of refining meshes in conventional FEM, an extended finite element method (XFEM) is proposed in this paper with which thermal analysis with pipes can be performed on coarse meshes instead of refined conforming meshes. The enrichment shape function is constructed in the proposed XFEM based on analytical solution and the Dirichlet boundary condition of the cooling pipe is numerically approximated by in-element Robin boundary condition. Thermal analysis of a concrete slab with a cooling pipe in the center is performed by XFEM with coarse meshes and by conventional FEM with refined conforming meshes respectively. The results demonstrate that the temperature field obtained by XFEM has a comparable accuracy level with the conventional FEM. The accuracy and reliability are further validated with three other numerical cases including the varying cooling boundary case, the internal heat source case and the multiple-pipe case. [ABSTRACT FROM AUTHOR]

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: In this paper, we use quartic B-spline to construct an approximating function to agree with the given integral values of a univariate real-valued function over the same intervals. It is called integro quartic spline interpolation. Our interpolation method is new and easy to implement. Moreover, it can work successfully even without any boundary conditions. The interpolation errors are studied. The super convergence (sixth order and fourth order, respectively) in approximating function values and second-order derivative values at the knots is proved. Numerical examples illustrate that our method is very effective and our integro-interpolating quartic spline has higher approximation ability than others. [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: In this paper, we study the eventual periodicity of the initial boundary value problem (IBVP) for Korteweg-de Vries equation posed on a bounded domain. We show that if the boundary forcing is periodic of period τ, then the solution u of the IBVP at each spatial point becomes eventually time-periodic of period τ. In order to exhibit eventual periodicity, we approximate the solution of the IBVP using the Adomian decomposition method. We compare our work with the approximate solution of IBVP obtained by the homotopy perturbation method and present numerical experiments using Mathematica. [Copyright &y& Elsevier]

Abstract: In this paper the laminar free convection in a hot outer square enclosure containing a cold inner sinusoidal circular cylinder at different amplitudes and number of undulations and different angle, is investigated by means of thermal lattice Boltzmann method (TLBM). Both square enclosure and cylinder are maintained at constant temperature and air filled the enclosure. Boundary conditions for fluid flow and temperature fields at curved boundaries are second order schemes. The buoyancy effect is modeled by applying the Boussinesq approximation to the density distribution function. The computed results show that by increasing amplitude or number of undulations or changing the angle might change the heat transfer coefficient and has profound influence on the temperature and fluid field. [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, quadratic B-spline functions are used for solution of 2-D elastic problems. Because B-spline functions are directly used as basis function, there is no need to use meshes and nodes in function approximation. In order to improve the computational efficiency, different scales are used for sub-domains of entire problem domain in function approximation. The modified variational form and Lagrange multipliers method are used for coupling of different scale in function approximation. Compared with meshless methods and other wavelet based methods, this multi-scale B-spline-based method is simple and easy to work with for numerical analysis. Furthermore, the computational efficiency of the multi-scale method is much higher than that of single scale approach. The numerical examples of 2-D elastic problems indicate that the present method is effective and stable for solving complicated problems. [Copyright &y& Elsevier]

Abstract: In this paper we introduce a new boundary condition that can be used when reconstructing an image from observed blurred and noisy data. Our approach uses information from the observed image to enforce boundary conditions that continue image features such as edges and texture across the boundary. Because of its similarity to methods used in texture synthesis, we call our approach synthetic boundary conditions. We provide an efficient algorithm for implementing the new boundary condition, and provide a linear algebraic framework for the approach that puts it in the context of more classical and well known image boundary conditions, including zero, periodic, reflective, and anti-reflective. Extensive numerical experiments show that our new synthetic boundary conditions provide a more accurate approximation of the true image scene outside the image boundary, and thus allow for better reconstructions of the unknown, true image scene. [Copyright &y& Elsevier]

Abstract: The paper focuses on the numerical study of electromagnetic scattering from two-dimensional (2D) large partly covered cavities, which is described by the Helmholtz equation with a nonlocal boundary condition on the aperture. The classical five-point finite difference method is applied for the discretization of the Helmholtz equation and a linear approximation is used for the nonlocal boundary condition. We prove the existence and uniqueness of the numerical solution when the medium in the cavity is -direction layered or the number of the mesh points on the aperture is large enough. The fast algorithm proposed in Bao and Sun (2005) for open cavity models is extended to solving the partly covered cavity problem with (vertically) layered media. A preconditioned Krylov subspace method is proposed to solve the partly covered cavity problem with a general medium, in which a layered medium model is used as a preconditioner of the general model. Numerical results for several types of partly covered cavities with different wave numbers are reported and compared with those by ILU-type preconditioning algorithms. Our numerical experiments show that the proposed preconditioning algorithm is more efficient for partly covered cavity problems, particularly with large wave numbers. [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: Lotfi et al. [Solving a full fuzzy linear programming using lexicography method and fuzzy approximate solution, Appl. Math. Modell. 33 (2009) 3151–3156] pointed out that there is no method in literature for finding the fuzzy optimal solution of fully fuzzy linear programming (FFLP) problems and proposed a new method to find the fuzzy optimal solution of FFLP problems with equality constraints. In this paper, a new method is proposed to find the fuzzy optimal solution of same type of fuzzy linear programming problems. It is easy to apply the proposed method compare to the existing method for solving the FFLP problems with equality constraints occurring in real life situations. To illustrate the proposed method numerical examples are solved and the obtained results are discussed. [ABSTRACT FROM AUTHOR]

