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This paper is devoted to the study of the existence of extremal solutions to a first-order initial value problem on an interval of an arbitrary time scale. We prove the existence of extremal solutions for problems satisfying Carathéodory's conditions. Moreover, they are approximated uniformly by a sequence of lower and upper solutions to this problem, respectively. We also can warrant the existence and approximation of extremal solutions for the problem by relaxing their continuity properties. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we develop discontinuous Galerkin (DG) methods based on non-polynomial approximation spaces for numerically solving time dependent hyperbolic and parabolic and steady state hyperbolic and elliptic partial differential equations (PDEs). The algorithm is based on approximation spaces consisting of non-polynomial elementary functions such as exponential functions, trigonometric functions, etc., with the objective of obtaining better approximations for specific types of PDEs and initial and boundary conditions. It is shown that L 2 stability and error estimates can be obtained when the approximation space is suitably selected. It is also shown with numerical examples that a careful selection of the approximation space to fit individual PDE and initial and boundary conditions often provides more accurate results than the DG methods based on the polynomial approximation spaces of the same order of accuracy. [Copyright &y& Elsevier]

Abstract: This paper presents the formulation and numerical analysis of circular cylindrical shells by the local adaptive differential quadrature method (LaDQM), which employs both localized interpolating basis functions and exterior grid points for boundary treatments. The governing equations of motion are formulated using the Goldenveizer–Novozhilov shell theory. Appropriate management of exterior grid points is presented to couple the discretized boundary conditions with the governing differential equations instead of using the interior points. The use of compactly supported interpolating basis functions leads to banded and well-conditioned matrices, and thus, enables large-scale computations. The treatment of boundary conditions with exterior grid points avoids spurious eigenvalues. Detailed formulations are presented for the treatment of various shell boundary conditions. Convergence and comparison studies against existing solutions in the literature are carried out to examine the efficiency and reliability of the present approach. It is found that accurate natural frequencies can be obtained by using a small number of grid points with exterior points to accommodate the boundary conditions. [Copyright &y& Elsevier]

Abstract: Non-polynomial spline is used for the numerical solutions of the sixth order linear special case boundary value problems. The method presented in this paper has also been proved to be second order convergent. Two examples are considered for the numerical illustrations of the method developed. The method is also compared with those developed by El-Gamel et al. [M. El-Gamel, J.R. Cannon, J. Latour, A.I. Zayed, Sinc-Galerkin method for solving linear sixth-order boundary-value problems, Math. Comput. 73 (247) (2003) 1325–1343] and Siddiqi and Twizell [S.S. Siddiqi, E.H. Twizell, Spline solutions of linear sixth-order boundary value problems, Int. J. Comput. Math. 60 (1996) 295–304], as well and is observed to be better. [Copyright &y& Elsevier]

In this paper, we establish the existence of an eigenvalue interval of solutions to the singular discrete boundary value problem ... where our nonlinearity may be singular in its dependent variable and is allowed to change sign. [ABSTRACT FROM AUTHOR]

In this paper a sextic spline is defined for interpolation at equally spaced knots along with the end conditions required to complete the definition of the spline. These conditions are in terms of given functional values at the knots and lead to uniform convergence of O(h7) throughout the interval of interpolation. The main objective of defining the end conditions for the sextic spline is to use the sextic spline not only for interpolation purposes, but also for the solution of the fifth-order boundary value problem, with the change consistent with the boundary value problem.†Dedicated to the memory of Dr. M. Rafique. [ABSTRACT FROM AUTHOR]

In this paper we establish some existence results for nonlocal multivalued boundary-value problems. Our approach is based on existence results for operator inclusions involving a suitable closed-valued multifunction; see [2, 3]. Some applications are given. [ABSTRACT FROM AUTHOR]

Abstract: Non-standard finite difference methods (NSFDMs), now-a-days, are playing an important role in solving the real life problems governed by ODEs and/or by PDEs. Many differential models of sciences and engineerings for which the existing methodologies do not give reliable results, these NSFDMs are solving them competitively. To this end, in this paper we consider, second order, linear, singularly perturbed differential difference equations. Using the second of the five non-standard modeling rules of Mickens [R.E. Mickens, Nonstandard Finite Difference Models of Differential Equations, World Scientific, Singapore, 1994], the new finite difference methods are obtained for the particular cases of these problems. This rule suggests us to replace the denominator function of the classical second order derivative with a positive function derived systematically in such a way that it captures most of the significant properties of the governing differential equation(s). Both theoretically and numerically, we show that these NSFDMs are ε-uniformly convergent. [Copyright &y& Elsevier]

