A Simple Solver for Linear Equations Containing Nonlinear Operators

This paper presents a simple equation solver. The solver finds solutions for sets of linear equations extended with several nonlinear operators, including integer division and modulus, sign extension, and bit slicing. The solver uses a new technique called {\em balancing}, which can eliminate some nonlinear operators from a set of equations before applying Gaussian elimination. The solver's principal advantages are its simplicity and its ability to handle some nonlinear operators, including nonlinear functions of more than one variable. The solver is part of an application generator that provides encoding and decoding of machine instructions based on equational specifications. The solver is presented not as pseudo code but as a literate program, which guarantees that the code shown in the paper is the same code that is actually used. Using real code exposes more detail than using pseudocode, but literate-programming techniques help manage the detail. The detail should benefit readers who want to implement their own solvers based on the techniques presented here.     

Solution of Polynomial Systems Derived from Differential Equations

Nonlinear two-point boundary value problems arise in numerous areas of application. The existence and number of solutions for various cases has been studied from a theoretical standpoint. These results generally rely upon growth conditions of the nonlinearity. However, in general, one cannot forecast how many solutions a boundary value problem may possess or even determine the existence of a solution. In recent years numerical continuation methods have been developed which permit the numerical approximation of all complex solutions of systems of polynomial equations. In this paper, numerical continuation methods are adapted to numerically calculate the solutions of finite difference discretizations of nonlinear two-point boundary value problems. The approach taken here is to perform a homotopy deformation to successively refine discretizations. In this way additional new solutions on finer meshes are obtained from solutions on coarser meshes. The complicating issue which the complex polynomial system setting introduces is that the number of solutions grows with the number of mesh points of the discretization. To counter this, the use of filters to limit the number of paths to be followed at each stage is considered.     

Expansions of Differential Operators and Nonsmooth Solutions of Differential Equations

Elements of F-space theory are presented, which made it possible to develop a promising procedure of extending differential operators.     

Symbolic Computation on Complex Polynomial Solution of Differential Equations

A symbolic computation scheme, based on the Lanczos τ-method, is proposed for obtaining exact polynomial solutions to some perturbed differential equations with suitable boundary conditions. The automated τ-method uses symbolic Faber polynomials as the perturbation terms for arbitrary circular sections of the complex plane and has advantages of avoiding rounding error and easy manipulation over the numerical counterpart. The method is illustrated by applying it to the modified Bessel function of the first kindI₀(z) and the quality of the approximation is discussed.     

Factoring and Solving Linear Partial Differential Equations

The problem of factoring a linear partial differential operator is studied. An algorithm is designed which allows one to factor an operator when its symbol is separable, and if in addition the operator has enough right factors then it is completely reducible. Since finding the space of solutions of a completely reducible operator reduces to the same for its right factors, we apply this approach and execute a complete analysis of factoring and solving a second-order operator in two independent variables. Some results on factoring third-order operators are exhibited.AMS Subject Classifications: 35A25, 35C05, 35G05.     

Interior Penalties for Summation-by-Parts Discretizations of Linear Second-Order Differential Equations

This work focuses on simultaneous approximation terms (SATs) for multidimensional summation-by-parts (SBP) discretizations of linear second-order partial differential equations with variable coefficients. Through the analysis of adjoint consistency and stability, we present several conditions on the SAT penalties for general operators, including those operators that do not have nodes on their boundary or do not correspond with a collocation discontinuous Galerkin method. Based on these conditions, we generalize the modified scheme of Bassi and Rebay and the symmetric interior penalty Galerkin method to SBP-SAT discretizations. Numerical experiments are carried out on unstructured grids with triangular elements to verify the theoretical results.     

多项式方程的迭代方法

基于韦达定理,给出了求解高次代数方程迭代方法,可同时迭代出所有实解。对其收敛性作了初步讨论。给出了实例以及MATLAB源程序.     

Minimal Realization of Completely Decidable Linear Differential Equations

For a linear system S in total differentials (Pfaff system), proposed was a procedure of constructing its minimal realization S ₀, that is, the Pfaff vector equation such that its phase space has the least dimensionality if the set of its output functions coincides with the family of the outputs of S.     

Polynomial Time-Marching for Three-Dimensional Wave Equations

As a continuation of the efficient and accurate polynomial interpolation time-marching technique in one-dimensional and two-dimensional cases, this paper proposes a method to extend it to three-dimensional problems. By stacking all two-dimensional (x, y) slice matrices along Z direction, three-dimensional derivative matrices are constructed so that the polynomial interpolation time-marching can be performed in the same way as one-dimensional and two-dimensional cases. Homogeneous Dirichlet and Neumann boundary conditions are also incorporated into the matrix operators in a similar way as in one- and two-dimensional problems. A simple numerical example of scalar wave propagation in a closed-cube has validated this extended method.     

