微分方程的Hamilton化与解法

提出一种求解微分方程的力学方法.首先,将一类常微分方程化成一个Hamilton方程,在特殊情况下化成Hamilton原来的方程,在一般情况下化成带非保守力的Hamilton方程.其次,利用Hamilton系统的Noether理论求守恒量.如果找到足够多的守恒量,便找到了方程的解.最后,举例说明结果的应用.     

Characterising solution sets of LTI differential equations

We prove that a set of smooth trajectories is LTID (i.e., the solution set of a constant coefficient linear differential equation) if and only if it is the direct sum of a linear time-invariant closed controllable part and a linear time-invariant finite-dimensional part. This characterisation does not directly involve derivatives in its formulation. It solves a problem opened by Willems (1991). We also characterise morphisms between LTID sets as linear time-invariant maps which do not increase support.     

The network method for a fast and reliable solution of ordinary differential equations: Applications to non-linear oscillators

The network method is used for designing electric circuits that implement any kind of linear and non-linear ordinary differential equations of any grade and any order. The model simulations are carried out in a suitable computational code without any other mathematical manipulation. The addends of the equation are assumed to be electric currents that balance each other in the only common node of the main loop. Four special kinds of controlled source, named voltage-controlled (or current-controlled) voltage-source and voltage-controlled (or current-controlled) current-source, in conjunction with auxiliary simple circuits are used to implement derivative terms other kinds of non-linear character terms. Applications are shown which demonstrate the reliability and power of the proposed method.     

Numerical solution of differential equations using Haar wavelets

Haar wavelet techniques for the solution of ODE and PDE is discussed. Based on the Chen–Hsiao method [C.F. Chen, C.H. Hsiao, Haar wavelet method for solving lumped and distributed-parameter systems, IEE Proc.—Control Theory Appl. 144 (1997) 87–94; C.F. Chen, C.H. Hsiao, Wavelet approach to optimising dynamic systems, IEE Proc. Control Theory Appl. 146 (1997) 213–219] a new approach—the segmentation method—is developed. Five test problems are solved. The results are compared with the result obtained by the Chen–Hsiao method and with the method of piecewise constant approximation [C.H. Hsiao, W.J. Wang, Haar wavelet approach to nonlinear stiff systems, Math. Comput. Simulat. 57 (2001) 347–353; S. Goedecker, O. Ivanov, Solution of multiscale partial differential equations using wavelets, Comput. Phys. 12 (1998) 548–555].     

Numerical solution of time-fractional fourth-order partial differential equations

A quintic B-spline collocation technique is employed for the numerical solution of time-fractional fourth-order partial differential equations. These equations occur in many applications in real-life problems such as modelling of plates and thin beams, strain gradient elasticity and phase separation in binary mixtures, which are basic elements in engineering structures and are of great practical significance to civil, mechanical and aerospace engineering. The time-fractional derivative is described in the Caputo sense. Backward Euler scheme is used for time discretization and the quintic B-spline-based numerical method is used for space discretization. The stability and convergence properties related to the time discretization are discussed and theoretically proven. The given problem is solved with three different boundary conditions, including clamped-type condition, simply supported-type condition, and a transversely supported-type condition. Numerical results are considered to investigate the accuracy and efficiency of the proposed method.     

Tau approximate solution of fractional partial differential equations

In this study, we improve the algebraic formulation of the fractional partial differential equations (FPDEs) by using the matrix-vector multiplication representation of the problem. This representation allows us to investigate an operational approach of the Tau method for the numerical solution of FPDEs. We introduce a converter matrix for the construction of converted Chebyshev and Legendre polynomials which is applied in the operational approach of the Tau method. We present the advantages of using the method and compare it with several other methods. Some experiments are applied to solve FPDEs including linear and nonlinear terms. By comparing the numerical results obtained from the other methods, we demonstrate the high accuracy and efficiency of the proposed method.     

