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The goal of this paper is to design a stabilizing feedback boundary control for a reaction–diffusion partial differential equation (PDE), where the boundary control is subject to a constant delay while the equation may be unstable without any control. For this system, which is equivalent to a parabolic equation coupled with a transport equation, a prediction-based control is explicitly computed by splitting the infinite-dimensional system into two parts: a finite-dimensional unstable part and a stable infinite-dimensional part. A finite-dimensional delayed controller is computed for the unstable part, and it is shown that this controller stabilizes the whole PDE. The proof is based on an explicit expression of the classical Artstein transformation combined with an adequately designed Lyapunov function. A numerical simulation illustrates the constructive feedback design method. [ABSTRACT FROM AUTHOR]

This paper is concerned with the global exponential stability of a $2\times 2$ linear hyperbolic system with a sampled-data boundary feedback control designed by means of the backstepping method for a nominal continuous input. We show that there exists a sufficiently small intersampling time (that encompasses both periodic and aperiodic sampling) for which the global exponential stability of the closed-loop system is guaranteed. In addition, we provide easily tractable sufficient stability conditions that can be used to find an upper bound of the maximum intersampling time. The results rely on the combination of the Lyapunov method and looped-functionals. The effectiveness of the proposed results is illustrated with a numerical example. [ABSTRACT FROM AUTHOR]

This paper provides an effective technique for removing the rigid body motions in the interior traction BVP (boundary value problem) of plane elasticity. In the technique, we assume three additional node support conditions. After substituting those conditions in the algebraic equations derived from discretization of BIE (boundary integral equation), we will obtain the algebraic equations where the number of the equations is larger than the number of unknowns. The mentioned algebraic equations can be solved by using the least square technique. Finally, the displacement vector can be evaluated. A stress component derived from the suggested technique is compared to the exact one. Excellent accuracy has been found in the suggested technique. [ABSTRACT FROM AUTHOR]

Abstract: Various processes in the natural sciences and engineering lead to the nonclassical parabolic initial boundary value problems which involve nonlocal integral terms over the spatial domain. The integral term may appear in the boundary conditions. It is the reason for which such problems gained much attention in recent years, not only in engineering but also in the mathematics community. In this paper the problem of solving the one-dimensional parabolic partial differential equation subject to given initial and nonlocal boundary conditions is considered. Several approaches for the numerical solution of this boundary value problem which have been considered in the literature, are reported. New finite difference techniques are proposed for the numerical solution of the one-dimensional heat equation subject to the specification of mass. Numerical examples are given at the end of this paper to compare the efficiency of the new techniques. Some specific applications in various engineering models are introduced. [Copyright &y& Elsevier]

Abstract: Sinc method is widely used for solving the boundary value problems because of the ease with which it may handle the presence of singularities or unbounded domains. In this paper, the domain decomposition method is used to deal with problems with interior layers and problems posed on irregular domains. With boundary reduction technique, the functional values on internal points can be eliminated. The scheme is based on a direct approach based on the corresponding compatibility matching conditions between sub-domains. Numerical experiments show that rational Sinc domain decomposition method based on the interpolation of the highest derivatives (RSIHD-DDM) is effective for treating problems with interior layers or boundary layers on irregular domains. [Copyright &y& Elsevier]

Abstract: The modified Craig–Sneyd (MCS) scheme is a promising splitting scheme of the ADI type for multi-dimensional pure diffusion equations having mixed spatial-derivative terms. In this paper we investigate the extension of the MCS scheme to two-dimensional convection–diffusion equations with a mixed derivative. Both necessary and sufficient conditions on the parameter θ of the scheme are derived concerning unconditional stability in the von Neumann sense. [Copyright &y& Elsevier]

Abstract: The regularized meshless method is a novel boundary-type meshless method but by now has largely been confined to homogeneous problems. In this paper, we apply the regularized meshless method to the nonhomogeneous problems in conjunction with the dual reciprocity technique in the evaluation of the particular solution. Numerical experiments of three benchmark nonhomogeneous problems demonstrate the accuracy and efficiency of the present strategy. [ABSTRACT FROM AUTHOR]

Abstract: The radial basis functions (RBFs) is an efficient tool in multivariate approximation, but it usually suffers from an ill-conditioned interpolation matrix when interpolation points are very dense or irregularly spaced. The RBFs with variable shape parameters can usually improve the interpolation matrix condition number. In this paper a new shape parameter variation scheme is implemented. Comparison studies with the constant shaped RBF on convergence and stability are made. Results show that under the same accuracy level, the interpolation matrix condition number by our scheme grows much slower than that of the constant shaped RBF interpolation matrix with increase in the number of interpolation points. As an application example, the dual reciprocity method equipped with the new RBF is combined with the boundary face method to solve boundary value problems governed by Poisson equations. Numerical results further demonstrate the robustness and better stability of the new RBF. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we present a method for the numerical solution of Love''s equation in a particular physical system. The resolution protocol is based on the Boubaker Polynomials Expansion Scheme (BPES), followed by array analyses. The obtained results are compared with some recently published ones. The accuracy and the asymptotic behaviors of the solutions are discussed. [Copyright &y& Elsevier]

Abstract: Based on the use of Green''s function, we propose in this paper a new approach for solving specific classes of inverse source identification problems. Effective numerical algorithms are developed to recover both the intensities and locations of unknown point sources from scattered boundary measurements. For numerical verification, several boundary value problems defined on both bounded and unbounded regions of regular shape are given. Due to the use of closed analytic form of Green''s function, the efficiency and accuracy of the proposed method can be guaranteed. [Copyright &y& Elsevier]

