Convergence of the Swartztrauber-Sweet Method for Parabolic Problems on a Disk with Singularities

In this article, we study the convergence analysis for the initial and boundary value problem of parabolic equations on a disk with singular solutions. It is assumed that the exact solution performs singular properties that its derivatives go to infinity at the boundary of the disk. We propose a fully implicit time-stepping numerical scheme. A stretching polynomial-like function with a parameter is used to construct a local grid refinement. Over the nonuniform partition, we combine the Swartztrauber-Sweet scheme and the backward Euler method in spatial and temporal discretization, respectively. We carry out convergence analysis and analyze the effects of the parameter. It is shown that our numerical scheme is of first order accuracy for temporal discretization and of almost second order accuracy for spatial discretization. Numerical experiments are performed to illustrate our analysis results and show that there exists an optimal value for the parameter to obtain a best approximate solution.     

A boundary value approach to the numerical solution of initial value problems by multistep methos †

A boundary value appraoch to the numerical solution of initial value problems by means of linear multistep methods is presented. This theory is based on the study of linear difference equations when their general solution is computed by imposing boundary conditions. All the main stability and convergence properties of the obtained methods are investigated abd compared to those of the classical multistep methods. Then, as an example, new itegration formulas, called extended trapezoidal rules, are derived. For any order they have the same stability properties (in the sense of the definitions given in this paper) of the trapezoidal rule, which is the first method in this class. Some numerical examples are presented to confirm the theoretical expectations and to allow us to trust a future code based on boundary value methods.     

Infinite Systems of Quasilinear Differential Difference Inequalities and Applications

The article deals with the initial boundary value problem for an infinite system of first order quasilinear functional differential equations. A comparison result concerning infinite systems of differential difference inequalities is proved. A function satisfying such inequalities is estimated by a solution of a suitable Cauchy problem for an ordinary functional differential system. The comparison result is used in an existence theorem and in the investigation of the stability of the numerical method of lines for the original problem. A theorem on the error estimate of the method is given. The infinite system of first order functional differential equations contains, as particular cases, equations with a deviated argument and integral differential equations of the Volterra type.     

Banach空间中非线性脉冲微分-积分方程的极值解

该文用单调迭代法结合一些新的技巧,研究了Ｂａｎａｃｈ空间中一阶非线性脉冲微分,积分方程初值问题极值解,在较宽松的条件下,建立了新的存在性定理,对文（３）在Ｌ３＝０的结果作了本质性的改进和推广。     

Novel numerical techniques based on Fokas transforms,for the solution of initial boundary value problems

The unified transform method of A. S. Fokas has led to important new developments, regarding the analysis and solution of various types of linear and nonlinear PDE problems. In this work we use these developments and obtain the solution of time-dependent problems in a straightforward manner and with such high accuracy that cannot be reached within reasonable time by use of the existing numerical methods. More specifically, an integral representation of the solution is obtained by use of the A. S. Fokas approach, which provides the value of the solution at any point, without requiring the solution of linear systems or any other calculation at intermediate time levels and without raising any stability problems. For instance, the solution of the initial boundary value problem with the non-homogeneous heat equation is obtained with accuracy 10⁻¹⁵, while the well-established Crank–Nicholson scheme requires 2048 time steps in order to reach a 10⁻⁸ accuracy.     

Fast Fourier-Galerkin methods for first-kind logarithmic-kernel integral equations on open arcs

We propose a fully discrete fast Fourier-Galerkin method for solving an integral equation of the first kind with a logarithmic kernel on a smooth open arc,which is a reformulation of the Dirichlet problem of the Laplace equation in the plane.The optimal convergence order and quasi-linear complexity order of the proposed method are established.A precondition is introduced.Combining this method with an efficient numerical integration algorithm for computing the single-layer potential defined on an open arc,we...     

