Accuracy of multivalue methods during change of time‐step size: Adams‐Moulton

Multivalue methods are a class of time‐stepping procedures for numerical solution of differential equations that progress to a new time level using the approximate solution for the function of interest and its derivatives at a single time level. The methods differ from multistep procedures, which make use of solutions to the differential equation at multiple time levels to advance to the new time level. Multistep methods are difficult to employ when a change in time‐step is desired, because the standard formulas (e.g., Adams‐Moulton or Gear) must be modified to accommodate the change. Multivalue methods seem to possess the desirable feature that the time‐step may be changed arbitrarily as one proceeds, since the solution proceeds from a single time level. However, in practice, changes in the time‐step introduce lower order errors or alter the coefficient in the truncation error term. Here, the multivalue Adams‐Moulton method is presented based on a general interpolation procedure. Modifications required to retain the high‐order accuracy of these methods during a change in time‐step are developed. Additionally, a formula for the unknown initial derivatives is presented. Finally, two examples are provided to illustrate the potential merit of the modification to the standard multivalue methods. © 2000 John Wiley & Sons, Inc. Numer Methods Partials Differential Eq 16: 312–326, 2000     

Numerical Solution of Arbitrary-Order Ordinary Differential and Integro-Differential Equations with Separated Boundary Conditions Using Optimal Control Technique

In this paper, a new method for solving arbitrary order ordinary differential equations and integro-differential equations of Fredholm and Volterra kind is presented. In the proposed method, these equations with separated boundary conditions are converted to a parametric optimization problem subject to algebraic constraints. Finally, control and state variables will be approximated by a Chebychev series. In this method, a new idea has been used, which offers us the ability of applying the mentioned method for almost all kinds of ordinary differential and integro-differential equations with different types of boundary conditions. The accuracy and efficiency of the proposed numerical technique have been illustrated by solving some test problems.     

A new numerical algorithm to solve fractional differential equations based on operational matrix of generalized hat functions

In this paper, we propose a new numerical algorithm for solving linear and non linear fractional differential equations based on our newly constructed integer order and fractional order generalized hat functions operational matrices of integration. The linear and nonlinear fractional order differential equations are transformed into a system of algebraic equations by these matrices and these algebraic equations are solved through known computational methods. Further some numerical examples are given to illustrate and establish the accuracy and reliability of the proposed algorithm. The results obtained, using the scheme presented here, are in full agreement with the analytical solutions and numerical results presented elsewhere.     

Parallel preconditioners and multigrid solvers for stochastic polynomial chaos discretizations of the diffusion equation at the large scale

This paper presents parallel preconditioners and multigrid solvers for solving linear systems of equations arising from stochastic polynomial chaos formulations of the diffusion equation with random coefficients. These preconditioners and solvers are extensions of the preconditioner developed in an earlier paper for strongly coupled systems of elliptic partial differential equations that are norm equivalent to systems that can be factored into an algebraic coupling component and a diagonal differential component. The first preconditioner, which is applied to the norm equivalent system, is obtained by sparsifying the inverse of the algebraic coupling component. This sparsification leads to an efficient method for solving these systems at the large scale, even for problems with large statistical variations in the random coefficients. An extension of this preconditioner leads to stand‐alone multigrid methods that can be applied directly to the actual system rather than to the norm equivalent system. These multigrid methods exploit the algebraic/differential factorization of the norm equivalent systems to produce variable‐decoupled systems on the coarse levels. Moreover, the structure of these methods allows easy software implementation through re‐use of robust high‐performance software such as the Hypre library package. Two‐grid matrix bounds will be established, and numerical results will be given. Copyright © 2015 John Wiley & Sons, Ltd.     

Numerical comparison of methods for solving linear differential equations of fractional order

In this article, we implement relatively new analytical techniques, the variational iteration method and the Adomian decomposition method, for solving linear differential equations of fractional order. The two methods in applied mathematics can be used as alternative methods for obtaining analytic and approximate solutions for different types of fractional differential equations. In these schemes, the solution takes the form of a convergent series with easily computable components. This paper will present a numerical comparison between the two methods and a conventional method such as the fractional difference method for solving linear differential equations of fractional order. The numerical results demonstrates that the new methods are quite accurate and readily implemented.     

Order and stability properties of explicit multivalue methods

Recent papers by Burrage and Moss [1] and Burrage [2] have studied in some detail the order properties of implicit multivalue (or general linear) methods and certain classes of these methods were proposed as being suitable for solving stiff differential equations. In this present paper we study the order and stability of explicit multivalue methods with a view to deriving new families of methods suitablefor solving non stiff problems.     

Jet spaces of varieties over differential and difference fields

We prove some new structural results on finite-dimensional differential algebraic varieties and difference algebraic varieties in characteristic zero, using elementary methods involving jet spaces. Some partial results and problems are given in the positive characteristic cases. The impact of these methods and results on proofs of the Mordell-Lang conjecture for function fields will also be discussed.     

Ordered differential fields

The paper is devoted to the study of a new algebraic structure - an oredred differential field. The structure is a differential field and an ordered field, the order being connected with a derivation. This paper is the first one in a series devoted to new algebraic methods in analysis. The essence of the methods is to exploit the connection between a linear order and a derivation for the study of function sets and operators on these sets.     

Discontinuous solutions of neutral delay differential equations

It is well known that the solutions of delay differential and implicit and explicit neutral delay differential equations (NDDEs) may have discontinuous derivatives, but it has not been appreciated (sufficiently) that the solutions of NDDEs—and, therefore, solutions of delay differential algebraic equations—need not be continuous. Numerical codes for solving differential equations, with or without retarded arguments, are generally based on the assumption that a solution is continuous. We illustrate and explain how the discontinuities arise, and present some methods to deal with these problems computationally. The investigation of a simple example is followed by a discussion of more general NDDEs and further mathematical detail.     

