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This paper deals with an adaptation of the Poincare-Lindstedt method for the determination of periodic orbits in three-dimensional nonlinear differential systems. We describe here a general symbolic algorithm to implement the method and apply it to compute periodic solutions in a three-dimensional Lotka-Volterra system modeling a chain food interaction. The sufficient conditions to make secular terms disappear from the approximate series solution are given in the paper.

In this paper, we propose a composite Laguerre spectral method for the nonlinear Fokker–Planck equations modelling the relaxation of fermion and boson gases. A composite Laguerre spectral scheme is constructed. Its convergence is proved. Numerical results show the efficiency of this approach and coincide well with theoretical analysis. Some results on the Laguerre approximation and techniques used in this paper are also applicable to other nonlinear problems on the whole line. Copyright © 2016 John Wiley & Sons, Ltd.

A discrete dynamical system named real - valued pendulum is presented in the paper. The system is developed for modeling of particles relocations on abstract graph with formalized rules of behavior (competition for space) and movement plans logistics. Logistics is given in a formal statement by the real numbers of the unit closed interval, presented in calculus with base equals to the graph power N. The central question is to study the behavior of the pendulum depending on its architecture. In the previous papers, with investigation of logistic pendulum, we have studied chaotic pendulum, where the particle plan for “tomorrow” is played “today”. A particular case of logistic pendulum, such that particle plans are time shifts of N−ary digital representation of single generating number, is called phase pendulum. In this paper, we are considering a dynamical system with generation of particles plans using a some mapping F, defined on the set of system states. Copyright © 2016 John Wiley & Sons, Ltd.

In this paper, we apply the boundary integral equation technique and the dual reciprocity boundary elements method (DRBEM) for the numerical solution of linear and nonlinear time-fractional partial differential equations (TFPDEs). The main aim of the present paper is to examine the applicability and efficiency of DRBEM for solving TFPDEs. We employ the time-stepping scheme to approximate the time derivative, and the method of linear radial basis functions is also used in the DRBEM technique. This method is improved by using a predictor–corrector scheme to overcome the nonlinearity that appears in the nonlinear problems under consideration. To confirm the accuracy of the new approach, several examples are presented. The convergence of the DRBEM is studied numerically by comparing the exact solutions of the problems under investigation. Copyright © 2015 John Wiley & Sons, Ltd.

This paper reports a modified homotopy perturbation algorithm, called the domain decomposition homotopy perturbation method (DDHPM), for solving two-point singular boundary value problems arising in science and engineering. The essence of the approach is to split the domain of the problem into a number of nonoverlapping subdomains. In each subdomain, a method based on a combination of HPM and integral equation formalism is implemented. The boundary condition at the right endpoint of each inner subdomain is established before deriving an iterative scheme for the components of the solution series. The accuracy and efficiency of the DDHPM are demonstrated by 4 examples (2 nonlinear and 2 linear). In comparison with the traditional HPM, the proposed domain decomposition HPM is highly accurate.

In this paper we provide a detailed convergence analysis for fully discrete second order (in both time and space) numerical schemes for nonlocal Allen-Cahn (nAC) and nonlocal Cahn-Hilliard (nCH) equations. The unconditional unique solvability and energy stability ensures $\ell^4$ stability. The convergence analysis for the nAC equation follows the standard procedure of consistency and stability estimate for the numerical error function. For the nCH equation, due to the complicated form of the nonlinear term, a careful expansion of its discrete gradient is undertaken and an $H^{-1}$ inner product estimate of this nonlinear numerical error is derived to establish convergence. In addition, an a-priori $W^{1,\infty}$ bound of the numerical solution at the discrete level is needed in the error estimate. Such a bound can be obtained by performing a higher order consistency analysis by using asymptotic expansions for the numerical solution. Following the technique originally proposed by Strang (e.g., 1964), instead of the standard comparison between the exact and numerical solutions, an error estimate between the numerical solution and the constructed approximate solution yields an $O( s^3 + h^4)$ convergence in $\ell^\infty (0, T; \ell^2)$ norm, which leads to the necessary bound under a standard constraint $s \le C h$. Here, we also prove convergence of the scheme in the maximum norm under the same constraint.