Solving second order initial value problems by a hybrid multistep method without predictors

A three-step seventh order hybrid linear multistep method (HLMM) with three non-step points is proposed for the direct solution of the special second order initial value problems (IVPs) of the form y″ = f(x, y) with an extension to y″ = f(x, y, y′). The main method and additional methods are obtained from the same continuous scheme derived via interpolation and collocation procedures.The stability properties of the methods are discussed by expressing them as a one-step method in higher dimension. The methods are then applied in block form as simultaneous numerical integrators over non-overlapping intervals. Numerical results obtained using the proposed block form reveal that it is highly competitive with existing methods in the literature.     

Finite Order Rank-One Convex Envelopes and Computation of Microstructures with Laminates in Laminates

Finite order rank-one convex envelopes are introduced and it is shown that the i-th order laminated microstructures, or laminates in laminates, can be solved by any of the k-th order rank-one convex envelopes with k i. It is also shown that in finite element approximations of microstructures, replacing the non-quasiconvex potential energy density by its k-th order rank-one convex envelope, one can generally obtain sharper numerical results. Especially, for crystalline microstructures with laminates in laminates of order no greater than k + 1, numerical results with up to the computer precision can be obtained. Numerical examples on the first and second order rank-one convex envelopes for the Ericksen-James two-dimensional model for elastic crystals are given. A numerical example on finite element approximations of a crystalline microstructure by using the first order rank-one convex envelope and the periodic relaxation method is also presented. The methods turn out to be very successful for microstructures with laminates in laminates.     

A subdivision collocation method for solving two point boundary value problems of order three

In this paper, we propose a method for the numerical solution of self adjoint singularly perturbed third order boundary value problems in which the highest order derivative is multiplied by a small parameter $\varepsilon$. In this method, first we introduce the derivatives of two scale relations satisfied by the subdivision schemes. After that we use these derivatives to construct the subdivision collocation method for the numerical solution of singularly perturbed boundary value problems. Convergence of the subdivision collocation method is also discussed. Numerical examples are presented to illustrate the proposed method.     

A scheduling policy for adjusting economic lot quantities to a feasible solution

A heuristic scheduling policy is introduced for a multi-item, single-machine production facility. The scheduling policy uses the presumed optimal order quantities derived from solving an Economic Lot Size Problem and checks that the quantities obtain a feasible production schedule according to current inventory levels and expected demand rates. If not, the scheduling policy modifies the order quantities to achieve a possible solution without shortages. The scheduling policy is inspired by modification of the similar heuristic Dynamic Cycle Lengths Policy by Leachman and Gascon from 1988, 1991. The main characteristics of this scheduling policy are successive batches of the same item are treated explicitly, due to that it is quite possible that one item be manufactured several times before one other item is manufactured once more; the batches are ordered in increasing run-out time; if the existing situation creates stock-outs with ordinary order quantities, then the order quantities are decreased with a common scaling factor to try to prevent inventory shortages; in case the decrease of the order quantities changes expected run-out times, the batches are reordered after new run-out times; no filling up to an explicit inventory level is done, the filling up is done by the desirable order quantity; to prevent possible excess inventory the policy suggests time periods where no production should be performed. The scheduling policy contains no economical evaluation; this is supposed to be done when the order quantities are calculated, the policy prevents shortages and excess inventory. A numerical example illustrates the suggested scheduling policy. Finally, it is discussed as to how the policy can also take into account stochastic behaviour of the demand rates and compensate the schedule by applying appropriate safety times.     

Numerical method for a singularly perturbed convection-diffusion problem with delay

This paper deals with the singularly perturbed boundary value problem for a linear second-order delay differential equation. For the numerical solution of this problem, we use an exponentially fitted difference scheme on a uniform mesh which is accomplished by the method of integral identities with the use of exponential basis functions and interpolating quadrature rules with weight and remainder term in integral form. It is shown that one gets first order convergence in the discrete maximum norm, independently of the perturbation parameter. Numerical results are presented which illustrate the theoretical results.     

