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This paper presents a novel shooting method for solving two-point boundary value problems for second order ordinary differential equations. The method works as follows: first, a guess for the initial condition is made and an integration of the differential equation is performed to obtain an initial value problem solution; then, the end value of the solution is used in a simple iteration formula to correct the initial condition; the process is repeated until the second boundary condition is satisfied. The iteration formula is derived utilizing an auxiliary function that satisfies both boundary conditions and minimizes the H 1 semi-norm of the difference between itself and the initial value problem solution. [ABSTRACT FROM AUTHOR]

Abstract: Traditional approach to effectiveness factor calculations uses finite difference methods to discretize two-point boundary value differential equations. The finite difference method (FDM) approximates derivatives using only nearby concentration values; however, an approach that takes into account the influence of concentrations away from a point is highly intuitive and requires careful consideration. In this paper we extend the integral formulation for the case when external limitations are present. We also propose a new approach based on the use of two-dimensional Green''s function for effectiveness factor calculations in two-dimensional geometries, and provide a generic procedure for concentration profile calculation in one- and two-dimensional geometries. The proposed approach retains the representation rigor of analytical approach and reduces the computational complexities associated with FDM. It has been shown that the presence of external transport limitation shall modify the Green''s function obtained considering only internal transport limitations. The problem of very high sensitivity of the boundary contribution term in the Green''s Function Solution for two-dimensional cylindrical particle is addressed by variable transformation. It has been shown in this paper that Green''s Function method provides superior performance for larger step size when compared with the FDM approach, thus providing computationally efficient approach. [Copyright &y& Elsevier]

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: In this paper, we show differentiability of solutions with respect to the given boundary value data for nonlinear singularly perturbed boundary value problems and its corresponding asymptotic expansion of small parameter. This result fills the gap caused by the solvability condition in Esipova’s result so as to lay a rigorous foundation for the theory of boundary function method on which a guideline is provided as to how to apply this theory to the other forms of singularly perturbed nonlinear boundary value problems and enlarge considerably the scope of applicability and validity of the boundary function method. A third-order singularly perturbed boundary value problem arising in the theory of thin film flows is revisited to illustrate the theory of this paper. Compared to the original result, the imposed potential condition is completely removed by the boundary function method to obtain a better result. Moreover, an improper assumption on the reduced problem has been corrected. [Copyright &y& Elsevier]

Abstract: This paper deals with stability properties of Runge–Kutta methods for the initial value problem in nonlinear neutral delay differential equationsThe new concepts of -stability, -stability and Weak -stability are introduced, and it is shown that -algebraically stable Runge–Kutta methods with piecewise linear interpolation are -, - and Weakly -stable. Two numerical examples are given in the end of this paper which confirm our results. [Copyright &y& Elsevier]

Abstract: When solving a system of first order differential equations involving boundary conditions using collocation at k Gauss points that are the zeros of , the kth degree Legendre polynomial, to produce a continuous solution over the problem interval, it is well known that the global error at the mesh points is and at other points the global error is . In addition, it is known that the error in the derivative of the solution at non-mesh points is . This paper shows that the error in the solution at the zeros of in each subinterval is and that the error in the derivative of the solution at the zeros of is . This higher order accuracy can be used in various interpolation schemes (Hermite–Birkhoff interpolation in the case that includes derivative data) to produce global errors in the solution without any additional collocation and without any additional function evaluations. Only the evaluation of the collocation solution or its derivative at certain special points is necessary. Hence, this increased accuracy is almost free compared to other methods which give global errors but at considerable extra expense. Some numerical results illustrating the higher accuracy at these special points are given at the end of the paper along with comparisons, in the form of diagrams, of results produced from the package COLSYS and the results produced from the interpolation schemes (which use COLSYS as a starting point). [Copyright &y& Elsevier]