Abstract: In this paper we study the efficiency and the reliability of an anisotropic a posteriori error estimator in the case of the Poisson problem supplied with mixed boundary conditions. The error estimator may be classified as a residual-based one, but its novelty is twofold: firstly, it employs anisotropic estimates of the interpolation error for linear triangular finite elements and, secondly, it makes use of the Zienkiewicz–Zhu recovery procedure to approximate the gradient of the exact solution. Finally, we describe the adaptive procedure used to obtain a numerical solution satisfying a given accuracy, and we include some numerical test cases to assess the robustness of the proposed numerical algorithm. [Copyright &y& Elsevier]

The multiadaptive continuous/discontinuous Galerkin methods mcG(q) and mdG(q) for the numerical solution of initial value problems for ordinary differential equations are based on piecewise polynomial approximation of degree q on partitions in time with time steps which may vary for different components of the computed solution. In this paper, we prove general order a priori error estimates for the mcG(q) and mdG(q) methods. To prove the error estimates, we represent the error in terms of a discrete dual solution and the residual of an interpolant of the exact solution. The estimates then follow from interpolation estimates, together with stability estimates for the discrete dual solution. [ABSTRACT FROM AUTHOR]

In this paper we consider the impedance boundary value problem for the Helmholtz equation in a half-plane with piecewise constant boundary data, a problem which models, for example, outdoor sound propagation over inhomogeneous flat terrain. To achieve good approximation at high frequencies with a relatively low number of degrees of freedom, we propose a novel Galerkin boundary element method, using a graded mesh with smaller elements adjacent to discontinuities in impedance and a special set of basis functions so that, on each element, the approximation space contains polynomials (of degree ν) multiplied by traces of plane waves on the boundary. We prove stability and convergence and show that the error in computing the total acoustic field is O(N-(ν+1) log1/2 N), where the number of degrees of freedom is proportional to N log N. This error estimate is independent of the wavenumber, and thus the number of degrees of freedom required to achieve a prescribed level of accuracy does not increase as the wavenumber tends to infinity. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we have applied restrictive Taylor approximation classical explicit finite difference method to the Burgers’ equation with a set of initial and boundary conditions to obtain its numerical solution. The stability region and truncation error of the new method are discussed. The accuracy of the proposed method is demonstrated by the two test problems. The numerical results are found in good agreement with the exact solutions. [Copyright &y& Elsevier]

Abstract: This paper presents a truly meshfree method referred to as radial point interpolation collocation method (RPICM) for solving partial differential equations. This method is different from the existing point interpolation method (PIM) that is based on the Galerkin weak-form. Because it is based on the collocation scheme no background cells are required for numerical integration. Radial basis functions are used in the work to create shape functions. A series of test examples were numerically analysed using the present method, including 1-D and 2-D partial differential equations, in order to test the accuracy and efficiency of the proposed schemes. Several aspects have been numerically investigated, including the choice of shape parameter c with can greatly affect the accuracy of the approximation; the enforcement of additional polynomial terms; and the application of the Hermite-type interpolation which makes use of the normal gradient on Neumann boundary for the solution of PDEs with Neumann boundary conditions. Particular emphasis was on an efficient scheme, namely Hermite-type interpolation for dealing with Neumann boundary conditions. The numerical results demonstrate that good improvement on accuracy can be obtained after using Hermite-type interpolation. The h-convergence rates are also studied for RPICM with different forms of basis functions and different additional terms. [Copyright &y& Elsevier]

