Adams-Type Methods for the Numerical Solution of Stochastic Ordinary Differential Equations

The modelling of many real life phenomena for which either the parameter estimation is difficult, or which are subject to random noisy perturbations, is often carried out by using stochastic ordinary differential equations (SODEs). For this reason, in recent years much attention has been devoted to deriving numerical methods for approximating their solution. In particular, in this paper we consider the use of linear multistep formulae (LMF). Strong order convergence conditions up to order 1 are stated, for both commutative and non-commutative problems. The case of additive noise is further investigated, in order to obtain order improvements. The implementation of the methods is also considered, leading to a predictor-corrector approach. Some numerical tests on problems taken from the literature are also included.     

Numerical Solution of Systems of Loaded Ordinary Differential Equations with Multipoint Conditions

A system of loaded ordinary differential equations with multipoint conditions is considered. The problem under study is reduced to an equivalent boundary value problem for a system of ordinary differential equations with parameters. A system of linear algebraic equations for the parameters is constructed using the matrices of the loaded terms and the multipoint condition. The conditions for the unique solvability and well-posedness of the original problem are established in terms of the matrix made up of the coefficients of the system of linear algebraic equations. The coefficients and the righthand side of the constructed system are determined by solving Cauchy problems for linear ordinary differential equations. The solutions of the system are found in terms of the values of the desired function at the initial points of subintervals. The parametrization method is numerically implemented using the fourth-order accurate Runge–Kutta method as applied to the Cauchy problems for ordinary differential equations. The performance of the constructed numerical algorithms is illustrated by examples.     

Numerical Solution of the Eigenvalue Problem for Discontinuous Linear Ordinary Differential Equations

The eigenvalue problem for linear ordinary differential equationswith discontinuous terms is considered. It is shown that onecan effectively use Atkinson's product integration techniquestogether with a Green's function to solve the problem numerically.     

一种偏微分方程数值求解的自适应神经网络模型

基于NN-PDE模型提出了一种改进的模型自适应PDE-Net.数值实例验证了该方法的有效性.     

Stability of Numerical Methods for Ordinary Differential Equations

Variable stepsize stability results are found for three representative multivalue methods. For the second order BDF method, a best possible result is found for a maximum stepsize ratio that will still guarantee A(0)-stability behaviour. It is found that under this same restriction, A()-stability holds for 70°. For a new two stage two value first order method, which is L-stable for constant stepsize, A(0)-stability is maintained for stepsize ratios as high as aproximately 2.94. For the third order BDF method, a best possible result of (1/2)(1+ ) is found for a ratio bound that will still guarantee zero-stability.     

Numerical Methods for Partial Differential Equations

Numerical methods for partial differential equation have experienced steady developments in the past several decades.Most of the remaining challenges fall into two categories,both of which are associated with the curse of the dimensionality.In the first category,the func-tion of the partial differential equation is high dimensional,with examples from many-body quantum mechanics,molecular dynamics,to kinetic equations and control problems.In the second category,the objective of the interest is the high dimensional map from the parameters of the equation to the solution or vice versa,with examples such as inverse problems and un-certainty quantifications.The curse of the dimensionality has been the bottleneck for further breakthroughs in the numerical solution of partial differential equation problems.     

一阶微分方程数值解的一种误差估计方法

针对于微分方程数值解,介绍了一种新的误差估计方法.方法证实了伪谱方法具有精度高速度快的优点,进而引出了修正的伪谱方法.     

The Numerical Solution of Parabolic Partial Differential Equations Using the Method of Lanczos

Several types of parabolic equations are solved, subject tovarious boundary conditions. A polynomial solution is sought,which is the exact solution of a perturbed version of the originaldifferential equation. The method is developed for the heatequation and then extended to a wide range of problems.     

逆谱问题中偏微分方程数值解的稳定性

讨论一类偏微分方程数值解的稳定性,这种方程源于Sturm-Liouville算子逆谱问题中变换算子法.证明这类偏微分方程差分格式解的存在性、唯一性、收敛性定理及稳定性定理成立.     

A Collocation Method for the Numerical Solution of Non-linear Parabolic Partial Differential Equations

Lagrange interpolation formulae are used to obtain a new algorithmfor the approximate polynomial solution of linear and non-linearparabolic equations. The algorithm is described for problemsin one space dimension, although it is applicable to problemsdefined in two or more dimensions. It is also shown how thealgorithm may be adapted to solve a moving boundary (Stefan)problem.     

