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We investigate a new kind of anti-periodic type boundary value problems of sequential fractional differential equation of order q ∊ (2; 3]. We make use of Banach's contraction mapping principle to obtain the uniqueness result while the existence of solutions is established via Krasnoselskii's fixed point theorem and Leray-Schauder nonlinear alternative. The paper concludes with some illustrative examples. [ABSTRACT FROM AUTHOR]

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: 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: Deriving high precision schemes to compute turbulent flows on fixed or moving complex grids is becoming a central issue in the direct numerical simulation (DNS) and large eddy simulation (LES) community. The step between classical DNS/LES codes on fixed structured grids and future methods on moving unstructured grids is a significant evolution in terms of numerical methods. For reacting flows, this evolution must also include more precise descriptions of multispecies flows and boundary conditions. This paper describes the development of a method for unsteady multispecies reacting flows on moving grids. The target field of application of this method is DNS and LES but this paper focuses on the method development and elementary test cases. The theoretical basis for the numerical method, the boundary conditions and the moving grid extension are first discussed. Various tests of the method are then provided on fixed and moving grids for simple reacting and non-reacting flows to demonstrate the precision and power of the method in simple reference laminar cases. [Copyright &y& Elsevier]

In wave propagation problems thin structures are often replaced by boundaries of zero thickness, in order to reduce the numerical mesh. The reduction of thickness has to be incorporated in the approximate boundary condition applied to the interface. This paper presents a modification of an approximate boundary condition originally introduced by Mitzner. The modified Mitzner condition can be applied to the boundary of zero thickness and it is more general and accurate than the commonly used impedance boundary condition. For frequency domain solvers the condition is as easy to implement as the impedance boundary condition. It is tested and compared to the impedance boundary condition for planar and cylindrical structures. The new condition is exact at normal incidence on a planar structure. [ABSTRACT FROM AUTHOR]

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: 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]

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]

In this paper we present a new kind of P-stable eighteenth-order six-step method for periodic initial-value problems. We add the fourth derivatives to our previous P-stable six-step method to increase the accuracy. We apply two classes of well-known problems to our new method and compare it with the previous methods. The numerical results show that the new method is much more stable, accurate and efficient than the previous methods. [ABSTRACT FROM AUTHOR]

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: 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]

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]

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]

Metastable behaviour, which refers to an asymptotically exponentially slow time dependent motion to the limiting steady‐state solution, is often associated with certain exponentially ill‐conditioned singularly perturbed problems. As a result of this severe ill‐conditioning, little is known concerning the convergence and stability of the numerical schemes that compute metastable behaviour. In this paper, a rigorous uniform convergence analysis is given for several finite difference schemes applied to a boundary layer resonance problem, which is the simplest linear exponentially ill‐conditioned boundary value problem (BVP). It is found that the numerical computation of this problem does not cause any more difficulties than other standard singular perturbation problems, provided that we can use sufficiently high precision arithmetic. The qualitative results from the detailed study of this specific problem are shown numerically also to be valid for other exponentially ill‐conditioned BVPs and their corresponding time‐dependent equations. [ABSTRACT FROM PUBLISHER]

In this paper, an adaptive mesh method is employed to solve a class of singular boundary value problems. The approach is based on an area-preserving map and some mesh shape control in two-dimensional space. Two benchmark problems, which both involve singularities in physical domains, are tested. © 1998 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

A matrix-theoretic approach for the numerical solution of z J′ν (z) + HJν (z) = 0, an equation of the classical boundary value problem, for z ≠ 0 given complex parameters H and ν, and for ν given H and z ≠ 0, is proposed and analyzed. In each case, the problem can be reformulated as an eigenvalue problem for an infinite complex symmetric tridiagonal matrix posed in the classical Hilbert space of all square-summable infinite sequences. The paper justifies the approximate solution by truncation, giving extremely accurate asymptotic error estimates. Computer experiments confirm the theoretical results. © 1997 Scripta Technica, Inc. Electron Comm Jpn Pt 3, 80(7): 44–54, 1997 [ABSTRACT FROM AUTHOR]

In this paper, an H1-Galerkin mixed finite element method is proposed and analyzed for parabolic partial differential equations with nonselfadjoint elliptic parts. Compared to the standard H1-Galerkin procedure, C1-continuity for the approximating finite dimensional subspaces can be relaxed for the proposed method. Moreover, it is shown that the finite element approximations have the same rates of convergence as in the classical mixed method, but without LBB consistency condition and quasiuniformity requirement on the finite element mesh. Finally, a better rate of convergence for the flux in L2-norm is derived using a modified H1-Galerkin mixed method in two and three space dimensions, which confirms the findings in a single space variable and also improves upon the order of convergence of the classical mixed method under extra regularity assumptions on the exact solution. [ABSTRACT FROM AUTHOR]

This paper is concerned with several refinement techniques of finite element meshes for treating elliptic boundary value problems in domains with re-entrant edges and corners. A priori mesh grading is explained, and it is combined with the well-known adaptive finite element method. For two representative examples the numerically determined error norms are recorded, and the different strategies are compared. [ABSTRACT FROM AUTHOR]

With respect to a given boundary value problem, the corresponding conventional boundary integral equation is shown to yield non-equivalent solutions, which are dependent upon Poisson's ratio and geometry. In the paper a systematic method for establishing a necessary and sufficient boundary integral formulation has been proposed for two-dimensional elastostatic problems. Numerical analyses show that the conventional boundary integral equation yields incorrect results when the scale in the fundamental solution approaches a degenerate scale value. However, the results of the necessary and sufficient boundary integral equation are in good agreement with analytical solutions of the boundary value problem. [ABSTRACT FROM AUTHOR]