Abstract: Feature modeling systems usually employ a boundary representation (b-rep) to store the shape information on a product. It has, however, been shown that a b-rep has a number of shortcomings, and that a cellular representation can be a valuable alternative. A cellular model stores additional shape information on features, including the feature faces that are not on the boundary of the product. Such information can be profitably used for several purposes. A major operation in every feature modeling system is boundary evaluation, which computes the geometric model of a product, i.e. either the b-rep or the cellular model, from the features that have been specified by the user. Since boundary evaluation has to be executed each time a feature is added, removed or modified, its efficiency is of paramount importance. In this paper, boundary evaluation for a cellular model is described in some detail. Its efficiency is compared to the efficiency of boundary evaluation for a b-rep, on the basis of both complexity analysis and performance measurements for the two types of evaluation. It turns out that boundary evaluation for a cellular model is, in fact, more efficient than for a b-rep, which makes cellular models even more attractive as an alternative to b-reps. [Copyright &y& Elsevier]

In this paper, we describe how to analyze boundary value problems for third-order nonlinear ordinary differential equations over an infinite interval. Several physical problems of interest are governed by such systems. The seminumerical schemes described here offer some advantages over solutions obtained by using traditional methods such as finite differences, shooting method, etc. These techniques also reveal the analytic structure of the solution function. For illustrative purposes, several physical problems, mainly drawn from fluid mechanics, are considered; they clearly demonstrate the efficiency of the techniques presented here. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we propose a numerical method for computing minimal surfaces with fixed boundaries. The level set method is used to evolve a codimension-1 surface with fixed codimension-2 boundary in under mean curvature flow. For n =3 the problem has been approached in D.L. Chopp, 1993 and L.-T. Cheng [D.L. Chopp, Computing minimal surfaces via level set curvature flow, J. Comput. Phys. 106(1) (1993) 77–91 and L.-T. Cheng, The level set method applied to geometrically based motion, materials science, and image processing, UCLA CAM Report, 00-20] using the level set method, but with a more complicated boundary conditions. The method we present can be generalized straightforward to arbitrary dimension, and the framework in which it is presented is dimension independent. Examples are shown for n =2, 3, 4. [Copyright &y& Elsevier]

Imposition of essential boundary conditions in meshfree methods is made difficult because the shape functions used do not possess the ‘delta’ property. Various procedures have been proposed including penalty, Lagrange multipliers and collocation. It is shown in this paper that the success of a procedure depends on the arrangement of the nodal points. Certain methods of imposing boundary conditions are shown not to work for unstructured nodal arrangements. Much previous work has been demonstrated using structured grids thus hiding these drawbacks. Copyright © 2005 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, a new numerical method is presented for solving the linear Poisson–Boltzmann equation. A mortar finite element method with fundamental solution and artificial boundary condition are used to deal with the difficulties in numerical simulation and optimal error estimates are obtained. [Copyright &y& Elsevier]

This paper discovers and researches problems on numerical oscillations of the solution in element-free Galerkin method (EFGM) when it uses high order polynomial basis, and puts forward the meshless method based on orthogonal basis (MLMBOB), which is composed of essential boundary conditions with Penalty method, then gets the numerical solutions of the partial differential equations. This method holds nearly all qualities of EFGM and removes many drawbacks of it, and it has high accuracy when high order orthogonal basis is used. Therefore, it is fit for many problems in engineering computational electromagnetics. Examples are given to prove the proposed method. [ABSTRACT FROM AUTHOR]

In this paper, a novel numerical solution technique, the moving least squares differential quadrature (MLSDQ) method is employed to study the free vibration problems of generally laminated composite plates based on the first order shear deformation theory. The weighting coefficients used in MLSDQ approximation are obtained through a fast computation of the MLS shape functions and their partial derivatives. By using this method, the governing differential equations are transformed into sets of linear homogeneous algebraic equations in terms of the displacement components at each discrete point. Boundary conditions are implemented through discrete grid points by constraining displacements, bending moments and rotations of the plate. Combining these algebraic equations yields a typical eigenvalue problem, which can be solved by a standard eigenvalue solver. Detailed formulations and implementations of this method are presented. Convergence and comparison studies are carried out to verify the reliability and accuracy of the numerical solutions. The applicability, efficiency, and simplicity of the present method are all demonstrated through solving several sample problems. [Copyright &y& Elsevier]

Abstract: We present a computational method for the solution of the third-order boundary value problem characterized by the well-known Falkner–Skan equation on a semi-infinite domain. Numerical treatments of this problem reported in the literature thus far are based on shooting and finite differences. While maintaining the simplicity of the shooting approach, the method presented in this paper uses a technique known as automatic differentiation, which is neither numerical nor symbolic. Using automatic differentiation, a Taylor series solution is constructed for the initial value problems by calculating the Taylor coefficients recursively. The effectiveness of the method is illustrated by applying it successfully to various instances of the Falkner–Skan equation. [Copyright &y& Elsevier]