This paper presents 2 new classes of the Bessel functions on a compact domain [0,T] as generalized-tempered Bessel functions of the first- and second-kind which are denoted by GTBFs-1 and GTBFs-2. Two special cases corresponding to the GTBFs-1 and GTBFs-2 are considered. We first prove that these functions are as the solutions of 2 linear differential operators and then show that these operators are self-adjoint on suitable domains. Some interesting properties of these sets of functions such as orthogonality, completeness, fractional derivatives and integrals, recursive relations, asymptotic formulas, and so on are proved in detail. Finally, these functions are performed to approximate some functions and also to solve 3 practical differential equations of fractionalorders.

In this paper, we devote ourselves to establishing the unconditionally stable and absolutely convergent classical finite difference Crank-Nicholson (CN) implicit (CFDCNI) scheme and optimized finite difference CN-extrapolated implicit (OFDCNEI) scheme containing very few degrees of freedom but holding fully second-order accuracy for the two-dimensional viscoelastic wave via the proper orthogonal decomposition technique, analyzing the existence, stability, and convergence of the CFDCNI and OFDCNEI solutions, and using the numerical simulations to verify that the OFDCNEI scheme is far more superior than the CFDCNI scheme.

As far as the numerical solution of boundary value problems defined on an infinite interval is concerned, in this paper, we present a test problem for which the exact solution is known. Then we study an a posteriori estimator for the global error of a nonstandard finite difference scheme previously introduced by the authors. In particular, we show how Richardson extrapolation can be used to improve the numerical solution using the order of accuracy and numerical solutions from 2 nested quasi-uniform grids. We observe that if the grids are sufficiently fine, the Richardson error estimate gives an upper bound of the global error.

In this paper, we investigate a backward problem for a space-fractional partial differential equation. The main purpose is to propose a modified regularization method for the inverse problem. The existence and the uniqueness for the modified regularized solution are proved. To derive the gradient of the optimization functional, the variational adjoint method is introduced, and hence, the unknown initial value is reconstructed. Finally, numerical examples are provided to show the effectiveness of the proposed algorithm. Copyright © 2017 John Wiley & Sons, Ltd.

In this paper, a second-order fast explicit operator splitting method is proposed to solve the mass-conserving Allen–Cahn equation with a space–time-dependent Lagrange multiplier. The space–time-dependent Lagrange multiplier can preserve the volume of the system and keep small features. Moreover, we analyze the discrete maximum principle and the convergence rate of the fast explicit operator splitting method. The proposed numerical scheme is of spectral accuracy in space and of second-order accuracy in time, which greatly improves the computational efficiency. Numerical experiments are presented to confirm the accuracy, efficiency, mass conservation, and stability of the proposed method. Copyright © 2017 John Wiley & Sons, Ltd.

In this paper, we first formulate a linear quasi-static poroelastic shell model of Naghdi's type. The model is given in three unknowns: displacement u˜ of the middle surface, infinitesimal rotation ω˜ of the cross section of the shell, and the pressure π. The model has the structure of the quasi-static Biot's system and can be seen as a system of the shell equation with pressure term as forcing and the parabolic type equation for the pressure with divergence of the filtration velocity as forcing term. On the basis of the ideas of the operator splitting methods, we formulate two sequences of approximate solutions, corresponding to ‘undrained split’ and ‘fixed stress split’ methods. We show that these sequences converge to the solution of the poroelastic shell model. Therefore, the iterations constitute two numerical methods for the model. Moreover, both methods are optimized in a certain sense producing schemes with smallest contraction coefficient and thus faster convergence rates. Also, these convergences imply existence of solutions for the model. Copyright © 2017 John Wiley & Sons, Ltd.