Abstract: This paper presents a meshless method, which replaces the inhomogeneous biharmonic equation by two Poisson equations in terms of an intermediate function. The solution of the Poisson equation with the intermediate function as the right-hand term may be written as a sum of a particular solution and a homogeneous solution of a Laplace equation. The intermediate function is approximated by a series of radial basis functions. Then the particular solution is obtained via employing Kansa’s method, while the homogeneous solution is approximated by using the boundary radial point interpolation method by means of boundary integral equations. Besides, the proposed meshless method, in conjunction with the analog equation method, is further developed for solving generalized biharmonic-type problems. Some numerical tests illustrate the efficiency of the method proposed. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we develop a set of numerical techniques for the simulation of the profile evolution of a valley glacier in the framework of isothermal shallow ice approximation models. The different mathematical formulations are given in terms of a highly nonlinear parabolic equation. A first nonlinearity comes from the free boundary problem associated with the unknown basal extension of the glacier region. This feature is treated using a fixed domain complementarity formulation which is solved numerically by a duality method. The nonlinear diffusive term is explicitly treated in the time marching scheme. A convection dominated problem arises, so a characteristic scheme is proposed for the time discretization, while piecewise linear finite elements are used for the spatial discretization. The presence of infinite slopes in polar regimes motivates an alternative formulation based on a prescribed flux boundary condition at the head of the glacier instead a homogeneous Dirichlet one. Finally, several numerical examples illustrate the performance of the proposed methods. [ABSTRACT FROM AUTHOR]

Abstract: This paper investigates the forced Duffing equation with integral boundary conditions. Its approximate solution is developed by combining the homotopy perturbation method (HPM) and the reproducing kernel Hilbert space method (RKHSM). HPM is based on the use of the traditional perturbation method and the homotopy technique. The HPM can reduce nonlinear problems to some linear problems and generate a rapid convergent series solution in most cases. RKHSM is also an analytical technique, which can solve powerfully linear boundary value problems. Therefore, the forced Duffing equation with integral boundary conditions can be solved using advantages of these two methods. Two numerical examples are presented to illustrate the strength of the method. [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 this paper, a new numerical approach, based on mortar discretization of interface fields, is proposed to solve lubricated contact problems between deformable solid bodies. The Reynolds approximation to the thin film fluid equations is used to describe the fluid behavior on the interface, while the equations of small strain elasticity (to which finite elements are applied) are assumed to govern the solid body behavior. The result of this approach is a monolithic scheme in which the fluid and solid mechanics are solved simultaneously, without the need for iterative or staggered approaches between the fluid and solid phases. Key features of the approach are that the fluid film thickness, directly related to the deformation of the solid phase, is computed from a least squares projection based on dual basis functions (as pioneered by Wohlmuth [B.I. Wohlmuth, Discretization Methods and Iterative Solvers Based on Domain Decomposition, Springer-Verlag, Heidelberg, 2001]). The free boundary problem for the fluid phase, corresponding to the boundary between lubricated and cavitated regions, is regularized with a penalty method and thus resolved as part of the solution. Several numerical examples are given to demonstrate the approach. [Copyright &y& Elsevier]

Abstract: The equations considered in this paper are mixed-type functional equations (sometimes known as forward–backward differential equations) that take the form We consider basic existence and uniqueness properties when and we seek solutions that satisfy for prescribed functions absolutely continuous, respectively, on . With arbitrary boundary conditions specified in this way, the problem turns out to be ill-posed and so existence and uniqueness questions have an important role to play in developing numerical schemes. We consider, with , numerical approximations of a solution when it exists. The numerical methods that we consider are linear -methods and we investigate computationally their effectiveness through some illustrative examples. [Copyright &y& Elsevier]

Abstract: Based on the geometric grid information as geometric coordinates, an algebraic multigrid (AMG) method with the interpolation reproducing the rigid body modes (namely the kernel elements of semi-definite operator arising from linear elasticity) is constructed, and such method is applied to the linear elasticity problems with a traction free boundary condition and crystal problems with free boundary conditions as well. The results of various numerical experiments in two dimensions are presented. It is shown from the numerical results that the constructed AMG method is robust and efficient for such semi-definite problems, and the convergence is uniformly bounded away from one independent of the problem size. Furthermore, the AMG method proposed in this paper has better convergence rate than the commonly used AMG methods. Simultaneously, an AMG method that can preserve the quotient space, which means that if the exact solution of original problem belongs to the quotient space of discrete operator considered, then the numerical solution of AMG method is convergent in the same quotient space, is obtained using the technique of orthogonal decomposition. [Copyright &y& Elsevier]