A note on the numerical solution of an identification problem for observing two‐phase flow in capillaries

In this paper, a right‐hand side identification problem for a parabolic equation with an overdetermined condition on an observation point is considered. A first and second order of accuracy difference schemes are constructed for obtaining approximate solutions of the problem that arises in two‐phase flow in capillaries. Stability estimates and numerical results are also established. Copyright © 2013 John Wiley & Sons, Ltd.     

A parameter-uniform numerical method for a Sobolev problem with initial layer

The present study is concerned with the numerical solution, using finite difference method of a one-dimensional initial-boundary value problem for a linear Sobolev or pseudo-parabolic equation with initial jump. In order to obtain an efficient method, to provide good approximations with independence of the perturbation parameter, we have developed a numerical method which combines a finite difference spatial discretization on uniform mesh and the implicit rule on Shishkin mesh(S-mesh) for the time variable. The fully discrete scheme is shown to be convergent of order two in space and of order one expect for a logarithmic factor in time, uniformly in the singular perturbation parameter. Some numerical results confirming the expected behavior of the method are shown.      

Simplified immersed interface methods for elliptic interface problems with straight interfaces

In this article, we propose simplified immersed interface methods for elliptic partial/ordinary differential equations with discontinuous coefficients across interfaces that are few isolated points in 1D, and straight lines in 2D. For one‐dimensional problems or two‐dimensional problems with circular interfaces, we propose a conservative second‐order finite difference scheme whose coefficient matrix is symmetric and definite. For two‐dimensional problems with straight interfaces, we first propose a conservative first‐order finite difference scheme, then use the Richardson extrapolation technique to get a second‐order method. In both cases, the finite difference coefficients are almost the same as those for regular problems. Error analysis is given along with numerical example. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 28: 188–203, 2012     

Taylor's Decomposition on two points for one‐dimensional Bratu problem

In this article, Taylor's Decomposition method is introduced for solving one‐dimensional Bratu problem. The numerical scheme is based on the application of the Taylor's decomposition to the corresponding first order differential equation system. The technique is illustrated with different eigenvalues and the results show that the method converges rapidly and hence approximate the exact solution very accurately for relatively large step sizes. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010     

一阶双曲问题间断有限元的后验误差分析

一阶双曲问题的有限元后验误差估计至今没有得到很好的解决.本文对d维区域上一阶双曲问题的k次间断有限元逼近提出了一种新的后验误差分析方法, 进而建立了间断有限元解在DG范数下(强于L₂范数)基于误差余量型的后验误差估计. 数值计算验证了本文理论分析的有效性. 本文方法也适用于其他变分问题有限元逼近的后验误差分析.     

Explicit One-Step P-Stable Methods for Second Order Periodic Initial Value Problems

In this paper, we present an explicit one-step method for solving periodic initial value problems of second order ordinary differential equations. The method is P-stable, and of first algebraic order and high phase-lag order. To improve the algebraic order, we give a composition second order scheme with the proposed method and its adjoint. We report some numerical results to illustrate the efficiency of our methods.     

Convergence analysis of a modified weak Galerkin finite element method for Signorini and obstacle problems

In this article, we apply a modified weak Galerkin method to solve variational inequality of the first kind which includes Signorini and obstacle problems. Optimal order a priori error estimates in the energy norm are derived. We also provide some numerical experiments to validate the theoretical results.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 1459–1474, 2017     

NUMERICAL SOLUTIONS OF DISCONTINUOUS BOUNDARY VALUE PROBLEMS FOR GENERAL ELLIPTIC COMPLEX EQUATIONS OF FIRST ORDER

1Formulati0nofDiscontinuousBoundaryValueProblemsLetDbeanN 1-connectedb0undeddomaininthez=x iy-planeCwiththeboundaryFEC:(0相似文献   

On the concept of solution for fractional differential equations with uncertainty

We consider a differential equation of fractional order with uncertainty and present the concept of solution. It extends, for example, the cases of first order ordinary differential equations and of differential equations with uncertainty. Some examples are presented.