In this paper, we consider a Riesz–Feller space-fractional backward diffusion problem with a time-dependent coefficient ut(x,t)=l(t)xDθγu(x,t)+f(x,t),(x,t)∈R×(0,T). We show that this problem is ill-posed; therefore, we propose a convolution regularization method to solve it. New error estimates for the regularized solution are given under a priori and a posteriori parameter choice rules, respectively. Copyright © 2016 John Wiley & Sons, Ltd.

In this paper, a numerical solution of fractional partial differential equations (FPDEs) for electromagnetic waves in dielectric media will be discussed. For the solution of FPDEs, we developed a numerical collocation method using an algorithm based on two-dimensional shifted Legendre polynomials approximation, which is proposed for electromagnetic waves in dielectric media. By implementing the partial Riemann–Liouville fractional derivative operators, two-dimensional shifted Legendre polynomials approximation and its operational matrix along with collocation method are used to convert FPDEs first into weakly singular fractional partial integro-differential equations and then converted weakly singular fractional partial integro-differential equations into system of algebraic equation. Some results concerning the convergence analysis and error analysis are obtained. Illustrative examples are included to demonstrate the validity and applicability of the technique. Copyright © 2017 John Wiley & Sons, Ltd.

This paper concerns new continuum phenomenological model for epitaxial thin-film growth with three different forms of the Ehrlich–Schwoebel current. Two of these forms were first proposed by Politi and Villain 1996 and then studied by Evans, Thiel, and Bartelt 2006. The other one is completely new. Energy structure and properties of the new model are studied. Following the techniques used in Li and Liu 2003, we present rigorous analysis of the well-posedness, regularity, and time stability for the new model. We also studied both the global and the local behavior of the surface roughness in the growth process. By using a convex–concave time-splitting scheme, one can naturally build unconditionally stable semi-implicit numerical discretizations with linear implicit parts, which is much easier to implement than conventional models requiring nonlinear implicit parts. Despite this fundamental difference in the model, numerical experiments show that the nonlinear morphological instability of the new model agrees well with results of other models published before which indicates that the new model correctly captures the essential morphological states in the thin-film growth process. Copyright © 2017 John Wiley & Sons, Ltd.

This paper deals with an inverse problem of determining the diffusion coefficient, spacewise dependent source term, and the initial value simultaneously for a one-dimensional heat equation based on the boundary control, boundary measurement, and temperature distribution at a given single instant in time. By a Dirichlet series representation for the boundary observation, the identification of the diffusion coefficient and initial value can be transformed into a spectral estimation problem of an exponential series with measurement error, which is solved by the matrix pencil method. For the identification of the source term, a finite difference approximation method in conjunction with the truncated singular value decomposition is adopted, where the regularization parameter is determined by the generalized cross-validation criterion. Numerical simulations are performed to verify the result of the proposed algorithm. Copyright © 2016 John Wiley & Sons, Ltd.

This paper presents a computational technique for the solution of the nonlinear mixed Volterra–Fredholm integral equations of the second kind. Using the properties of three-dimensional modification of hat functions, these are types of equations to a nonlinear system of algebraic equations. Also, convergence results and error analysis are discussed. The efficiency and accuracy of the proposed method is illustrated by numerical examples. Copyright © 2016 John Wiley & Sons, Ltd.

The evaluation of the diagonal of matrix functions arises in many applications and an efficient approximation of it, without estimating the whole matrix f(A), would be useful. In the present paper, we compare and analyze the performance of three numerical methods adjusted to attain the estimation of the diagonal of matrix functions f(A), where A∈Rp×p is a symmetric matrix and f a suitable function. The applied numerical methods are based on extrapolation and Gaussian quadrature rules. Various numerical results illustrating the effectiveness of these methods and insightful remarks about their complexity and accuracy are demonstrated. Copyright © 2016 John Wiley & Sons, Ltd.