Abstract: We construct and analyze non-standard finite difference methods for a class of singularly perturbed differential equations. The class consists of two types of problems: (i) those having solutions with layer behavior and (ii) those having solutions with oscillatory behavior. Since no fitted mesh method can be designed for the latter type of problems, other special treatment is necessary, which is one of the aims being attained in this paper. The main idea behind the construction of our method is motivated by the modeling rules for non-standard finite difference methods, developed by Mickens. These rules allow one to incorporate the essential physical properties of the differential equations in the numerical schemes so that they provide reliable numerical results. Note that the usual ways of constructing the fitted operator methods need the fitting factor to be incorporated in the standard finite difference scheme and then it is derived by requiring that the scheme be uniformly convergent. The method that we present in this paper is fairly simple as compared to the other approaches. Several numerical examples are given to support the predicted theory. [Copyright &y& Elsevier]

Abstract: The paper deals with the initial–boundary value problem for the linear wave equation where , are given constants and , , are given functions, and the unknown function and the unknown boundary value satisfy the following nonlinear integral equation: where , , , are given constants and , are given functions. In this paper, we consider three main parts. In Part 1 we prove a theorem of global existence and uniqueness of a weak solution of problem (1.1)–(1.5). The proof is based on a Galerkin method associated with a priori estimates, weak convergence and compactness techniques. For the case of , Part 2 is devoted the study of the asymptotic behavior of the solution as . Finally, in Part 3 we obtain an asymptotic expansion of the solution of the problem (1.1)–(1.5) up to order in four small parameters , , , . [Copyright &y& Elsevier]

Abstract: This paper revisits and extends the work that we started in 1972 on maximum polynomial degree (m.p.d) (k, p) Nordsieck–Gear methods for the numerical solution of the non-stiff ordinary differential equations initial value problem: f ∈ C k+2(N), N a neighborhood of the graph the exact solution values of y and its derivatives up to order p −1. In the notation (k, p) k is the number of scaled derivatives that are retained by the method, p is the order of the differential equation and k > p ⩾1. These methods belong to the class of linear multi-value predictor–corrector methods. They are fixed order, variable step and admit of accurate self-starting algorithms. In our first paper we proved that, given exact starting values for the p +1, p +2,…, k scaled derivative starting values, a vector of p Nordsieck–Gear (k, p) corrector–predictor coefficients must be recursively computed to preserve m.p.d k +1. Computational experiment showed that the sequence of coefficient vectors was well-defined in the (k,1) case for k =2,3,4,5. This led to the two-part conjecture: (1) The sequence of coefficient vectors is well-defined for all (k, p) and (2) These coefficient vectors cycle to their fixed values at most k − p +1 steps. Our second paper dealt with the construction of a starting method which yields starting scaled derivatives of order p +1, p +2,…, k, all accurate to O(h k+2), with exactly (k − p +1)(k − p +2)/2+2 evaluations of the derivative function f. In this paper we begin by proving the first conjecture and a slightly stronger version of the second. With these results in hand we then prove convergence in the (k,1) method. It is shown that the starting algorithm’s guaranteed O(h k+2) starting error for the (k,1) method can be preserved globally for sufficiently small step-size h by augmenting two corrector–predictor evaluations with a stability-preserving, computationally efficient refinement. We have coded a (5,1) Fortran 77 algorithm, DIFFEQ, to implement, test, and confirm the underlying theory. We compare its computational efficiency with the variable order, variable step Adams–Gear method contained in the 1995 edition of the International Mathematical and Statistical Library (IMSL). [Copyright &y& Elsevier]

Abstract: This paper deals with numerical solutions to a partial differential equation of fractional order. Generally this type of equation describes a transition from anomalous diffusion to transport processes. From a phenomenological point of view, the equation includes at least two fractional derivatives: spatial and temporal. In this paper we proposed a new numerical scheme for the spatial derivative, the so-called Riesz–Feller operator. Moreover, using the finite difference method, we show how to employ this scheme in the numerical solution of fractional partial differential equations. In other words, we considered an initial-boundary value problem in one-dimensional space. In the final part of this paper some numerical results and plots of simulations are shown as examples. [Copyright &y& Elsevier]

Abstract: In this paper, methods based on numerical differentiation and their regions of absolute stability of some classes of special multistep methods are derived. Some of the methods given in Henrici [P. Henrici, Discrete Variable Methods in Ordinary Differential Equations, Wiley, 1962] and some methods derived in this paper are applied to solve a second-order initial value problem hence establishing the superiority of the methods derived in this paper. [Copyright &y& Elsevier]

