Double chebyshev expansions for wave functions

A double Chebyshev expansion technique is suggested for evaluating positive energy solutions of the radial Coulomb equation. It is shown that, for a particle with a wide range of energies and in either an attractive or a repulsive Coulomb field, the two linearly independent radial solutions can be rapidly evaluated.     

Explicit solutions of a class of linear fractional BSDEs

We obtain explicit solutions for a class of linear backward stochastic differential equations driven by a fractional Brownian motion with arbitrary Hurst parameter via the solution of a partial differential equation and a fractional Itô formula.     

Alternative Method of Solution of the Regulator Equation: L2‐Space Approach

An alternative method for the proof of solvability of the differential equation that is a part of the regulator equation which arises from the solution of the output regulation problem. The proof uses the L²‐space based theory of solutions of partial differential equations for the case of the linear output regulation problem. In the nonlinear case, a sequence of linear equations is defined so that their solutions converge to the solution of the nonlinear problem. This is proved using the Banach Contraction Theorem. Copyright © 2011 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

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.     

Auxiliary equation method for time-fractional differential equations with conformable derivative

In this paper, the auxiliary equation method is applied to obtain analytical solutions of (2 + 1)-dimensional time-fractional Zoomeron equation and the time-fractional third order modified KdV equation in the sense of the conformable fractional derivative. Given equations are converted to the nonlinear ordinary differential equations of integer order; and then, the resulting equations are solved using a novel analytical method called the auxiliary equation method. As a result, some exact solutions for them are successfully established. The exact solutions obtained by the proposed method indicate that the approach is easy to implement and effective.     

First order linear fuzzy differential equations under generalized differentiability

First order linear fuzzy differential equations are investigated. We interpret a fuzzy differential equation by using the strongly generalized differentiability concept, because under this interpretation, we may obtain solutions which have a decreasing length of their support (which means a decreasing uncertainty). In several applications the behaviour of these solutions better reflects the behaviour of some real-world systems. Derivatives of the H-difference and the product of two functions are obtained and we provide solutions of first order linear fuzzy differential equations, using different versions of the variation of constants formula. Some examples show the rich behaviour of the solutions obtained.     

Dynamical low-rank approximation: applications and numerical experiments

Dynamical low-rank approximation is a differential-equation-based approach to efficiently compute low-rank approximations to time-dependent large data matrices or to solutions of large matrix differential equations. We illustrate its use in the following application areas: as an updating procedure in latent semantic indexing for information retrieval, in the compression of series of images, and in the solution of time-dependent partial differential equations, specifically on a blow-up problem of a reaction-diffusion equation in two and three spatial dimensions. In 3D and higher dimensions, space discretization yields a tensor differential equation whose solution is approximated by low-rank tensors, effectively solving a system of discretized partial differential equations in one spatial dimension.     

Discrete models in social sciences

Behaviors of solutions of a differential equation and its discretization sometimes differ greatly from each other. For example, even if a differential equation has no chaotic solution, its discretization happens to have a chaotic solution. In this paper, the meanings of this fact in economics are shown. Also, a finite difference model in sociology is given as a chaotic phenomenon.     

Simplest equation method for some time-fractional partial differential equations with conformable derivative

The conformable fractional derivative was proposed by R. Khalil et al. in 2014, which is natural and obeys the Leibniz rule and chain rule. Based on the properties, a class of time-fractional partial differential equations can be reduced into ODEs using traveling wave transformation. Then the simplest equation method is applied to find exact solutions of some time-fractional partial differential equations. The exact solutions (solitary wave solutions, periodic function solutions, rational function solutions) of time-fractional generalized Burgers equation, time-fractional generalized KdV equation, time-fractional generalized Sharma–Tasso–Olver (FSTO) equation and time-fractional fifth-order KdV equation, $(3+1)$-dimensional time-fractional KdV–Zakharov–Kuznetsov (KdV–ZK) equation are constructed. This method presents a wide applicability to solve some nonlinear time-fractional differential equations with conformable derivative.     

Artificial neural network approach for solving fuzzy differential equations

The current research attempts to offer a novel method for solving fuzzy differential equations with initial conditions based on the use of feed-forward neural networks. First, the fuzzy differential equation is replaced by a system of ordinary differential equations. A trial solution of this system is written as a sum of two parts. The first part satisfies the initial condition and contains no adjustable parameters. The second part involves a feed-forward neural network containing adjustable parameters (the weights). Hence by construction, the initial condition is satisfied and the network is trained to satisfy the differential equations. This method, in comparison with existing numerical methods, shows that the use of neural networks provides solutions with good generalization and high accuracy. The proposed method is illustrated by several examples.     

求一类非线性偏微分方程解析解的简洁构造算法

通过引入一个变换,利用齐次平衡原理和选准一个待定函数来构造求解一类非线性偏微分方程解析解的算法.作为实例,我们将该算法应用到了mKdV方程,KdV-Burgers方程和KdV-Burgers-Kuramoto方程.借助符号计算软件Mathematica获得了这些方程的解析解.不难看出,该方法不仅简洁,而且有望进一步扩展.     