In this paper, we propose a space-time spectral method for solving a class of time fractional convection diffusion equations. Because both fractional derivative and spectral method have global characteristics in bounded domains, we propose a space-time spectral-Galerkin method. The convergence result of the method is proved by providing a priori error estimate. Numerical results further confirm the expected convergence rate and illustrate the versatility of our method. Copyright © 2016 John Wiley & Sons, Ltd.

Our work is devoted to an inverse problem for three-dimensional parabolic partial differential equations. When the surface temperature data are given, the problem of reconstructing the heat flux and the source term is investigated. There are two main contributions of this paper. First, an adjoint problem approach is used for analysis of the Frechet gradient of the cost functional. Second, an improved conjugate gradient method is proposed to solve this problem. Based on Lipschitz continuity of the gradient, the convergence analysis of the conjugate gradient algorithm is studied. Copyright © 2016 John Wiley & Sons, Ltd.

In this paper, we consider the conductivity problem with piecewise-constant conductivity and Robin-type boundary condition on the interface of discontinuity. When the quantity of interest is the jump of the conductivity, we perform a local stability estimate for a parameterized non-monotone family of domains. We give also a quantitative stability result of local optimal solution with respect to a perturbation of the Robin parameter. In order to find an optimal solution, we propose a Kohn–Vogelius-type cost functional over a class of admissible domains subject to two boundary values problems. The analysis of the stability involves the computation of first-order and second-order shape derivative of the proposed cost functional, which is performed rigorously by means of shape-Lagrangian formulation without using the shape sensitivity of the states variables. Copyright © 2016 The Author. Mathematical Methods in the Applied Sciences Published by John Wiley & Sons Ltd.

In this paper, we are concerned with the backward problem of reconstructing the initial condition of a time-fractional diffusion equation from interior measurements. We establish uniqueness results and provide stability analysis. Our method is based on the eigenfunction expansion of the forward solution and the Tikhonov regularization to tackle the ill-posedness issue of the underlying inverse problem. Some numerical examples are included to illustrate the effectiveness of the proposed approach. Copyright © 2016 John Wiley & Sons, Ltd.

The aim of this paper is to propose mixed two-grid finite difference methods to obtain the numerical solution of the one-dimensional and two-dimensional Fitzhugh–Nagumo equations. The finite difference equations at all interior grid points form a large-sparse linear system, which needs to be solved efficiently. The solution cost of this sparse linear system usually dominates the total cost of solving the discretized partial differential equation. The proposed method is based on applying a family of finite difference methods for discretizing the spatial and time derivatives. The obtained system has been solved by two-grid method, where the two-grid method is used for solving the large-sparse linear systems. Also, in the proposed method, the spectral radius with local Fourier analysis is calculated for different values of h and Δt. The numerical examples show the efficiency of this algorithm for solving the one-dimensional and two-dimensional Fitzhugh–Nagumo equations. Copyright © 2016 John Wiley & Sons, Ltd.

In this paper, we develop an hp-adaptive procedure for the numerical solution of general, semilinear elliptic boundary value problems in 1d, with possible singular perturbations. Our approach combines both a prediction-type adaptive Newton method and an hp-version adaptive finite element discretization (based on a robust a posteriori residual analysis), thereby leading to a fully hp-adaptive Newton–Galerkin scheme. Numerical experiments underline the robustness and reliability of the proposed approach for various examples. Copyright © 2016 John Wiley & Sons, Ltd.

In this paper, boundary integral formulations for a time-harmonic acoustic scattering-resonance problem are analyzed. The eigenvalues of eigenvalue problems resulting from boundary integral formulations for scattering-resonance problems split in general into two parts. One part consists of scattering-resonances, and the other one corresponds to eigenvalues of some Laplacian eigenvalue problem for the interior of the scatterer. The proposed combined boundary integral formulations enable a better separation of the unwanted spectrum from the scattering-resonances, which allows in practical computations a reliable and simple identification of the scattering-resonances in particular for non-convex domains. The convergence of conforming Galerkin boundary element approximations for the combined boundary integral formulations of the resonance problem is shown in canonical trace spaces. Numerical experiments confirm the theoretical results. Copyright © 2016 John Wiley & Sons, Ltd.