An implicit method for numerical solution of differential equations

The paper presents an implicit method for numerical solution of differential equations, based on the use of the derivatives of the right-hand side jointly with a directed motion in the discrepancy space on passing to the next integration layer. Two examples of solving equations by the proposed method are considered.     

Numerical solution of stiff differential equations via Haar wavelets

A simple and effective algorithm based on Haar wavelets is proposed to the solution of stiff problems in this article. It can integrate the stiff equation with very accurate results for any length of time. The simulation result shows that the single-term Haar wavelet method is better than the improved Runge–Kutta–Fehlberg method, while the terms of both expansions are the same.     

Explicit solution to a class of functional differential equations

An explicit solution for tho transition matrix of a class of functional differential equations ia found in the form of an infinite series. Methods are given showing how to find the coefficients of this infinite series in the form of a recursive formula, or in the form of a tree structure. An example is given to illustrate the method.     

Finite difference solution of differential equations with stochastic inputs

This paper discusses application of two numerical methods (central difference and predictor corrector) for the solution of differential equations with deterministic as well as stochastic inputs. The methods are applied to a second order linear differential equation representing a series RLC netowrk with step function, sinusoidal and stochastic inputs. It is shown that both methods give correct answers for the step function and sinusoidal inputs. However, the central-difference method of solution is recommended for stochastic inputs. This statement is justified by comparing the auto-correlation and cross-correlation functions of the central-difference solution (with stochastic inputs) with the corresponding theoretical values of a continuous system. It is further shown that the more common predictor-corrector methods, although suitable for solution of differential equations with regular inputs, diverge for stochastic inputs. The reason is that these methods, by the application of several point integral formulas, use a high degree of smoothing on the variable and its derivatives. Inherent in the derivation of these integral formulas is the assumption of the continuity of the variable and its derivatives, a condition which is not satisfied in problems with stochastic inputs.Note that the second order differential equation chosen here for numerical experiments can be solved by classical methods for all of the given inputs, including the probabilistic inputs. The classical methods, however, unlike the numerical solutions, can not be extended to nonlinear differential equations which frequently arise in the digital simulation of engineering problems.     

Numerical solution of fuzzy differential equations by predictor-corrector method

In this paper three numerical methods to solve “The fuzzy ordinary differential equations” are discussed. These methods are Adams-Bashforth, Adams-Moulton and predictor-corrector. Predictor-corrector is obtained by combining Adams-Bashforth and Adams-Moulton methods. Convergence and stability of the proposed methods are also proved in detail. In addition, these methods are illustrated by solving two fuzzy Cauchy problems.     

Numerical solution of nonlinear differential equations using computer algebra

This paper describes a numerical method for finding periodic solutions to nonlinear ordinary differential equations. The solution is approximated by a trigonometric series. The series is substituted into the differential equation using the FORMAC computer algebra system for the resulting lengthy algebraic manipulations. This lead to a set of nonlinear algebraic equations for the series coefficients. Modern search methods are used to solve for the coefficients. The method is illustrated by application to Duffing’ equation.     

Accelerating numerical solution of stochastic differential equations with CUDA

Numerical integration of stochastic differential equations is commonly used in many branches of science. In this paper we present how to accelerate this kind of numerical calculations with popular NVIDIA Graphics Processing Units using the CUDA programming environment. We address general aspects of numerical programming on stream processors and illustrate them by two examples: the noisy phase dynamics in a Josephson junction and the noisy Kuramoto model. In presented cases the measured speedup can be as high as 675× compared to a typical CPU, which corresponds to several billion integration steps per second. This means that calculations which took weeks can now be completed in less than one hour. This brings stochastic simulation to a completely new level, opening for research a whole new range of problems which can now be solved interactively.

