Note on Kovacic's Algorithm

Algorithms exist to find Liouvillian solutions of second order homogeneous linear differential equations (Kovacic, 1986, Singer and Ulmer, 1993b). In this paper, we show how, by carefully combining the techniques of those algorithms, one can find the Liouvillian solutions of an irreducible second order linear differential equation by computing only rational solutions of some associated linear differential equations. The result is an easy-to-implement simplified version of the Kovacic algorithim, based as much on the computation of rational solutions of linear differential equations.     

How to solve linear differential equations: An outline

There are several definitions of closed form solutions to linear differential equations. In this paper, we look for the so-called Liouvillian solutions. Through examples, we give an overview of how the differential Galois theory leads to algorithms to find the Liouvillian solutions. We will outline the general ideas and results, but will give examples instead of proofs. This article was submitted by the author in English.     

Vanishing Moment Method and Moment Solutions for Fully Nonlinear Second Order Partial Differential Equations

This paper concerns with numerical approximations of solutions of fully nonlinear second order partial differential equations (PDEs). A new notion of weak solutions, called moment solutions, is introduced for fully nonlinear second order PDEs. Unlike viscosity solutions, moment solutions are defined by a constructive method, called the vanishing moment method, and hence, they can be readily computed by existing numerical methods such as finite difference, finite element, spectral Galerkin, and discontinuous Galerkin methods. The main idea of the proposed vanishing moment method is to approximate a fully nonlinear second order PDE by a higher order, in particular, a quasilinear fourth order PDE. We show by various numerical experiments the viability of the proposed vanishing moment method. All our numerical experiments show the convergence of the vanishing moment method, and they also show that moment solutions coincide with viscosity solutions whenever the latter exist. This work was partially supported by the NSF grants DMS-0410266 and DMS-0710831.     

Volterra series and geometric control theory

It is shown here that controlled differential equations which are analytic in the state and linear in the control have solutions which can be expanded in a Volterra series provided there is no finite escape time. The Volterra kernels are computed in terms of the power series expansion of the functions defining the differential equation. We also give necessary and sufficient conditions for a Volterra series to be realizable by a linear-analytic system. These conditions are particularly easy to test if the Volterra series is finite; a complete theory is worked out for this case. In the final section some applications are considered to singular control, multilinear realization theory, etc.     

Solving Difference Equations in Finite Terms

We define the notion of a Liouvillian sequence and show that the solution space of a difference equation with rational function coefficients has a basis of Liouvillian sequences iff the Galois group of the equation is solvable. Using this we give a procedure to determine the Liouvillian solutions of such a difference equation.     

Search for Liouvillian solutions of linear recurrence equations in the MAPLE computer algebra system

The paper considers implementation of the Singer-Hendriks algorithm in the MAPLE computer algebra system. The algorithm finds Liouvillian solutions of linear recurrence equations with coefficients in the form of rational functions.     

求解非线性悬梁方程行波解的变分算法Mountain Pass算法

§１．引言 经典的求解微分方程初一边值问题的算法，无不要求我们事先对解的某些性质有所了解．例如利用Ｒｕｎｇｅ－Ｋｕｔｔａ法解四阶常微分方程，我们至少需要知道解及其１—３阶导数的初值；又如广义牛顿法则对于初始点的选取有较高的要求，等等．如果事先对所求之解没有足够的了解，就给求解一般（特别是非线性）问题带来困难． １９７３年由 Ａｍｂｒｏｓｅｔｔｉ和 Ｒａｂｉｎｏｗｉｔｚ提出的 Ｍｏｕｎｔａｉｎ Ｐａｓｓ理论（一译“爬山理论”，又译“山径理论”）现己发展成为讨论非线性泛函临界值问题的一个重要方法之一．其几…     

Formal Solutions of Linear PDEs and Convex Polyhedra

The Newton polygon construction for ODEs, and Malgrange–Ramis polygon for partial differential equations in one variable are generalized in order to give an algorithm to find solutions of a linear partial differential equation at a singularity. The solutions found involve exponentials, logarithms and Laurent power series with exponents contained in a strongly convex cone.     