This paper presents a numerical approach to study the electrical properties of the Earth's atmosphere. The finite-difference time-domain (FDTD) technique is applied to model the Earth's atmosphere in order to determine Schumann resonant frequencies of the Earth. Three-dimensional spherical coordinates are employed and the conductivity profile of the atmosphere versus height is introduced. Periodic boundary conditions are implemented in order to exploit the symmetry in rotation of the Earth and decrease computational requirements dramatically. For the first time, very accurate FDTD results are obtained, not only for the fundamental mode but also for higher order modes of Schumann resonances. The proposed method constitutes a useful tool to obtain Schumann resonant frequencies, therefore to validate electrical models for the terrestrial atmosphere, or atmospheres of other celestial bodies. [ABSTRACT FROM AUTHOR]

The paper considers the problem of justifying the asymptotic solution of the phase-field system. [ABSTRACT FROM AUTHOR]

The main aim of this paper is to highlight that, when dealing with DG methods for linear hyperbolic equations or advection-dominated equations, it is much more convenient to write the upwind numerical flux as the sum of the usual (symmetric) average and a jump penalty. The equivalence of the two ways of writing is certainly well known (see e.g. Ref. 4); yet, it is very widespread not to consider upwinding, for DG methods, as a stabilization procedure, and too often in the literature the upwind form is preferred in proofs. Here, we wish to underline the fact that the combined use of the formalism of Ref. 3 and the jump formulation of upwind terms has several advantages. One of them is, in general, to provide a simpler and more elegant way of proving stability. The second advantage is that the calibration of the penalty parameter to be used in the jump term is left to the user (who can think of taking advantage of this added freedom), and the third is that, if a diffusive term is present, the two jump stabilizations (for the generalized upwinding and for the DG treatment of the diffusive term) are often of identical or very similar form, and this can also be turned to the user's advantage. [ABSTRACT FROM AUTHOR]

In this paper a Galerkin method is used to construct series solutions of a homogeneous Sturm-Liouville problem defined on [0, &pie;]. The series constructed are shown to converge to a specified du Bois-Reymond function f in L² [0,&Pie;]. It is then shown that the series solutions can be made to converge uniformly to the specified du Bois-Reymond function when averaged by the Cesaro-one summability method. Therefore, in the Cesaro-one sense, every continuous function / o n [0, &Pie;] is the uniform limit of solutions of non-homogeneous Sturm-Liouville problems. [ABSTRACT FROM AUTHOR]

Abstract: An alternative to control of the global error of a numerical solution to a boundary value problem (BVP) for ordinary differential equations (ODEs) is control of its residual, the amount by which it fails to satisfy the ODEs and boundary conditions. Among the methods used by codes that control residuals are collocation, Runge-Kutta methods with continuous extensions, and shooting. Specific codes that concern us are bvp4c of the Matlab problem solving environment and the FORTRAN code MIRKDC for general scientific computation. The residual of a numerical solution is related to its global error by a conditioning constant. In this paper, we investigate a conditioning constant appropriate for BVP solvers that control residuals and show how to estimate it numerically at a modest cost. Codes that control residuals can compute pseudosolutions, numerical solutions to BVPs that do not have solutions. That is, a “well-behaved” approximate solution is computed for an ill-posed mathematical problem. The estimate of conditioning is used to improve the robustness of bvp4c and MIRKDC and in particular, help users identify when a pseudosolution may have been computed. [Copyright &y& Elsevier]

This paper deals with methods for the solution of boundary value problems for linear systems of stochastic differential equations. We investigate numerical algorithms, the problem of existence and uniqueness of solutions and other specific problems (including stationary boundary value problems, reduction of a boundary value problem to a Cauchy problem, extended boundary value problems, active and passive boundary conditions, etc.). [ABSTRACT FROM AUTHOR]

