High order approximations using spline-based differential quadrature method: Implementation to the multi-dimensional PDEs

A new differential quadrature method based on cubic B-spline is developed for the numerical solution of differential equations. In order to develop the new approach, the B-spline basis functions are used on the grid and midpoints of a uniform partition. Some error bounds are obtained by help of cubic spline collocation, which show that the method in its classic form is second order convergent. In order to derive higher accuracy, high order perturbations of the problem are generated and applied to construct the numerical algorithm. A new fourth order method is developed for the numerical solution of systems of second order ordinary differential equations. By solving some test problems, the performance of the proposed methods is examined. Also the implementation of the method for multi-dimensional time dependent partial differential equations is presented. The stability of the proposed methods is examined via matrix analysis. To demonstrate the applicability of the algorithms, we solve the 2D and 3D coupled Burgers’ equations and 2D sine-Gordon equation as test problems. Also the coefficient matrix of the methods for multi-dimensional problems is described to analyze the stability.     

Generalization of the simplest equation method for nonlinear non-autonomous differential equations

It is known that the simplest equation method is applied for finding exact solutions of autonomous nonlinear differential equations. In this paper we extend this method for finding exact solutions of non-autonomous nonlinear differential equations (DEs). We applied the generalized approach to look for exact special solutions of three Painlevé equations. As ODE of lower order than Painlevé equations the Riccati equation is taken. The obtained exact special solutions are expressed in terms of the special functions defined by linear ODEs of the second order.     

一类二阶变系数线性微分方程的积分因子解法

通过寻求积分因子f1(x),f2(x),求解一类二阶线性微分方程,包括二阶常系数线性微分方程和二阶Euler方程.     

Symmetry analysis and some exact solutions of cylindrically symmetric null fields in general relativity

The symmetry reduction method based on the Fréchet derivative of the differential operators is applied to investigate symmetries of the Field equations in general relativity corresponding to cylindrically symmetric space–time, that is a coupled system of nonlinear partial differential equations of second order. More specifically, this technique yields invariant transformation that reduce the given system of partial differential equations to a system of nonlinear ordinary differential equations. Some of the reduced systems are further studied for exact solutions.     

一类二阶抛物型方程初边值问题解的存在定理

本文研究了一类二阶非线性抛物型方程解的存在唯一性问题.利用非线性分析中的吸引盆理论和同胚理论,获得了相应的二阶非线性抛物型方程初边值问题解的大范围存在唯一性定理.     

Simplified Order Conditions of Some Canonical Difference Schemes

1.IntroductionIn[5-8],explicitcanonicaldifferenceschemesuptothefourth0rderarec0n-structedforseparableHamiltoniansystems(i-e.,systernswiththeHamilt0nianfunctionH(P,q)=U(P)+V(q)).Butunfortunatelywecannotfindthegeneral0rderc0nditi0nsf0rthismethodwhethersnalgebraicorLiemethodisusedtogetordercondition8forsomeschemeofadefinitestagenumber.InthispaPer)wewilluseP-seriesilltroducedin[4]andtreemethodologyusedbySanz-Sernain[2ltogetthegeneral0rderc0ndi-tionsfortheexplicitcanonicalmethodandthensimplify…     

二阶常微初值问题的单步格式

讨论二阶常微分方程初值问题utt+au=f,u(0)=u0，ut(0)=v0 的一种单步格式，采用u及v=ut为未知量，计算简单. 证明了此格式的稳定性及对u，v皆有二阶精度. 此格式可用于双曲问题.     

On preserving long-time features of a linear stochastic oscillator

A method for the numerical solution of stochastic differential equations is presented. The method has mean-square order equal to 1/2 when it is applied to a general stochastic differential equation and equal to 1 if the equation has additive noise. In addition, it is shown that the method captures some long-time properties of a linear stochastic oscillator: It reproduces exactly the growth rate of the second moment and the oscillation property of the solution. AMS subject classification (2000)  60H10, 34F05, 65U05, 60K40     

Equivariant degree of convex-valued maps applied to set-valued BVP

An equivariant degree is defined for equivariant completely continuous multivalued vector fields with compact convex values. Then it is applied to obtain a result on existence of solutions to a second order BVP for differential inclusions carrying some symmetries.     

On a recurrence method for solving a singularly perturbed Cauchy problem for a second order equation

In the present article, the method of deviating argument is applied to solving a singularly perturbed Cauchy problem for an ordinary differential equation of the second order with variable coefficients.     