In this paper, multiscale Galerkin method is presented to approximate the solutions of second-order boundary value problems of Fredholm integro-differential equation. The method is based on traditional Galerkin method and uses the multiscale orthonormal bases to discretize the equations. The proposed method is proved to be stable and have the optimal convergence order. Numerical examples are presented to confirm the theoretical results and show that the method is computationally stable, valid and accurate. [ABSTRACT FROM AUTHOR]

Abstract: The scaled boundary method is a semi-analytical method of analysis which can be used in computational mechanics. This method uses a normalised radial coordinate system, introducing shape functions to weaken the governing equations in the circumferential direction and solving the resulting differential equations analytically in the radial direction. This paper presents a new Element-free Galerkin (EFG) scaled boundary method in which the EFG approach is used in the circumferential direction. The proposed model is verified by application to a number of standard problems of elasticity. The numerical solutions show that the new method has higher accuracy (for any particular number of nodes) and better convergence than scaled boundary finite element methods, and an accurate smooth stress field can be obtained directly without the necessity of using a stress recovery procedure. [Copyright &y& Elsevier]

Abstract: In this paper we study the blow-up phenomenon for the non-local -laplacian equation with a reaction term, with Dirichlet conditions ( in ) or Neumann conditions (). Those problems are the non-local analogous to the equation with the corresponding conditions. We determine in both cases which are the global existence exponents, that coincide with the exponents of the corresponding local problem for Neumann boundary conditions. However, we observe differences with respect to the global existence exponents of the local Dirichlet problem. Moreover, we show that the blow-up rate is the same as the one that holds for the ODE , that is, . We also find some differences between the local and the non-local models concerning the blow-up sets. Precisely we show that regional blow-up is not possible for non-local problems. Finally, we include some numerical experiments which illustrate our results. [Copyright &y& Elsevier]

Abstract: In the this paper, we establish sufficient conditions for the existence and nonexistence of positive solutions to a general class of integral boundary value problems for a coupled system of fractional differential equations. The differential operator is taken in the Riemann–Liouville sense. Our analysis rely on Banach fixed point theorem, nonlinear differentiation of Leray–Schauder type and the fixed point theorems of cone expansion and compression of norm type. As applications, some examples are also provided to illustrate our main results. [Copyright &y& Elsevier]

Abstract: In this paper, we show the existence of solutions to a nonlinear singular second order ordinary differential equation, subject to periodic boundary conditions, where is a given constant, is a parameter, and the nonlinearity satisfies the local Carathéodory conditions on . Here, we study the case that a well-ordered pair of lower and upper functions does not exist and therefore the underlying problem cannot be treated by well-known standard techniques. Instead, we assume the existence of constant lower and upper functions having opposite order. Analytical results are illustrated by means of numerical experiments. [Copyright &y& Elsevier]

Abstract: In this paper we obtain a result on existence of extremal quasisolutions and solutions for second-order functional differential equations with functional boundary conditions and past, present and future dependence. In order to do that, we develop a generalized monotone method in presence of lower and upper solutions. Some examples of application are provided too. [Copyright &y& Elsevier]

Abstract: In this paper, the Adomian decomposition method with Green’s function is applied to solve linear and nonlinear sixth-order boundary value problems. The numerical results obtained with a small amount of computation are compared with the exact solutions to show the efficiency of the method. The results show that the decomposition method is of high accuracy, more convenient and efficient for solving high-order boundary value problems. [Copyright &y& Elsevier]

Abstract: A modified continuum model of electrically actuated nanobeams is presented by incorporating surface elasticity in this paper. The classical beam theory is adopted to model the bulk, while the bulk stresses along the surfaces of the bulk substrate are required to satisfy the surface balance equations of the continuum surface elasticity. On the basis of this modified beam theory the governing equation of an electrically actuated nanobeam is derived and a powerful technology, analog equation method (AEM) is applied to solve this complex problem. Beams made from two materials: aluminum and silicon are chosen as examples. The numerical results show that the pull-in phenomena in electrically actuated nanobeams are size-dependent. The effects of the surface energies on the static and dynamic responses, pull-in voltage and pull-in time are discussed. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we establish the existence of a positive solution to a singular boundary value problem of nonlinear fractional differential equation. Our analysis rely on nonlinear alternative of Leray–Schauder type and Krasnoselskii’s fixed point theorem in a cone. [Copyright &y& Elsevier]