In this paper, a numerical method is suggested to solve a class of boundary value problems (BVPs) for a weakly coupled system of singularly perturbed second-order ordinary differential equations of convection-diffusion type. First, in this method, an asymptotic expansion approximation of the solution of the BVP is constructed by using the basic ideas of a well known perturbation method namely Wentzal, Kramers and Brillouin (WKB). Then, some initial value problems (IVPs) are constructed such that their solutions are the terms of this asymptotic expansion. These problems happen to be singularly perturbed problems and, therefore, exponentially fitted finite difference schemes are used to solve these problems. As the BVP is converted into a set of IVPs and an asymptotic expansion approximation is used, the present method is termed as asymptotic initial-value method. The necessary error estimates are derived and examples provided to illustrate the method. [ABSTRACT FROM AUTHOR]

In this paper, a fourth order finite difference method for a class of singular boundary value problems is presented. The original differential equation is modified at the singular point. The fourth order finite difference method is then employed to solve the boundary value problem. Some model problems are solved, and the numerical results are compared with exact solution. [Copyright &y& Elsevier]

Enhanced models such as second gradient models have been mainly theoretically developed during the sixties and the seventies. New attention has been paid to these models recently when it has been recognized that classical inelastic constitutive equations, especially the ones not obeying a normality rule, induce ill-posed boundary value problems. However little work has been done to study the well-posedness properties of boundary value problems involving these enhanced models. The aim of this paper is a partial study of this point. Only uniqueness of solution of numerical discretization of boundary value problems involving second gradient models is considered. We extend to second gradient models a numerical search method to find non-uniqueness, already established for classical media. In the case of loss of uniqueness for the underlying model, our numerical results show clearly that in some cases uniqueness is lost for second gradient model as well. It seems that second gradient extensions of classical models do not restore the uniqueness properties, but only the objectivity of the numerical computation. Finally our numerical experiments show once more that post-peak behavior is a structural effect. They also suggest that some constitutive equations often used in geomaterials modeling may be questionable, and may not be complete because even adding second gradient enhancement to these models yields non-unique solutions. [Copyright &y& Elsevier]

In this paper we describe a Monte Carlo method for permeability calculations in complex digitized porous structures. The relation between the permeability and the diffusion penetration depth is established. The corresponding Dirichlet boundary value problem is solved by random walk algorithms. The results of computational experiments for some random models of porous media confirm the log-normality hypothesis for the permeability distribution. [ABSTRACT FROM AUTHOR]

In this paper, we use parametric cubic spline function to develop a numerical method, which is fourth order for a specific choice of the parameter, for computing smooth approximations to the solution for second order boundary value problems. Some numerical evidence is also included to demonstrate the superiority of our method. [Copyright &y& Elsevier]

This paper deals with a class of singularly perturbed nonlinear elliptic problems (Pℇ) with subcritical nonlinearity. The coefficient of the linear part is assumed to concentrate in a point of the domain, as ℇ→0, and the domain is supposed to be unbounded and with unbounded boundary. Domains that enlarge at infinity, and whose boundary flattens or shrinks at infinity, are considered. It is proved that in such domains problem (Pℇ) has at least 2 solutions. [ABSTRACT FROM AUTHOR]

Abstract: Singularly perturbed two-point boundary value problems (BVPs) for fourth-order ordinary differential equations (ODEs) with a small positive parameter multiplying the highest derivative are considered. A numerical method is suggested in this paper to solve such problems. In this method, the given BVP is transformed into a system of two ODEs subject to suitable boundary conditions. Then, the domain of definition of the differential equation (a closed interval) is divided into two nonoverlapping subintervals, which we call “inner region” (boundary layer) and “outer region”. Then, the DE is solved in these intervals separately. The solutions obtained in these regions are combined to give a solution in the entire interval. To obtain terminal boundary conditions (boundary values inside this interval) we use mostly zero-order asymptotic expansion of the solution of the BVP. First, linear equations are considered and then nonlinear equations. To solve nonlinear equations, Newton''s method of quasilinearization is applied. The present method is demonstrated by providing examples. The method is easy to implement. [Copyright &y& Elsevier]

Abstract: This paper deals with various aspects of the theory and implementation of finite element methods for elliptic boundary value problems whose variational formulation is posed on anisotropic Sobolev spaces. The theory is applied to the Onsager pancake equation which arises in the study of high speed gas centrifuges. [Copyright &y& Elsevier]