On Chebyshev Solution of Parabolic Partial Differential Equations

Now at Mathemarics Department, Assiut University Egypt A method is presented to transform parabolic equations to asystem of ordinary differential equations for the solution atthe Chebyshev points. The system may be solved analyticallyor by numerical methods and the Chebyshev coefficients are computed.We have the exact solution of a perturbed problem.     

The Numerical Solution of the Fourth Boundary Value Problem for Parabolic Partial Differential Equations

A direct method for the numerical solution of the implicit finitedifference equations derived from a parabolic differential equationwith periodic spatial boundary conditions is presented in algorithmicfrom. Consideration is given to the stability of the roundingerrors involved in the solution process and numerical resultsare derived which compare favourably with those obtained fromthe analytical solution and a matrix spectral resolution methodwhich is closely allied to the method of lines.     

The Numerical Approximation of Stochastic Partial Differential Equations

The numerical solution of stochastic partial differential equations (SPDEs) is at a stage of development roughly similar to that of stochastic ordinary differential equations (SODEs) in the 1970s, when stochastic Taylor schemes based on an iterated application of the Itô formula were introduced and used to derive higher order numerical schemes. An Itô formula in the generality needed for Taylor expansions of the solution of a SPDE is however not available. Nevertheless, it was shown recently how stochastic Taylor expansions for the solution of a SPDE can be derived from the mild form representation of the SPDE, which avoid the need of an Itô formula. A brief review of the literature is given here and the new stochastic Taylor expansions are discussed along with numerical schemes that are based on them. Both strong and pathwise convergence are considered.     

Numerical Solution of Arbitrary-Order Ordinary Differential and Integro-Differential Equations with Separated Boundary Conditions Using Optimal Control Technique

In this paper, a new method for solving arbitrary order ordinary differential equations and integro-differential equations of Fredholm and Volterra kind is presented. In the proposed method, these equations with separated boundary conditions are converted to a parametric optimization problem subject to algebraic constraints. Finally, control and state variables will be approximated by a Chebychev series. In this method, a new idea has been used, which offers us the ability of applying the mentioned method for almost all kinds of ordinary differential and integro-differential equations with different types of boundary conditions. The accuracy and efficiency of the proposed numerical technique have been illustrated by solving some test problems.     

Stability and Dynamics of Numerical Methods for Nonlinear Ordinary Differential Equations

Stability of numerical methods for nonlinear autonomous ordinarydifferential equations is approached from the point of viewof dynamical systems. It is proved that multistep methods (withnonlinear algebraic equations exactly solved) with bounded trajectoriesalways produce correct asymptotic behaviour, but this is notthe case with Runge-Kutta. Examples are given of Runge-Kuttaschemes converging to wrong solutions in a deceptively ‘smooth’manner and a characterization of such two-stage methods is presented.PE(CE)^m schemes are examined as well, and it is demonstratedthat they, like Runge-Kutta, may lead to false asymptotics.     

Reducible Ordinary Differential Equations

The class of reducible differential equations under consideration here includes the class of symmetric systems, and examples show that the inclusion is proper. We first discuss reducibility, as well as the stronger concept of complete reducibility, from the viewpoint of Lie algebras of vector fields and their invariants, and find Lie algebra conditions for reducibility which generalize the conditions in the symmetric case. Completely reducible equations are shown to correspond to a special class of abelian Lie algebras. Then we consider the inverse problem of determining all vector fields which are reducible by some given map. We find conditions imposed on the vector fields by the map, and present an algorithmic access for a given polynomial or local analytic map to Next, reducibility of polynomial systems is discussed, with applications to local reducibility near a stationary point. We find necessary conditions for reducibility, including restrictions for possible reduction maps to a one-dimensional equation.     

A Note on the Numerical Solution of Quasi-linear Parabolic Partial Differential Equations with Error Estimates

A method is considered for the numerical solution of quasi-linearpartial differential equations. The partial differential equationis reduced to a set of ordinary differential equations usinga Chebyshev series expansion. The exact solution of this setof ordinary differential equations is shown to be the solutionof a perturbed form of the original equation. This enables errorestimates to be found for linear and mildly non-linear problems.     

从偏导数恒等式变换到偏微分方程求解

从高等数学教材课后习题的偏导数恒等式变换求解,引导学生讨论一类偏微分方程的求解.在拓展课程内容、应用和常微分方程变量分离方法的基础上,巩固多元复合函数求导法则,常系数线性微分方程求解方法和傅里叶级数的相关理论与方法.