Abstract: Many scientific and engineering problems are described using Ordinary Differential Equations (ODEs), where the analytic solution is unknown. Much research has been done by the scientific community on developing numerical methods which can provide an approximate solution of the original ODE. In this work, two approaches have been considered based on BDF and Piecewise-linearized Methods. The approach based on BDF methods uses a Chord–Shamanskii iteration for computing the nonlinear system which is obtained when the BDF schema is used. Two approaches based on piecewise-linearized methods have also been considered. These approaches are based on a theorem proved in this paper which allows to compute the approximate solution at each time step by means of a block-oriented method based on diagonal Padé approximations. The difference between these implementations is in using or not using the scale and squaring technique. Five algorithms based on these approaches have been developed. MATLAB and Fortran versions of the above algorithms have been developed, comparing both precision and computational costs. BLAS and LAPACK libraries have been used in Fortran implementations. In order to compare in equality of conditions all implementations, algorithms with fixed step have been considered. Four of the five case studies analyzed come from biology and chemical kinetics stiff problems. Experimental results show the advantages of the proposed algorithms, especially when they are integrating stiff problems. [Copyright &y& Elsevier]

Abstract: In this paper, the variational iteration method is applied to solve fourth order boundary-value problems. Comparison with numerical solutions shows that the method is very effective and convenient. [Copyright &y& Elsevier]

Abstract: In this paper, a new modification of the homotopy analysis method (HAM) is presented for solving systems of second-order boundary-value problems (BVPs). The main advantage of the modified HAM (MHAM) is that one can avoid the uncontrollability problems of the nonzero endpoint conditions encountered in the standard HAM. Numerical comparisons show that the MHAM is more efficient than the standard HAM. [Copyright &y& Elsevier]

The strong γk-γcl H∞ stabilization problem is addressed in this paper. Based on the bounded real lemma, an improved sufficient condition on strong H∞, stabilization as well as its dual form is derived, which extends the ARE constraint in the former results into an inequality constraint. Two path-following algorithms are proposed to solve the associated optimization problems with BMI constraints, which lead to full order controllers. A numerical example is presented to demonstrate the effectiveness of the result. [ABSTRACT FROM AUTHOR]

This paper presents the steady-state and transient analysis of the fluid squeezed between two long parallel plates. The governing coupled partial differential equations have been discretized by element free Galerkin method and implemented using variational approach. Penalty and Lagrange multiplier techniques have been utilized to enforce the essential boundary conditions. Four point Gauss quadrature has been used to evaluate the viscous terms in the coefficient matrix whereas reduced integration scheme (i.e. one point Gauss quadrature) has been used to evaluate the penalty terms over two-dimensional domain (Ω). Cubicspline, exponential and rational weight functions have been used in the present work. The results obtained by EFG method are compared with those obtained by finite element and analytical methods. The effect of scaling and penalty parameters on EFG results has been discussed in detail. [Copyright &y& Elsevier]

Abstract: This paper reports a new boundary-integral-equation method (BIEM) for numerically solving biharmonic problems with Dirichlet boundary conditions. For the solution of these problems in convex polygons, it was found that the accuracy of the conventional BIEM is significantly reduced, and spurious oscillatory behaviour is often observed in the boundary solutions especially for areas near corners (Mai-Duy N, Tanner RI. An effective high order interpolation scheme in BIEM for biharmonic boundary value problems. Eng Anal Bound Elem 2005;29:210–23). In this study, a new treatment for these difficulties is proposed. The unknown functions in boundary integrals are approximated using a domain-type interpolation scheme rather than traditional boundary-type interpolation schemes. Two test problems are considered to validate the formulation and to demonstrate the attractiveness of the proposed method. [Copyright &y& Elsevier]

Abstract: New Runge–Kutta–Nyström methods especially designed for the numerical integration of perturbed oscillators are presented in this paper. They are capable of exactly integrating the harmonic or unperturbed oscillator. We construct an embedded 4(3) RKN pair that is based on the FSAL property. The new method is much more efficient than previously derived RKN methods for some subclasses of problems. [Copyright &y& Elsevier]

Abstract: It is shown in this paper that self-consistent boundary conditions for numerical methods based on blurred derivatives can be derived from a suitable change of variables of the fundamental blurred approximation of the differential equation, followed by application of Leibnitz theorem for differentiation of an integral. The simplest scheme obtained in this way resembles the weak Local Petrov-Galerkin approximation, although interpretation of the operators appearing in the final equations is quite different—as is the derivation itself. Subsequent transformation leads to integral equations similar to the starting point for boundary integral methods of solution. In this way, a number of well-known computational methods are shown to be derivable from adequate manipulation of the blurred derivative technique. However, other approximations, which are not derivable with standard methods can also be obtained, hinting at a greater generality of blurred derivatives. [Copyright &y& Elsevier]

In this paper, the natural boundary integral method and some related numerical methods for solving exterior boundary value problems are discussed. These different numerical methods are compared, and some numerical examples are given. Those methods based on the natural boundary reduction are very efficient and have many extraordinary advantages for exterior problems. Numerical results accord with theoretical analysis very well. [Copyright &y& Elsevier]

A common method for numerically solving wave problems in unbounded domains is based on truncating the infinite domain via an artificial boundary B, thus defining a finite computational domain, and using a special non-reflecting boundary condition (NRBC) on B. Low-order local NRBCs have been constructed and practiced since the 1970s. Exact non-local NRBCs were introduced in the 1980s. Only recently high-order local NRBCs have been devised. These NRBCs, despite being of an arbitrarily high-order, do not involve high derivatives owing to the use of specially defined auxiliary variables. This paper reviews the latter approach, explains its advantages compared to previous approaches, and discusses the different schemes which have been proposed in this context. [Copyright &y& Elsevier]