A dimension splitting method (DSM) with Crank–Nicolson time discrete strategy for a three-dimensional heat equation is proposed. The basic idea is to simulate the three-Dimensional problem by numerically solving a series of two-dimensional problems in parallel fashion. Convergence and error estimation for the DSM scheme are derived in the paper. Numerical experiments demonstrate the feasibility and efficiency of the DSM scheme. Copyright © 2016 John Wiley & Sons, Ltd.

In this paper, we apply the dual reciprocity boundary elements method for the numerical solution of two-dimensional linear and nonlinear time-fractional modified anomalous subdiffusion equations and time-fractional convection–diffusion equation. The fractional derivative of problems is described in the Riemann–Liouville and Caputo senses. We employ the linear radial basis function for interpolation of the nonlinear, inhomogeneous and time derivative terms. This method is improved by using a predictor–corrector scheme to overcome the nonlinearity which appears in the nonlinear problems under consideration. The accuracy and efficiency of the proposed schemes are checked by five test problems. The proposed method is employed for solving some examples in two dimensions on unit square and also in complex regions to demonstrate the efficiency of the new technique. Copyright © 2016 John Wiley & Sons, Ltd.

In this paper, we develop a spectral method for mixed inhomogeneous Dirichlet/Neumann/Robin boundary value problems defined on rectangle. Some results on two-dimensional Legendre approximation in Jacobi-weighted Sobolev space are established. As examples of applications, spectral schemes are provided for two model problems with mixed inhomogeneous boundary conditions. The spectral accuracy in space of proposed algorithms are proved. Efficient implementations are presented. Numerical results demonstrate their high accuracy and confirm the theoretical analysis well. Copyright © 2016 John Wiley & Sons, Ltd.

In this paper, we consider a high order finite volume approximation of one-dimensional nonlocal reactive flows of parabolic type. The method is obtained by discretizing in space by arbitrary order vertex-centered finite volumes, followed by a modified Simpson quadrature scheme for the time stepping. Compared to the existed finite volume methods, this new finite volume scheme could achieve the desired accuracy with less data storage by employing higher-order trial spaces. The finite volume approximations are proved to possess optimal order convergence rates in the H1-norm and L2-norm, which are also confirmed by numerical tests. Copyright © 2015 John Wiley & Sons, Ltd.

This paper presents a superconvergence result of the stationary natural convection equations. The superconvergence result is obtained by applying a coarse mesh projection for finite element approximation. This projection method is actually a postprocessing procedure that constructs a new approximation based on a high order polynomial on coarse mesh, under some regularity assumption. Finally, numerical experiment is presented for testing, which confirms the theoretic analysis. Copyright © 2015 John Wiley & Sons, Ltd.

In this paper, we consider a class of parabolic partial differential equations with a time delay. The first model equation is the mixed problems for scalar generalized diffusion equation with a delay, whereas the second model equation is a delayed reaction-diffusion equation. Both of these models have inherent complex nature because of which their analytical solutions are hardly obtainable, and therefore, one has to seek numerical treatments for their approximate solutions. To this end, we develop a fitted Galerkin spectral method for solving this problem. We derive optimal error estimates based on weak formulations for the fully discrete problems. Some numerical experiments are also provided at the end. Copyright © 2015 John Wiley & Sons, Ltd.

Our aim in this paper is to study, in term of finite dimensional exponential attractors, the Willmore regularization, (depending on a small regularization parameter β > 0), of two phase-field equations, namely, the Allen–Cahn and the Cahn–Hilliard equations. In both cases, we construct robust families of exponential attractors, that is, attractors that are continuous with respect to the perturbation parameter. Copyright © 2015 John Wiley & Sons, Ltd.