This paper considers extrapolation of rational methods for the numerical solution of initial value problems (IVPs) in ordinary differential equations (ODEs). The extrapolation code, DIFEX2 of Fatunla [1], which is a modification of the automatic extrapolation code, DIFEXI of Deuflhard [2], is further modified to accommodate the basic integrators. The extrapolation method which we refer to as DIFEX2+ for reference purposes is compared with results from DIFEXI and DIFEX2, and GBS extrapolation method. [Copyright &y& Elsevier]

In this paper, a generalization of the Schwarz–Neumann method to more than two domains is given. We prove the convergence and the numerical stability of the algorithm. The results apply to both bounded and unbounded domains, and are given for the weak solution of an elliptic problem with mixed boundary conditions. Numerical results are given for both bounded and unbounded domains. [ABSTRACT FROM PUBLISHER]

We study the boundary-value problems for singularly perturbed differential-difference equations with small shifts. Similar boundary-value problems are associated with expected first-exit time problems of the membrane potential in models for activity of neurons (SIAM J. Appl. Math. 1994; 54: 249–283; 1982; 42: 502–531; 1985; 45: 687–734) and in variational problems in control theory. In this paper, we present a numerical method to solve boundary-value problems for a singularly perturbed differential-difference equation of mixed type, i.e. which contains both type of terms having negative shifts as well as positive shifts, and consider the case in which the solution of the problem exhibits rapid oscillations. The stability and convergence analysis of the method is given. The effect of small shift on the oscillatory solution is shown by considering the numerical experiments. The numerical results for several test examples demonstrate the efficiency of the method. Copyright © 2004 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

Boundary value methods (BVMs) for ordinary differential equations require the solution of non-symmetric, large and sparse linear systems. In this paper, these systems are solved by using the generalized minimal residual (GMRES) method. A block-circulant preconditioner with circulant blocks (BCCB preconditioner) is proposed to speed up the convergence rate of the GMRES method. The BCCB preconditioner is shown to be invertible when the BVM is Ak1,k2-stable. The spectrum of the preconditioned matrix is clustered and therefore, the preconditioned GMRES method converges fast. Moreover, the operation cost in each iteration of the preconditioned GMRES method by using our BCCB preconditioner is less than that required by using block-circulant preconditioners proposed earlier. In numerical experiments, we compare the number of iterations of various preconditioners. Copyright © 2003 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the method of the numerical solution of boundary value problems in inhomogeneous domains composed of homogeneous multidimensional parallelepipeds. The method is the symbiosis of high-order difference schemes in homogeneous subdomains and multipoint one-dimensional boundary conditions at interfaces. Due to splitting, the boundary value problem reduces to systems of linear algebraic equations with matrices different from a tridiagonal matrix by the availability of separate 'long' rows with more than three nonzero elements. Two algorithms are investigated to solve these systems. The first algorithm is based on the immediate transformation of the system of equations to a tridiagonal form. The second one is a generalization of the known sweep parallelizing method. [ABSTRACT FROM AUTHOR]

We consider the solution of a system of ordinary differential equations (ODEs) by waveform relaxation (WR) iterations in conjunction with boundary value methods (BVMs). The WR method is a continuous-in-time analogue of the stationary method and it iterates with functions. In each WR iteration, we use BVMs to discretize systems of ODEs. BVMs are relatively new ODE solvers based on linear multistep formulae. In this paper, we discuss the use of the generalized minimal residual (GMRES) method with block-circulant–circulant-block preconditioners for solving the linear systems arising from the application of BVMs in each WR iteration. These preconditioners are effective in speeding up the convergence rate of the GMRES method. Numerical experiments are presented to illustrate the effectiveness of our methods. [Copyright &y& Elsevier]

In this paper, we develop a new non-iterative numerical method for solving inverse boundary value problems for linear elliptic equations of second order. Here we assume that the Dirichlet and Neumann boundary conditions are given only on part of the domain, we have to reconstruct the solution and its normal derivative on the unaccessible part of the domain, which is the well-known ill-posed Cauchy problem. We propose to solve such inverse problems directly by using the recently developed radial basis meshless method. Several numerical experiments are given to demonstrate the effectiveness and efficiency of the proposed method. Copyright © 2003 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