An Adaptive Version of the Boost by Majority Algorithm

We propose a new boosting algorithm. This boosting algorithm is an adaptive version of the boost by majority algorithm and combines bounded goals of the boost by majority algorithm with the adaptivity of AdaBoost.The method used for making boost-by-majority adaptive is to consider the limit in which each of the boosting iterations makes an infinitesimally small contribution to the process as a whole. This limit can be modeled using the differential equations that govern Brownian motion. The new boosting algorithm, named BrownBoost, is based on finding solutions to these differential equations.The paper describes two methods for finding approximate solutions to the differential equations. The first is a method that results in a provably polynomial time algorithm. The second method, based on the Newton-Raphson minimization procedure, is much more efficient in practice but is not known to be polynomial.     

Finite elements and characteristics applied to advection-diffusion equations

This paper deals with an algorithm for the solution of advection-diffusion equations based on the finite element method combined with the discretization of the total differential $\text{Dρ}\text{Dt}$. We give in the one dimensional case the finite difference analog of our Galerkin method. Diffusion of the scheme is studied in the two dimensional case by means of a classic example. We show that the scheme is stable, has no phase error and leads to simple problems at each time step.     

Normalized coprime factorizations for linear time-varying systems

In this paper we show that a finite dimensional linear time-varying continuous-time system admits normalized coprime factorizations if and only if it admits a stabilizable and detectable realization. We construct state-space formulas for these factorizations using the stabilizing solutions to standard Riccati differential equations. In the process, we give a simple proof that stabilizability and detectability are sufficient to ensure the existence of such solutions. Based on these results, and on recent advances in the theory of _∞ optimization, we present an algorithm to compute the distance between two systems in the gap metric.     

The numerical solution of fractional differential equations using the Volterra integral equation method based on thin plate splines

Finding the approximate solution of differential equations, including non-integer order derivatives, is one of the most important problems in numerical fractional calculus. The main idea of the current paper is to obtain a numerical scheme for solving fractional differential equations of the second order. To handle the method, we first convert these types of differential equations to linear fractional Volterra integral equations of the second kind. Afterward, the solutions of the mentioned Volterra integral equations are estimated using the discrete collocation method together with thin plate splines as a type of free-shape parameter radial basis functions. Since the scheme does not need any background meshes, it can be recognized as a meshless method. The proposed approach has a simple and computationally attractive algorithm. Error analysis is also studied for the presented method. Finally, the reliability and efficiency of the new technique are tested over several fractional differential equations and obtained results confirm the theoretical error estimates.

     

A Dual Consistent Finite Difference Method with Narrow Stencil Second Derivative Operators

We study the numerical solutions of time-dependent systems of partial differential equations, focusing on the implementation of boundary conditions. The numerical method considered is a finite difference scheme constructed by high order summation by parts operators, combined with a boundary procedure using penalties (SBP–SAT). Recently it was shown that SBP–SAT finite difference methods can yield superconvergent functional output if the boundary conditions are imposed such that the discretization is dual consistent. We generalize these results so that they include a broader range of boundary conditions and penalty parameters. The results are also generalized to hold for narrow-stencil second derivative operators. The derivations are supported by numerical experiments.     

Solving analysis and synthesis problems for a spatially two-dimensional distributed object represented with an infinite system of differential equations

We prove theorems that define an algorithm for passing from differential equations with partial derivatives with respect to two spatial variables and time to an infinite-dimensional system of ordinary differential equations in Cauchy form. We study the convergence of resulting solutions and show that it is possible to pass from an infinite system in Cauchy form to a finite one, which opens up the possibilities to use state space methods for controller design in distributed systems. Based on the quadratic quality criterion, we design a controller for the case when controlling influences are applied at the boundaries of the control object. We obtain the solution of this system analysis problem in the form of Fourier series with respect to spatial variables based on orthogonal systems of trigonometric functions and Bessel functions.     

Strong Convergence Analysis of the Stochastic Exponential Rosenbrock Scheme for the Finite Element Discretization of Semilinear SPDEs Driven by Multiplicative and Additive Noise

In this paper, we consider the numerical approximation of a general second order semilinear stochastic spartial differential equation (SPDE) driven by multiplicative and additive noise. Our main interest is on such SPDEs where the nonlinear part is stronger than the linear part also called stochastic reactive dominated transport equations. Most numerical techniques, including current stochastic exponential integrators lose their good stability properties on such equations. Using finite element for space discretization, we propose a new scheme appropriated on such equations, called stochastic exponential Rosenbrock scheme based on local linearization at every time step of the semi-discrete equation obtained after space discretization. We consider noise with finite trace and give a strong convergence proof of the new scheme toward the exact solution in the root-mean-square \(L^2\) norm. Numerical experiments to sustain theoretical results are provided.     

