A Multiquadric Interpolation Method for Solving Initial Value Problems

In this paper, an interpolation method for solving linear differential equations was developed using multiquadric scheme. Unlike most iterative formula, this method provides a global interpolation formula for the solution. Numerical examples show that this method offers a higher degree of accuracy than Runge-Kutta formula and the iterative multistep methods developed by Hyman (1978).     

The Discontinuous Galerkin Method for Two-Dimensional Hyperbolic Problems. Part I: Superconvergence Error Analysis

In this paper we investigate the superconvergence properties of the discontinuous Galerkin method applied to scalar first-order hyperbolic partial differential equations on triangular meshes. We show that the discontinuous finite element solution is O(h ^p+2) superconvergent at the Legendre points on the outflow edge for triangles having one outflow edge. For triangles having two outflow edges the finite element error is O(h ^p+2) superconvergent at the end points of the inflow edge. Several numerical simulations are performed to validate the theory. In Part II of this work we explicitly write down a basis for the leading term of the error and construct asymptotically correct a posteriori error estimates by solving local hyperbolic problems with no boundary conditions on more general meshes.     

基于神经网络的微分对策控制器设计

采用伴随－BP技术,将微分对策的两点边值求解问题转化两个神经网络的学习问题,训练后的两个神经网络分别作为对策双方的最优控制器在线使用,避免了直接求解复杂的两点边值问题,对追逃微分对策问题的仿真结果表明,该方法对初始条件和噪声具有较好的鲁棒性。     

求解最优控制问题的Chebyshev-Gauss伪谱法

提出了一种求解最优控制问题的Chebyshev-Gauss伪谱法, 配点选择为Chebyshev-Gauss点. 通过比较非线性规划问题的Kaursh-Kuhn-Tucker条件和伪谱离散化的最优性条件, 导出了协态和Lagrange乘子的估计公式. 在状态逼近中, 采用了重心Lagrange插值公式, 并提出了一种简单有效的计算状态伪谱微分矩阵的方法. 该法的独特优势是具有良好的数值稳定性和计算效率. 仿真结果表明, 该法能够高精度地求解带有约束的复杂最优控制问题.     

Deferred Correction Methods for Initial Boundary Value Problems

In this paper, we consider the deferred correction principle for initial boundary value problems. The method will here be applied to the discretization in time. We obtain a method of even order p by applying the implicit midpoint rule p/2 times in each time step. For the space discretization we will use a compact implicit difference scheme. We derive error estimates for the case of time dependent coefficients and present numerical experiments confirming the theoretical analysis.     

On a High Order Numerical Method for Solving Partial Differential Equations in Complex Geometries

Initial-boundary value problems for hyperbolic and parabolic partial differential equations with Diriclilet boundary conditions are considered by the method of lines approach in an arbitrarily given domain D in 2-D or 3-D. With D embedded in a rectangular domain, a new high order method for the space discretization problem is constructed in D by employing a Fourier collocation method in a uniform Cartesian system of gridpoints. Singularities are systematically removed by utilizing properties of the Bernoulli polynomials. Theoretical estimates for the accuracy of the method are established. The estimates are confirmed by numerical experiments for simple approximation problems.     

解最优控制问题结合同伦法的自适应拟谱方法

针对弱间断最优控制问题和Bang-Bang最优控制问题,提出一种结合同伦法的自适应拟谱方法.Chebyshev拟谱方法转换原问题成为非线性规划问题.基于同伦法思想,同伦参数改变路径约束的界限,得到一系列比较光滑的最优控制问题.通过解这些问题得到原问题的不光滑解.文中证明了弱间断情况下数值解的收敛性.依据这收敛性和同伦参数,误差指示量可以捕捉不光滑点.本文方法与其他方法在数值算例中的对比表明,本文方法在精度和效率上都有明显优势.     

Numerical method of bicharacteristics for hyperbolic partial functional differential equations

Classical solutions of mixed problems for first-order partial functional differential equations in several independent variables are approximated in this paper by solutions of a difference problem of Euler type. The mesh for the approximate solutions is obtained by the numerical solution of equations of bicharacteristics. Convergence of explicit difference schemes is proved by consistency and stability arguments. It is assumed that the given functions satisfy nonlinear estimates of Perron type. Differential equations with deviated variables and differential integral equations can be obtained from the general model by specifying the given operators.      

Some Aspects of the Method of Fundamental Solutions for Certain Harmonic Problems

The Method of Fundamental Solutions (MFS) is a boundary-type method for the solution of certain elliptic boundary value problems. The basic ideas of the MFS were introduced by Kupradze and Alexidze and its modern form was proposed by Mathon and Johnston. In this work, we investigate certain aspects of a particular version of the MFS, also known as the Charge Simulation Method, when it is applied to the Dirichlet problem for Laplace's equation in a disk.     