In this paper, a high-order accurate numerical method for two-dimensional semilinear parabolic equations is presented. We apply a Galerkin–Legendre spectral method for discretizing spatial derivatives and a spectral collocation method for the time integration of the resulting nonlinear system of ordinary differential equations. Our formulation can be made arbitrarily high-order accurate in both space and time. Optimal a priori error bound is derived in the L2-norm for the semidiscrete formulation. Extensive numerical results are presented to demonstrate the convergence property of the method, show our formulation have spectrally accurate in both space and time. John Wiley & Sons, Ltd.

In this paper, a new rational approximation based on a rational interpolation and collocation method is proposed for the solutions of generalized pantograph equations. A comprehensive error analysis is provided. The first part of the error analysis gives an upper bound for the absolute error. The second part is based on residual error procedure that estimates the absolute error. Some numerical examples are given to illustrate the method. The theoretical results support the numerical results. Copyright © 2015 John Wiley & Sons, Ltd.

This paper presents a shifted fractional-order Jacobi orthogonal function (SFJF) based on the definition of the classical Jacobi polynomial. A new fractional integral operational matrix of the SFJF is presented and derived. We propose the spectral Tau method, in conjunction with the operational matrices of the Riemann–Liouville fractional integral for SFJF and derivative for Jacobi polynomial, to solve a class of time-fractional partial differential equations with variable coefficients. In this algorithm, the approximate solution is expanded by means of both SFJFs for temporal discretization and Jacobi polynomials for spatial discretization. The proposed tau scheme, both in temporal and spatial discretizations, successfully reduced such problem into a system of algebraic equations, which is far easier to be solved. Numerical results are provided to demonstrate the high accuracy and superiority of the proposed algorithm over existing ones. Copyright © 2015 John Wiley & Sons, Ltd.

This paper deals with optimal control problems constrained by linear elliptic partial differential equations. The case where the right-hand side of the Neumann boundary is controlled, is studied. The variational discretization concept for these problems is applied, and discretization error estimates are derived. On polyhedral domains, one has to deal with edge and corner singularities, which reduce the convergence rate of the discrete solutions, that is, one cannot expect convergence order two for linear finite elements on quasi-uniform meshes in general. As a remedy, a local mesh refinement strategy is presented, and a priori bounds for the refinement parameters are derived such that convergence with optimal rate is guaranteed. As a by-product, finite element error estimates in the H1(Ω)-norm, L2(Ω)-norm and L2(Γ)-norm for the boundary value problem are obtained, where the latter one turned out to be the main challenge. Copyright © 2015 John Wiley & Sons, Ltd.

The purpose of the present paper is to propose an efficient numerical method for solving the differential equations of Bratu-type with fractional order in reproducing kernel Hilbert space. The exact solution is calculated in the form of a convergent series with easily computable components. Finally, some examples are given to illustrate the efficiency and applicability of the method. Copyright © 2015 John Wiley & Sons, Ltd.

In this paper, we deal with the system that has solutions and the periodicity character of the following systems of rational difference equations with order three with initial conditions x−2,x−1,x0,y−2,y−1, and y0 that are arbitrary nonzero real numbers. Some numerical examples will be given to illustrate our results. Copyright © 2015 John Wiley & Sons, Ltd.

This paper is concerned with the oscillation of numerical solution for the Nicholson's blowflies model. Using two kinds of θ-methods, namely, the linear θ-method and the one-leg θ-method, several conditions under which the numerical solution oscillates are derived. Moreover, it is shown that every non-oscillatory numerical solution tends to equilibrium point of the original continuous-time model. Finally, numerical experiments are provided to illustrate the analytical results. Copyright © 2015 John Wiley & Sons, Ltd.