Abstract: This paper proposes an original numerical method and studies its performance for solving high-frequency scattering problems involving elongated scatterers. The approach is based on coupling a high-order Padé-type non-reflecting boundary condition with plane wave finite element formulations. It is shown that for some numerical examples the approximate solution of suitable accuracy can be obtained using a small number of degrees of freedom. [Copyright &y& Elsevier]

Abstract: In this paper we are concerned about a singular boundary value problem for a quasilinear second-order ordinary differential equation, involving the one-dimensional -laplacian. Asymptotic expansions of the one-parameter families of solutions, satisfying the prescribed boundary conditions, are obtained in the neighborhood of the singular points and this enables us to compute numerical solutions using stable shooting methods. [Copyright &y& Elsevier]

Abstract: The parallel solution of initial value problems for ordinary differential equations (ODE-IVPs) has received much interest from many researchers in the past years. In general, the possibility of using parallel computing in this setting concerns different aspects of the numerical solution of ODEs, depending on the parallel platform to be used and/or the complexity of the problem to be solved. In particular, in this paper we examine possible extensions of a parallel method previously proposed in the mid-nineties [P. Amodio, L. Brugnano, Parallel implementation of block boundary value methods for ODEs, J. Comput. Appl. Math. 78 (1997) 197–211; P. Amodio, L. Brugnano, Parallel ODE solvers based on block BVMs, Adv. Comput. Math. 7 (1997) 5–26], and analyze its connections with subsequent approaches to the parallel solution of ODE-IVPs, in particular the “Parareal” algorithm proposed in [J.L. Lions, Y. Maday, G. Turinici, Résolution d''EDP par un schéma en temps “pararéel”, C. R. Acad. Sci. Paris, Ser. I 332 (2001) 661–668; Y. Maday, G. Turinici, A parareal in time procedure for the control of partial differential equations, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 387–392]. [Copyright &y& Elsevier]

Abstract: In this paper,we present a new algorithm to solve a kind of singular boundary-value problems of higher even-order based on the theory of numerical analysis in the reproducing kernel space . Singular boundary-value problem of sixth-order has been discussed for the problem in the case of . The numerical results are compared with both the exact solution and its -order derived functions in the example. It is demonstrated that our method is of high precision and significance. [Copyright &y& Elsevier]

Abstract: In this paper we consider a singularly perturbed quasilinear boundary value problem depending on a parameter. The problem is discretized using a hybrid difference scheme on Shishkin-type meshes. We show that the scheme is second-order convergent, in the discrete maximum norm, independent of singular perturbation parameter. Numerical experiments support these theoretical results. [Copyright &y& Elsevier]

Abstract: In this paper, we consider a perturbed risk process (in which the inter-occurrence times are generalized Erlang(n)-distributed) compounded by a geometric Brownian motion. Integro-differential equations with certain boundary conditions for the moment-generating function and the mth moment of the present value of all dividends until ruin are derived. We also derive integro-differential equations with boundary conditions for the Gerber–Shiu function. Some special cases are considered in details. [Copyright &y& Elsevier]

Abstract: In this paper, by using the Schauder fixed point theorem, we offer some existence criteria for a solution and a positive solution to the following fourth-order two-point boundary value problem on time scale : where and . [Copyright &y& Elsevier]

Abstract: This paper deals with boundary value problems in a bounded domain for the stationary one-dimensional Kawahara equation. For reasonable boundary conditions we prove existence and uniqueness of a strong solution and its asymptotics when the coefficient of a higher order derivative approaches zero. Formulations of well-posed and ill-posed problems for odd-order equations are discussed. [Copyright &y& Elsevier]