Compensating for order variation in mesh refinement for direct transcription methods II: computational experience

The numerical theory for Implicit Runge Kutta methods shows that there can be order reduction when these methods are applied to either stiff or differential algebraic equations. A previous paper introduced a way to try and compensate for this order reduction in designing mesh refinement strategies. This paper presents the results from a number of computational studies on the effectiveness of this approach. In addition, we present a new test problem which can be used to examine the efficiency of codes developed for a particular class of applications.     

Explicit solutions for differential equations

Advantages exist in use of the decomposition method [1, 2] for solutions of differential equations. Even for the trivial case of solution of first-order separable differential equations the decomposition solutions are more useful because of the resulting convenient computable explicit solutions. The same techniques and benefits apply to the algebraic equations obtained by transform methods in solving differential equations. A comparison is made also between solutions by integrating factor and decomposition, and it is shown that decomposition is an obvious recourse when an integrating factor is not available. To show advantages of the procedure, a differential equation solvable by several methods and involving a logarithmic nonlinearity is solved by Adomian's decomposition for comparisons. The decomposition method will also solve higher-order differential equations and partial differential equations with logarithmic or even composite nonlinearities [2] when the other methods fail.     

EXACT SOLUTIONS OF SOME FIFTH——ORDER NONLINEAR EQUATIONS

Abstract. To solve the nonlinear partial differential equations is changed into solving some alge-braic equations by using the function U to be expressed as linear independent functions. Thenew soliton and periodic solutions of some fifth-order nonlinear partial differential equations areobtained.     

Order conditions for General Linear Nyström methods

The purpose of this paper is to analyze the algebraic theory of order for the family of general linear Nyström (GLN) methods introduced in D’Ambrosio et al. (Numer. Algorithm 61(2), 331–349, 2012) with the aim to provide a general framework for the representation and analysis of numerical methods solving initial value problems based on second order ordinary differential equations (ODEs). Our investigation is carried out by suitably extending the theory of B-series for second order ODEs to the case of GLN methods, which leads to a general set of order conditions. This allows to recover the order conditions of numerical methods already known in the literature, but also to assess a general approach to study the order conditions of new methods, simply regarding them as GLN methods: the obtained results are indeed applied to both known and new methods for second order ODEs.     

数学机械化进展综述（迎接ICM2002特约文章）

本文介绍数学机械化理论,构造性代数几何,构造性微分代数几何,构造性实代数几何,方程求解,与几何自动推理的主要进展及其若干领域的应用,我们还提出了一些待解决的问题。     

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.     

A UNIFIED FRAMEWORK FOR THE CONSTRUCTION OF VARIOUS ALGEBRAIC MULTILEVEL PRECONDITIONING METHODS

1.IntroductionThediscretizationofmanysecondorderselfadjointellipticboundaryvalueproblemsbythefiniteelementmethodleadstolargesparsesystemsoflinearequationswithsymmetricpositivedefinite(SPD)coefficientmatrices.Fortheselinearsystems,algebraicmultilevelp...     

Numerical solutions of distributed order fractional differential equations in the time domain using the Müntz–Legendre wavelets approach

In this paper, a numerical method is presented to obtain and analyze the behavior of numerical solutions of distributed order fractional differential equations of the general form in the time domain with the Caputo fractional derivative. The suggested method is based on the Müntz–Legendre wavelet approximation. We derive a new operational vector for the Riemann–Liouville fractional integral of the Müntz–Legendre wavelets by using the Laplace transform method. Applying this operational vector and collocation method in our approach, the problem can be reduced to a system of linear and nonlinear algebraic equations. The arising system can be solved by the Newton method. Discussion on the error bound and convergence analysis for the proposed method is presented. Finally, seven test problems are considered to compare our results with other well‐known methods used for solving these problems. The results in the tabulated tables highlighted that the proposed method is an efficient mathematical tool for analyzing distributed order fractional differential equations of the general form.     

A survey of multilevel preconditioned iterative methods

We survey multilevel iterative methods applied for solving large sparse systems with matrices, which depend on a level parameter, such as arise by the discretization of boundary value problems for partial differential equations when successive refinements of an initial discretization mesh is used to construct a sequence of nested difference or finite element meshes.We discuss various two-level (two-grid) preconditioning techniques, including some for nonsymmetric problems. The generalization of these techniques to the multilevel case is a nontrivial task. We emphasize several ways this can be done including classical multigrid methods and a recently proposed algebraic multilevel preconditioning method. Conditions for which the methods have an optimal order of computational complexity are presented.On leave from the Institute of Mathematics, and Center for Informatics and Computer Technology, Bulgarian Academy of Sciences, Sofia, Bulgaria. The research of the second author reported here was partly supported by the Stichting Mathematisch Centrum, Amsterdam.     

Polynomial cost for solving IVP for high-index DAE

We show that the cost of solving initial value problems for high-index differential algebraic equations is polynomial in the number of digits of accuracy requested. The algorithm analyzed is built on a Taylor series method developed by Pryce for solving a general class of differential algebraic equations. The problem may be fully implicit, of arbitrarily high fixed index and contain derivatives of any order. We give estimates of the residual which are needed to design practical error control algorithms for differential algebraic equations. We show that adaptive meshes are always more efficient than non-adaptive meshes. Finally, we construct sufficiently smooth interpolants of the discrete solution. AMS subject classification (2000) 34A09, 65L80, 68Q25