Computation of All Polynomial Solutions of a Class of Nonlinear Differential Equations

A class of nonlinear differential equations is considered. We show that all their polynomial solutions can be computed in a systematic manner. Implementation of our method is also illustrated by means of Maple programs.     

The Control Problem for Stage-by-Stage Changing Linear Systems of Loaded Differential Equations

We consider the control problem for stage-by-stage changing linear differential equations and the optimal control problem with a quality criterion defined for the entire time interval. We formulate necessary and sufficient conditions for complete controllability and the existence of a program control and motion. We construct an explicit form of the control action for the control problem and propose a method for solving the optimal control problem. We give a solution of the control problem for a specific loaded system.     

Formal Solutions of Linear Ordinary Differential Equations Containing m-Hypergeometric Series

Let a linear homogeneous ordinary differential equation with polynomial coefficients over a field be given. For a singular point of the equation, the fundamental system of formal solutions that contain a finite number of power series with coefficients belonging to the algebraic extension of can be constructed by known algorithms. In this paper, an algorithm is suggested for construction of a space of formal solutions such that all series containing in these solutions have m-hypergeometric coefficients. The implementation of the algorithm in the computer algebra system Maple is discussed.     

Stability and Convergence Analysis of a Class of Continuous Piecewise Polynomial Approximations for Time-Fractional Differential Equations

Sparse regularization plays a central role in many recent developments in imaging and other related fields. However, it is still of limited use in numerical solvers for partial differential equations (PDEs). In this paper we investigate the use of \(\ell _1\) regularization to promote sparsity in the shock locations of hyperbolic PDEs. We develop an algorithm that uses a high order sparsifying transform which enables us to effectively resolve shocks while still maintaining stability. Our method does not require a shock tracking procedure nor any prior information about the number of shock locations. It is efficiently implemented using the alternating direction method of multipliers. We present our results on one and two dimensional examples using both finite difference and spectral methods as underlying PDE solvers.     

Approximate Solutions of Polynomial Equations

In this paper, we introduce “approximate solutions" to solve the following problem: given a polynomial F(x, y) over Q, where x represents an n -tuple of variables, can we find all the polynomials G(x) such that F(x, G(x)) is identically equal to a constant c in Q ? We have the following: let F(x, y) be a polynomial over Q and the degree of y in F(x, y) be n. Either there is a unique polynomial g(x)  ∈ Q [ x ], with its constant term equal to 0, such that F(x, y)  = ∑_j = 0ⁿc_j(y − g(x))^jfor some rational numbers c_j, hence, F(x, g(x)  + a)  ∈ Q for all a ∈ Q, or there are at most t distinct polynomials g₁(x),⋯ , g_t(x), t ≤ n, such that F(x, g_i(x))  ∈ Q for 1  ≤ i ≤ t. Suppose that F(x, y) is a polynomial of two variables. The polynomial g(x) for the first case, or g₁(x),⋯ , g_t(x) for the second case, are approximate solutions of F(x, y), respectively. There is also a polynomial time algorithm to find all of these approximate solutions. We then use Kronecker’s substitution to solve the case of F(x, y).     

Construction and Implementation of General Linear Methods for Ordinary Differential Equations: A Review

It it the purpose of this paper to review the results on the construction and implementation of diagonally implicit multistage integration methods for ordinary differential equations. The systematic approach to the construction of these methods with Runge–Kutta stability is described. The estimation of local discretization error for both explicit and implicit methods is discussed. The other implementations issues such as the construction of continuous extensions, stepsize and order changing strategy, and solving the systems of nonlinear equations which arise in implicit schemes are also addressed. The performance of experimental codes based on these methods is briefly discussed and compared with codes from Matlab ordinary differential equation (ODE) suite. The recent work on general linear methods with inherent Runge–Kutta stability is also briefly discussed     

Equivalence Classes,Symmetries and Solutions of Linear Third-order Differential Equations

 The subject of this article are third-order differential equations that may be linearized by a variable change. To this end, at first the equivalence classes of linear equations are completely described. Thereafter it is shown how they combine into symmetry classes that are determined by the various symmetry types. An algorithm is presented allowing it to transform linearizable equations by hyperexponential transformations into linear form from which solutions may be obtained more easily. Several examples are worked out in detail. Received February 18, 2002; revised May 10, 2002 Published online: October 24, 2002