Abstract: A boundary value problem for second order singularly perturbed delay differential equation is considered. When the delay argument is sufficiently small, to tackle the delay term, the researchers [M.K. Kadalbajoo, K.K. Sharma, Numerical analysis of singularly perturbed delay differential equations with layer behavior, Appl. Math. Comput. 157 (2004) 11–28, R.E. O’Malley, Jr., Singular Perturbation Methods for Ordinary Differential Equations, Springer-Verlag, New York, 1991] used Taylor’s series expansion and presented an asymptotic as well as numerical approach to solve such type boundary value problem. But the existing methods in the literature fail in the case when the delay argument is bigger one because in this case, the use of Taylor’s series expansion for the term containing delay may lead to a bad approximation. In this paper to short out this problem, we present a numerical scheme for solving such type of boundary value problems, which works nicely in both the cases, i.e., when delay argument is bigger one as well as smaller one. To handle the delay argument, we construct a special type of mesh so that the term containing delay lies on nodal points after discretization. The proposed method is analyzed for stability and convergence. To demonstrate the efficiency of the method and how the size of the delay argument and the coefficient of the delay term affects the layer behavior of the solution several test examples are considered. [Copyright &y& Elsevier]

Abstract: This paper deals with crack energy density (hereafter CED) as a possible fracture parameter in piezoelectricity under arbitrary electric boundary conditions on a notch-like crack surface. The definitions of CED and its derivatives are given first under exact boundary condition. Next, their path independent integrals are also derived and their approximate expressions are discussed under some restrictions on the crack surfaces. It is found that electrical terms along the notch-like crack surface do not vanish unlike in the case of impermeable crack. Then, we introduce evaluation methods of CED, and, with the help of the results of finite element analyses (FEA), we closely examine how electric boundary conditions along the notch surface and initial notch width influence CED and its derivatives. It is shown from the FEA results that because of the difficulties of computing path integral terms along the notch-like crack tip in the path independent expressions, the evaluation by the definitions of CED and its derivatives is preferable and more convenient than the evaluation of their path independent expressions. It is also found that all the parameters are significantly affected by both permittivity inside the electric inclusion and root radius of the notch. Finally, the possibility of mechanical CED as a governing fracture parameter is discussed. [Copyright &y& Elsevier]

Abstract: In this paper, a combined finite strip and state space approach is introduced to obtain three-dimensional solutions of laminated composite plates with simply supported ends. The finite strip method is used to present in-plane displacement and stress components, while the through-thickness components are obtained by using the method of state equation. The method can replace the traditional three-dimensional finite element solutions for structures that have regular geometric plans and simple boundary conditions, where a full three-dimensional finite element analysis is very often both extravagant and unnecessary. The new method provides results that show good agreement with available benchmark problems having different material compositions, thickness and boundary conditions. The new method provides a three-dimensional solution for laminated plates, while the advantages of using the traditional finite strip method are fully taken. This solution also yields a continuous transverse stress field across material interfaces that normally is not achievable by other numerical modelling of laminates, such as the traditional finite element method. [Copyright &y& Elsevier]

In this paper, we propose a new discontinuous finite element method to solve initial value problems for ordinary differential equations and prove that the finite element solution exhibits an optimal O(Δt p+1) convergence rate in the norm. We further show that the p-degree discontinuous solution of differential equation of order m and its first m −1 derivatives are O(Δt 2p+2−m ) superconvergent at the end of each step. We also establish that the p-degree discontinuous solution is O(Δt p+2) superconvergent at the roots of (p +1− m)-degree Jacobi polynomial on each step. Finally, we present several computational examples to validate our theory and construct asymptotically correct a posteriori error estimates. [Copyright &y& Elsevier]

Abstract: In this paper, the origins of spurious solutions occurring in the high-order finite difference methods are studied. Based on a uniform mesh, spurious modes are found in the high-order one-sided finite difference discretizations of many eigenvalue problems. Spurious modes are classified as spectral pollution and non-spectral pollution. The latter can be partially avoided by mesh refinement, while the former persists when the mesh is refined. Through numerical studies of some prototype eigenvalue problems, such as those of the Helmholtz and beam equations, we show that perfect central differentiation schemes do not produce any spurious modes. Nevertheless, high-order central difference schemes encounter difficulties in implementing complex boundary conditions. We further show that the central difference schemes will produce spurious modes as well, if asymmetric approximation is involved in boundary treatments. In general, central difference schemes are less likely to produce spurious modes than one-sided difference schemes. [Copyright &y& Elsevier]