A symbolic-numerical method for solving the differential equation describing the states of polarizable particle in Coulomb potential

Methods for solving the differential equation describing the wave functions of a polarizable particle in the Coulomb potential are discussed. Relations between the coefficients under which the general solution of this equation can be found in analytical form are obtained. For the case of zero polarizability, the general solution to this equation in terms of special functions is obtained; for the first values of the parameter j, plots of the corresponding solutions are presented. For nonzero polarizability and certain specially chosen values of the energy level parameter, solutions possessing the required physical properties for the varying parameter j are constructed on fairly large intervals of the argument values using numerical methods and functional objects of the type DifferentialRoot. Instructions in Mathematica are presented that allow computer-aided analysis using numerical and analytical methods and visualization of the resulting solutions.     

Burgers方程的精确解

引入一个变换,将二阶非线性偏微分方程—Burgers方程降阶为一阶的非线性方程,再直接求解该方程,得出了Burgers方程精确解的新形式,并与已有结果完全吻合.这种方法也适合于求解其他非线性偏微分方程.     

偏微分方程的MATLAB数值解法及可视化

偏微分方程的数值解法在数值分析中占有很重要的地位,很多科学技术问题的数值计算包括了偏微分方程的数值解问题。在学习初等函数时,总是先画出它们的图形,因为图形能帮助了解函数的性质。而对于偏微分方程,画出它们的图形并不容易,尤其是没有解析解的偏微分方程,画图就显得更加不容易了。为了从偏微分方程的数学表达式中看出其所表达的图形、函数值与自变量之间的关系,通过MATLAB编程,数值求解了泊松方程,并将其结果可视化,给出了解析解与数值解的误差。     

Iterative solutions to non-linear fourth order differential equations through multi-integral methods

Implicit solution to a certain class of non-linear fourth order differential equation is presented. Bounds on sup norm for the derivative of a certain function f involved in the equation with different boundary values, are computed. These bounds provide the rate of convergence of the iterative sequence of approximate solutions obtained by Picard method, to the exact solution.     

Delay optimal control and viscosity solutions to associated Hamilton–Jacobi–Bellman equations

In this article, optimal control problems of differential equations with delays are investigated for which the associated Hamilton–Jacobi–Bellman (HJB) equations are nonlinear partial differential equations with delays. This type of HJB equation has not been previously studied and is difficult to solve because the state equations do not possess smoothing properties. We introduce a new notion of viscosity solutions and identify the value functional of the optimal control problems as the unique solution to the associated HJB equations. An analytical example is given as application.     

Hamilton体系下中厚板静力弯曲和自由弯曲振动问题的一类模型及通解

对于中厚板的静力弯曲和自由弯曲振动问题,引入两个辅助函数,采用胡海昌在Reissner板理论基础上提出的中厚板微分方程及边界条件,将两类问题的控制方程引入Hamilton体系,分别得到Hamilton体系下中厚板静力弯曲和自由振动问题的微分方程组模型. 比较后得到了Hamilton体系下中厚板静力和振动问题的统一模型,其特点是: 微分方程组模型的统一形式中Hamilton矩阵在对角线位置有2个零子块矩阵. 对于中厚板静力和振动问题,比较了所得齐次微分方程组的特征根,给出齐次微分方程组的通解并进行了比较,从而使问题的求解更理性化和合理化,求解过程遵循一套统一的方法论,便于把这类解法推广到其它问题.     

Numerical solution of hybrid fuzzy differential equation IVPs by a characterization theorem

In this paper, we study hybrid fuzzy differential equation initial value problems (IVPs). We consider the problem of finding their numerical solutions by using a recent characterization theorem of Bede for fuzzy differential equations. We prove a corollary to Bede’s characterization theorem and give a characterization theorem for hybrid fuzzy differential equation IVPs. Then we prove that any suitable numerical method for ODEs can be applied piecewise to numerically solve hybrid fuzzy differential equation IVPs. Numerical examples are provided which connect the new results with previous findings.     

Some new results on algebraic Riccati equations arising in linear quadratic differential games and the stabilization of uncertain linear systems

This paper presents some new results on algebraic Riccati equations arising in linear quadratic differential games. The first result is a uniqueness result for solutions of the Riccati equations under consideration. The second result is concerned with Riccati equations which arise in differential games in which the weighting on the minimizing control is allowed to approach zero. It is shown that if a certain minimum phase condition is satisfied then the corresponding solution to the Riccati equation will also approach zero.     

A Hierarchy of Approximations of the Master Equation Scaled by a Size Parameter

Solutions of the master equation are approximated using a hierarchy of models based on the solution of ordinary differential equations: the macroscopic equations, the linear noise approximation and the moment equations. The advantage with the approximations is that the computational work with deterministic algorithms grows as a polynomial in the number of species instead of an exponential growth with conventional methods for the master equation. The relation between the approximations is investigated theoretically and in numerical examples. The solutions converge to the macroscopic equations when a parameter measuring the size of the system grows. A computational criterion is suggested for estimating the accuracy of the approximations. The numerical examples are models for the migration of people, in population dynamics and in molecular biology. Financial support has been obtained from the Swedish Foundation for Strategic Research and the Swedish Graduate School in Mathematics and Computing.