Mathematical models are of great value in epidemiology to help understand the dynamics of the various infectious diseases, as well as in the conception of effective control strategies. The classical approach is to use differential equations to describe, in a quantitative manner, the spread of diseases within a particular population. An alternative approach is to represent each individual in the population as a string or vector of characteristic data and simulate the contagion and recovery processes by computational means. This type of model, referred in the literature as MBI (models based on individuals), has the advantage of being flexible as the characteristics of each individual can be quite complex, involving, for instance, age, sex, pre-existing health conditions, environmental factors, social habits, etc. However, when it comes to simulations involving large populations, MBI may require a large computational effort in terms of memory storage and processing time. In order to cope with the problem of heavy computational effort, this paper proposes a parallel implementation of MBI using a graphics processor unit compatible with CUDA. It was found that, even in the case of a simple susceptible–infected–recovered model, the computational gains in terms of processing time are significant. Copyright © 2015 John Wiley & Sons, Ltd.

The problem of determining the manner in which an incoming acoustic wave is scattered by an elastic body immersed in a fluid is one of the central importance in detecting and identifying submerged objects. The problem is generally referred to as a fluid-structure interaction and is mathematically formulated as a time-dependent transmission problem. In this paper, we consider a typical fluid-structure interaction problem by using a coupling procedure that reduces the problem to a nonlocal initial-boundary problem in the elastic body with a system of integral equations on the interface between the domains occupied by the elastic body and the fluid. We analyze this nonlocal problem by the Lubich approach via the Laplace transform, an essential feature of which is that it works directly on data in the time domain rather than in the transformed domain. Our results may serve as a mathematical foundation for treating time-dependent fluid-structure interaction problems by convolution quadrature coupling of FEM and BEM. Copyright © 2015 John Wiley & Sons, Ltd.

In this paper, we consider the shape inverse problem of a body immersed in the incompressible fluid governed by thermodynamic equations. By applying the domain derivative method, we obtain the explicit representation of the derivative of solution with respect to the boundary, which plays an important role in the inverse design framework. Moreover, according to the boundary parametrization technique, we present a regularized Gauss–Newton algorithm for the shape reconstruction problem. Finally, numerical examples indicate the proposed algorithm is feasible and effective for the low Reynolds numbers. Copyright © 2017 John Wiley & Sons, Ltd.

In this paper, the Schrodinger operator where the potential q(x) is single-well with transition point is considered. Suppose {λn}n ≥ 1 is the set of eigenvalues for the previous Schrodinger operator and the potential q(x) possesses an additional condition, we show that the first two eigenvalues λ2 and λ1 satisfy Equality holds if and only if q(x) is a constant. Copyright © 2014 John Wiley & Sons, Ltd.

In this paper, we give error estimates for quadrature rules with maximal trigonometric degree of exactness with respect to an even weight function on ( − π,π) for integrand analytic in a certain domain of complex plane. Copyright © 2013 John Wiley & Sons, Ltd.

In this paper, the authors established a unified framework for deriving and analyzing a posteriori error estimators for finite volume methods for the Stokes equations. The a posteriori error estimators are residual based and are applicable to various finite volume methods for the Stokes equations. In particular, the unified theoretical analysis works well for finite volume schemes arising from using trial functions of conforming, nonconforming, and discontinuous finite element functions, yielding new results that are not seen in the existing literature. Copyright © 2013 John Wiley & Sons, Ltd.

In this paper, we investigate a class of generalized Burgers–Huxley equation by employing the bifurcation method of planar dynamical systems. Firstly, we reduce the equation to a planar system via the traveling wave solution ansatz; then by computing the singular point quantities, we obtain the conditions of integrability and determine the existence of one stable limit cycle from Hopf bifurcation in the corresponding planar system. From this, some new exact solutions and a special periodic traveling wave solution, which is isolated as a limit, are obtained. Copyright © 2016 John Wiley & Sons, Ltd.

A Newton method is presented for the approximate solution of the inverse problem to determine the shape of a sound-soft or perfectly conducting arc from a knowledge of the far-field pattern for the scattering of time-harmonic plane waves. Frechet differentiability with respect to the boundary is shown for the far-field operator, which for a fixed incident wave maps the boundary arc onto the far-field pattern of the scattered wave. For the sake of completeness, the first part of the paper gives a short outline on the corresponding direct problem via an integral equation method including the numerical solution.