Three dimensional elastodynamics of 2D quasicrystals: The derivation of the time-dependent fundamental solution

The time-dependent differential equations of elasticity for 2D quasicrystals with general structure of anisotropy (dodecagonal, octagonal, decagonal, pentagonal, hexagonal, triclinic) are considered in the paper. These equations are written in the form of a vector partial differential equation of the second order with symmetric matrix coefficients. The fundamental solution (matrix) is defined for this vector partial differential equation. A new method of the numerical computation of values of the fundamental solution is suggested. This method consists of the following: the Fourier transform with respect to space variables is applied to vector equation for the fundamental solution. The obtained vector ordinary differential equation has matrix coefficients depending on Fourier parameters. Using the matrix computations a solution of the vector ordinary differential equation is numerically computed. Finally, applying the inverse Fourier transform numerically we find the values of the fundamental solution. Computational examples confirm the robustness of the suggested method for 2D quasicrystals with arbitrary type of anisotropy.     

The 3/2th and 2nd order asymptotic efficiency of maximum probability estimators in non-regular cases

In this paper we consider the estimation problem on independent and identically distributed observations from a location parameter family generated by a density which is positive and symmetric on a finite interval, with a jump and a nonnegative right differential coefficient at the left endpoit. It is shown that the maximum probability estimator (MPE) is 3/2th order two-sided asymptotically efficient at a point in the sense that it has the most concentration probability around the true parameter at the point in the class of 3/2th order asymptotically median unbiased (AMU) estimators only when the right differential coefficient vanishes at the left endpoint. The second order upper bound for the concentration probability of second order AMU estimators is also given. Further, it is shown that the MPE is second order two-sided asymptotically efficient at a point in the above case only.Research supported by University of Tsukuba Project Research.     

On the symmetry classification and conservation laws for quasilinear differential equations of second order

We obtain a sufficient condition for the absence of tangent transformations admitted by quasilinear differential equations of second order and a sufficient condition for the linear autonomy of the operators of the Lie group of transformations admitted by weakly nonlinear differential equations of second order. We prove a theorem concerning the structure of conservation laws of first order for weakly nonlinear differential equations of second order. We carry out the classification by first-order conservation laws for linear differential equations of second order with two independent variables.     

Periodic solutions of second order differential equations in Banach spaces

We use operator-valued Fourier multiplier theorems to study second order differential equations in Banach spaces. We establish maximal regularity results in L^p and C^s for strong solutions of a complete second order equation. In the second part, we study mild solutions for the second order problem. Two types of mild solutions are considered. When the operator A involved is the generator of a strongly continuous cosine function, we give characterizations in terms of Fourier multipliers and spectral properties of the cosine function. The results obtained are applied to elliptic partial differential operators. The first author is supported in part by Convenio de Cooperación Internacional (CONICYT) Grant # 7010675 and the second author is partially financed by FONDECYT Grant # 1010675     

On Nonlinear Hyperbolic Functional Differential Equations

In this paper, existence of weak solutions of second order evolution equations is proved and some properties of the solutions are shown. The results are applied to higher order nonlinear hyperbolic functional differential equations.     

Asymptotic and convergent expansions for solutions of third-order linear differential equations with a large parameter

In previous papers [6-8,10], we derived convergent and asymptotic expansions of solutions of second order linear differential equations with a large parameter. In those papers we generalized and developed special cases not considered in Olver"s theory [Olver, 1974]. In this paper we go one step forward and consider linear differential equations of the third order: $y"+a\Lambda^2 y"+b\Lambda^3y=f(x)y"+g(x)y$, with $a,b\in\mathbb{C}$ fixed, $f"$ and $g$ continuous, and $\Lambda$ a large positive parameter. We propose two different techniques to handle the problem: (i) a generalization of Olver"s method and (ii) the transformation of the differential problem into a fixed point problem from which we construct an asymptotic sequence of functions that converges to the unique solution of the problem. Moreover, we show that this second technique may also be applied to nonlinear differential equations with a large parameter. As an application of the theory, we obtain new convergent and asymptotic expansions of the Pearcey integral $P(x,y)$ for large $|x|$.     

高阶偏微分方程与概率方法

二阶偏微分方程与扩散过程的是概率界众所财知的。前者为后者提供了分析依据，后者为前者的解给出了概率表示；如何把这种联系推广到高阶偏微分方程的情形，是很多概率学家近十几年来一直关心的问题。     

Spectral Estimates for Resolvent Differences of Self-Adjoint Elliptic Operators

The concept of quasi boundary triples and Weyl functions from extension theory of symmetric operators in Hilbert spaces is developed further and spectral estimates for resolvent differences of two self-adjoint extensions in terms of general operator ideals are proved. The abstract results are applied to self-adjoint realizations of second order elliptic differential operators on bounded and exterior domains, and partial differential operators with δ-potentials supported on hypersurfaces are studied.     

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.     

Contribution to error estimate

The error estimate of an approximate solution to a nonlinear ordinary differential equations of the second order is obtained. The differential equation is subject to either two-point boundary conditions or initial conditions. The independent variable interval may be finite or infinite. The theory is applied to five problems.