Abstract: In this paper, the fourth-order linear and nonlinear differential equations with separated boundary conditions is converted into an optimal control problem. Then a convergent approximate solution is constructed such that the exact boundary conditions are satisfied. The analysis is accompanied by numerical examples. The obtained results demonstrate reliability and efficiency of the proposed scheme. [Copyright &y& Elsevier]

This paper concerns differential equations with boundary conditions. Given are sufficient conditions under which such problems with deviated arguments have a unique solution in a corresponding sector. To obtain existence results we apply a monotone iterative method. [Copyright &y& Elsevier]

Abstract: In this paper we introduce a new general formula that generates the stability functions of a group of methods, which the author has collectively named implicit advanced step-point (IAS) methods. This group of methods encompasses primarily the two implicit advanced step-point (TIAS) methods and the parallel implicit advanced step-point (PIAS) methods, but also the extended and modified extended BDF schemes (EBDF and MEBDF respectively). Furthermore we present, for the first time in a comprehensive way, the TIAS methods. To obtain the distinct stability functions for each of these implicit multistep methods is a quite time-consuming matter and thus an appropriate general formula can substantially facilitate stability analysis and further computational manipulation of these and analogous schemes. [Copyright &y& Elsevier]

Abstract: Finite difference method, and finite element method are widely used partly for their simplicity, though these methods can obtain first-order or second-order of accuracy. In this paper, we give two highly accurate (fourth-order and sixth-order of accuracy respectively), while still quite simple schemes for two point boundary value problems. We call them compact mixed methods, since they can obtain numerical solutions for unknown function and its first derivative simultaneously. To yield best numerical solutions, we combine our compact mixed schemes with a direct solver and an iterative solver. Then we employ Fourier method to analyze differencing errors shows that our compact mixed formulae are closer to the true wavenumber. Numerical experiments show compact mixed schemes are very efficient, and highly accurate techniques. [Copyright &y& Elsevier]

Abstract: In this paper, we study the nonlinear algebraic system of the formwhere λ >0 is a parameter, x and F(x) denote the column vectors:respectively with f k : R → R, k ∈{1,2,…, n}=[1, n] and n is a positive integer. A =(a ij ) n × n is an n × n matrix and all its entries are positive numbers. Many problems in various areas such as difference equations, boundary value problems, dynamical networks, stochastic process, numerical analysis etc. can be converted to system (E). Applying fixed point theorems, we prove results on existence, uniqueness, multiplicity and nonexistence of positive solutions for (E). [Copyright &y& Elsevier]

Abstract: This paper deals with the construction of numerical solutions of random initial value differential problems by means of a random Euler difference scheme whose mean square convergence is proved based on conditions expressed in terms of the mean square behavior of the right-hand side of the underlying random differential equation. A random mean value theorem is required and established. The concept of mean square modulus of continuity is also introduced and illustrative examples and possibilities are included. Expectation and variance of the approximating process are computed. [Copyright &y& Elsevier]

Abstract: New Runge–Kutta–Nyström methods especially designed for the numerical integration of periodic initial value problems are presented in this paper. They are able to integrate exactly the harmonic oscillator. Here we analyze the construction of an embedded 5(3) RKN pair with four stages. Numerical experiments indicate that the new pair is more efficient than some previously derived RKN pairs. [Copyright &y& Elsevier]

Abstract: Non-polynomial spline is used for the numerical solutions of a system of fourth-order boundary value problems associated with obstacle, unilateral and contact problems. In addition to the boundary conditions corresponding to the second derivatives, the boundary conditions corresponding to the first derivatives are also considered. Two examples (one for each case of boundary conditions) are considered for the numerical illustration of the method developed. It has been observed that the method developed in this paper is better than the other collocation, finite difference and spline methods. [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]

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]

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]