Program summary

Program title: SDECatalogue identifier: AEFG_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEFG_v1_0.htmlProgram obtainable from: CPC Program Library, Queen's University, Belfast, N. IrelandLicensing provisions: Gnu GPL v3No. of lines in distributed program, including test data, etc.: 978No. of bytes in distributed program, including test data, etc.: 5905Distribution format: tar.gzProgramming language: CUDA CComputer: any system with a CUDA-compatible GPUOperating system: LinuxRAM: 64 MB of GPU memoryClassification: 4.3External routines: The program requires the NVIDIA CUDA Toolkit Version 2.0 or newer and the GNU Scientific Library v1.0 or newer. Optionally gnuplot is recommended for quick visualization of the results.Nature of problem: Direct numerical integration of stochastic differential equations is a computationally intensive problem, due to the necessity of calculating multiple independent realizations of the system. We exploit the inherent parallelism of this problem and perform the calculations on GPUs using the CUDA programming environment. The GPU's ability to execute hundreds of threads simultaneously makes it possible to speed up the computation by over two orders of magnitude, compared to a typical modern CPU.Solution method: The stochastic Runge-Kutta method of the second order is applied to integrate the equation of motion. Ensemble-averaged quantities of interest are obtained through averaging over multiple independent realizations of the system.Unusual features: The numerical solution of the stochastic differential equations in question is performed on a GPU using the CUDA environment.Running time: < 1 minute     

Double Walsh series solution of first-order partial differential equations

A double Walsh series is introduced to represent approximately functions of two independent variables, and is then applied to analyse single as well as simultaneous first-order partial differential equations. The solutions for the coefficient matrices can be obtained directly from Kronecker product formulae, which are suitable for computer computation. An example for a single first-order partial differential equation is solved by a double Walsh series approximation with satisfactory results.     

Application of the Haar functions to solution of differential equations

In this paper, it is proposed that Haar functions should be used for solving ordinary differential equations of a time variable in facility. This is because integrated forms of Haar functions of any degree can be illustrated by linear- and linear segment-functions like as triangles. Fortunately, since they are placed where Haar functions are defined in a specified form respectively, these functions are computable by algebraic operations of quasi binary numbers. Therefore, when a given function is approximated in a form of stairsteps on a Haar function system their integration can be termwise executed by shift and add operations of coefficients of the approximation. The use of this system is comparable with an application using the midpoint rule in numerical integration. In this line, nonlinear differential equations can be solved like as linear differential equations.     

A new algorithm for numerical solution of ordinary differential equations

A numerical integration scheme which is particularly well suited to initial value problems having oscillatory or exponential solutions is proposed. The derivation of the algorithm is based on a representation of problems (that is problems having oscillatory or exponential solutions), the complex parameters have the real plane. The interpolating function has two complex parameters whose numerical estimates are obtained by using Newton-like scheme to solve three simultaneous nonlinear equations. For the above class of paoblems (that is problems having oscillatory or exponential solutions), the complex parameters have constant values throughout the interval of integration. Hence, the parameters are obtainable at the first integration step. As the approach is applicable to systems of equations, then for an initial value problem of order m, m sets of simultaneous equations have to be solved for the complex parameters.     

The BEM for numerical solution of partial fractional differential equations

A numerical method is presented for the solution of partial fractional differential equations (FDEs) arising in engineering applications and in general in mathematical physics. The solution procedure applies to both linear and nonlinear problems described by evolution type equations involving fractional time derivatives in bounded domains of arbitrary shape. The method is based on the concept of the analog equation, which in conjunction with the boundary element method (BEM) enables the spatial discretization and converts a partial FDE into a system of coupled ordinary multi-term FDEs. Then this system is solved using the numerical method for the solution of such equations developed recently by Katsikadelis. The method is illustrated by solving second order partial FDEs and its efficiency and accuracy is validated.     

Mean square power series solution of random linear differential equations

This paper deals with the construction of random power series solutions of linear differential equations containing uncertainty through the diffusion coefficient, the source term as well as the initial condition. Under appropriate hypotheses on the data, we establish that the constructed random power series solution is mean square convergent in a certain interval whose length depends on the mean square norm of the random variable coefficient. Also, the main statistical functions of the approximating stochastic process solution generated by truncation of the exact power series solution are given. Finally, we apply the proposed technique to several illustrative examples.