On a class of enclosure methods for initial value problems

This paper deals with the computation of interval enclosure for the solutions of initial value problems in systems of ordinary differential equations. We present a particular class of enclosure methods based on the Taylor series method. They are expressed in a single algorithm containing some parameters which may be chosen arbitrarily. However, a skillful choice is necessary in order to keep the widths of the enclosing intervals sufficiently small. We give some theoretical results about the behaviour of these widths.     

Mode shapes and frequencies by finite element method using consistent and lumped masses

The rate of convergence of the mode shapes and frequencies by the finite element method using consistent and lumped mass formulations has been established. Simple examples are given to demonstrate the results. It has been found that for a system of differential equations of second order such as the equations of equilibrium in terms of displacement in the theory of elasticity, membrane etc., a proper lumped mass formulation will not suffer any loss of rate of convergence utilizing simple elements. However, in the case of higher order differential equations or when the use of more complicated elements is required or desired, a consistent mass formulation often will provide a better rate of convergence.     

Nonstandard Local Discontinuous Galerkin Methods for Fully Nonlinear Second Order Elliptic and Parabolic Equations in High Dimensions

This paper is concerned with developing accurate and efficient numerical methods for fully nonlinear second order elliptic and parabolic partial differential equations (PDEs) in multiple spatial dimensions. It presents a general framework for constructing high order local discontinuous Galerkin (LDG) methods for approximating viscosity solutions of these fully nonlinear PDEs. The proposed LDG methods are natural extensions of a narrow-stencil finite difference framework recently proposed by the authors for approximating viscosity solutions. The idea of the methodology is to use multiple approximations of first and second order derivatives as a way to resolve the potential low regularity of the underlying viscosity solution. Consistency and generalized monotonicity properties are proposed that ensure the numerical operator approximates the differential operator. The resulting algebraic system has several linear equations coupled with only one nonlinear equation that is monotone in many of its arguments. The structure can be explored to design nonlinear solvers. This paper also presents and analyzes numerical results for several numerical test problems in two dimensions which are used to gauge the accuracy and efficiency of the proposed LDG methods.     

Energy control of the numerical solutions of an elastic oscillator

This paper points out how the energy may be used to select appropriate numerical schemes. For the purpose of illustration an elastic oscillator is considered. This oscillator is modeled by a system of differential equations (zero-dimensional oscillator) and by a system of partial differential equations (one-dimensional oscillator). For the case of free oscillator the energy is preserved in time by the exact solutions. It is shown here that the numerical schemes which preserve this property better give more accurate numerical solutions at a given time and for a given time step. The one-dimensional scheme applies equally well when shock waves are involved. The relation between the solutions obtained with the two models is also discussed.     

A comparison of finite-difference and finite-element methods for calculating free edge stresses in composites

This study considers the accuracy of the finite difference method in the solution of linear elasticity problems that involve either a stress discontinuity or a stress singularity. Solutions to three elasticity problems are discussed in detail: a semi-infinite plane subjected to a uniform load over a portion of its boundary; a bimetallic plate under uniform tensile stress; and a long, midplane symmetric, fiber-reinforced laminate subjected to uniform axial strain. Finite difference solutions to the three problems are compared with finite element solutions to corresponding problems. For the first problem a comparison with the exact solution is also made. The finite difference formulations for the three problems are based on second order finite difference formulas that provide for variable spacings in two perpendicular directions. Forward and backward difference formulas are used near boundaries where their use eliminates the need for fictitious grid points. Moreover, forward and backward finite difference formulas are used to enforced continuity of interlaminar stress components for the third problem. The study shows that the finite difference method employed in this investigation provides solutions to the three elasticity problems considered that are as accurate as the corresponding finite element solutions. Furthermore, the finite difference method appears to give a solution for the laminate problem that characterizes the stress distributions near an interface corner in a more realistic manner than the finite element method.