In this paper we obtain a second order finite difference method based on non-uniform mesh for a class of singular two point boundary value problems:(xαy′)′=f(x,y) 0h2 convergent for all α∈(0,1). [Copyright &y& Elsevier]

In this paper, a novel true meshless numerical technique – the Hermite–Cloud method, is developed. This method uses the Hermite interpolation theorem for the construction of the interpolation functions, and the point collocation technique for discretization of the partial differential equations. This technique is based on the classical reproducing kernel particle method except that a fixed reproducing kernel approximation is employed instead. As a true meshless technique, the present method constructs the Hermite-type interpolation functions to directly compute the approximate solutions of both the unknown functions and the first-order derivatives. The necessary auxiliary conditions are also constructed to generate a complete set of partial differential equations with mixed Dirichlet and Neumann boundary conditions. The point collocation technique is then used for discretization of the governing partial differential equations. Numerical results show that the computational accuracy of the Hermite–Cloud method at scattered discrete points in the domain is much refined not only for approximate solutions, but also for the first-order derivative of these solutions. [ABSTRACT FROM AUTHOR]

This paper introduces a fast numerical method for computing American option pricing problems governed by the Black-Scholes equation. The treatment of the free boundary is based on some properties of the solution of the Black-Scholes equation. An artificial boundary condition is also used at the other end of the domain. The finite difference method is used to solve the resulting problem. Computational results are given for some American call option problems. The results show that the new treatment is very efficient and gives better accuracy than the normal finite difference met hod. [ABSTRACT FROM AUTHOR]

This paper deals with the construction of stable discrete numerical solutions of strongly coupled mixed hyperbolic problems using difference schemes. By means of a discrete separation of variables method and solving the underlying discrete Sturm-Liouville type problem, the numerical solution of the discretized mixed problem is constructed. [Copyright &y& Elsevier]

In this paper we analyze a Galerkin procedure, based on a combination of finite and spectral elements, for approximating a time-harmonic acoustic wave scattered by a bounded inhomogeneity. The finite element method used to approximate the near field in the region of inhomogeneity is coupled with a nonlocal boundary condition, which consists in a linear integral equation. This integral equation is discretized by a spectral Galerkin approximation method. We provide error estimates for the Galerkin method, propose fully discrete schemes based on elementary quadrature formulas, and show that the perturbation due to this numerical integration gives rise to a quasi-optimal rate of convergence. We also suggest a method for implementing the algorithm using the preconditioned GMRES method and provide some numerical results. [ABSTRACT FROM AUTHOR]

The paper is concerned with a one-dimensional parabolic problem in a domain bounded by two lines x = 0 and x = kt, k > 0, (x, t) ∈ $\backslash Bbb\; R^2$, with the Neumann boundary condition on the line x = 0 and with dynamic boundary condition on the line x = kt. For the solution of this problem, a coercive estimate in a weighted Hölder norm is obtained. It is shown that this estimate can be useful for the analysis of parabolic free boundary problems. Bibliography: 7 titles. [ABSTRACT FROM AUTHOR]

For nonlinear hyperbolic problems, conservation of the numerical scheme is important for convergence to the correct weak solutions. In this paper the conservation of the wellknown compact scheme up to fourth order of accuracy on a single and uniform grid is studied, and a conservative interface treatment is derived for compact schemes on patched grids. For a pure initial value problem, the compact scheme is shown to be equivalent to a scheme in the usual conservative form. For the case of a mixed initial boundary value probiem, the compact scheme is conservative only if the rounding errors are small enough. For a patched grid interface, a conservative interface condition useful for mesh refinement and for parallel computation is derived and its order of local accuracy is analyzed. [ABSTRACT FROM AUTHOR]

In this article, we study the differential equation (- 1)n - px(n)(t) = f(t, x(t), x'(t, ..., x(n - 1)(t)), 0 < t <1, subject to the multi-point boundary conditions x(i) (0) = (0) for i = 0, 1, ..., p - 1, x(i) (1) = (0) for i = p + 1, ..., n - 1, ... where 1 ≤ p ≤ n - 1. We establish sufficient conditions for the existence of at least one solution at resonance and another at non-resonance. The emphasis in this paper is that f depends on all higher-order derivatives. Examples are given to illustrate the main results of this article. [ABSTRACT FROM AUTHOR]