On some 4-Point Spline Collocation Methods for Solving Ordinary Initial Value Problems

This paper discusses collocation schemes based on seventh C 3 -splines with four collocation points s_{i-1+alpha}=x_{i-1}+alpha h , x_{i-1+beta}=x_{i-1}+beta h , x_{i-1+theta}=x_{i-1}+theta h and x_i=x_{i-1}+h in each subinterval [ x i @ 1 , x i ], i = 1(1) N for solving initial value problems in ordinary differential equations including stiff equations. Here 0 < f < g < è < 1 are arbitrarily given. It is shown that the methods are convergent and the order of convergence is seven if: eqalign {&beta(theta -theta^2)+beta^2(theta^2-theta)+alpha^2[theta^2-theta +beta^2(1+4theta -5theta^2)+theta(4theta^2-4theta -1)]cr &qquad qquad qquad +alpha[theta -theta^2+beta (1+4theta -4theta^2)+beta^2(4theta^2+4theta -1)]le 0 and they are unstable if f , g , è < 0.7279115. Moreover, the absolute stability properties of the methods are considered. It shows that with 0.888035 h f < g < è < 1 the methods are A-stable while they are not if f h 0.5; on other hand, the sizes of regions of absolute stability increase when f , g , è M 1 m.     

Sinc-Galerkin method for solving nonlinear weakly singular two point boundary value problems

A study of Sinc-Galerkin method based on double exponential transformation for solving a class of nonlinear weakly singular two point boundary value problems with non-homogeneous boundary conditions is given. The properties of the Sinc-Galerkin approach are utilized to reduce the computation of nonlinear problem to nonlinear system of equations with unknown coefficients. This method tested on several test examples. We compare our numerical results with several numerical results of existing methods. The demonstrated results confirm that proposed method is considerably efficient, accurate nature and rapidly converge.     

Numerical solution of optimal control problem for a model of tumour growth with drug application

In this study, a combination of spectral and fixed point methods is used to solve an optimal control problem for a model of tumour growth. The growth of tumour is modelled using three first-order hyperbolic equations describing the evolution of cells and two second-order parabolic equations describing the diffusion of nutrient and drug concentration. In the optimal control problem, four control variables are employed to control the concentration of nutrient and drug on the boundary and inside the tumour. Since the problem is nonlinear, applying the fixed point method, in each step of iteration, the problem is changed to a linear one and the parabolic equations are solved using the spectral method. The convergence and stability of method are proven. Some examples are considered to illustrate the efficiency of method. Finally, some figures are provided to reflect the effects of control on the densities of tumourcells.     

A formula for frequency responses of distributed systems with one spatial variable

We develop a method for the exact determination of frequency responses for a class of infinite dimensional systems. In particular, we consider distributed systems in which a spatial independent variable belongs to a finite interval, and in which the inputs and outputs are spatially distributed over the same interval. We show that an explicit formula for the Hilbert–Schmidt norm of the operator-valued frequency response can be obtained whenever the underlying operators are represented by a forced two point boundary value state-space realizations (TPBVSR). This formula involves finite dimensional computations with matrices whose dimension is at most four times larger than the order of the underlying differential operator. Two examples are provided to illustrate the procedure.     

The Use of the Ritz Method and Laplace Transform for Solving 2D Fractional‐Order Optimal Control Problems Described by the Roesser Model

In this article a numerical solution is presented for a class of two‐dimensional fractional‐order optimal control problems (2D‐FOOCPs) with one input and two outputs. To implement the numerical method, the Legendre polynomial basis is used with the aid of the Ritz method and the Laplace transform. By taking the Ritz method as a basic scheme into account and applying a new constructed fractional operational matrix to estimate the fractional and integer order derivatives of the basis, the given 2D‐FOOCP is reduced to a system of algebraic equations. One of the advantages of the proposed method is that it provides greater flexibility in which the given initial and boundary conditions of the problem are imposed. Moreover, satisfactory results are obtained in just a small number of polynomials order. The convergence of the method is extensively investigated and finally two illustrative examples are included to show the validity and applicability of the novel proposed technique in the current work.     

An A Priori Error Analysis of the hp-Version of the Continuous Galerkin FEM for Nonlinear Initial Value Problems

A continuous Galerkin finite element time-stepping method for the approximation of nonlinear initial value problems is analyzed within an hp-context. We derive a priori error bounds in the L²- and H¹-norm that are explicit with respect to the time steps and the approximation orders. In particular, it is shown that, for analytic solutions (with certain possible start-up singularities) exponential convergence rates can be achieved. Moreover, we prove that the scheme superconverges at the nodal points of the time partition. Numerical experiments illustrate the performance of the method.     

Convergence of relaxation methods for two-point boundary-value linear problems

The solution of linear differential problems, with explicit two-point boundary conditions, can sometimes be obtained by a relaxation method of computation. This paper shows that the convergence of iterations is linked to the spectral radius value of an integral operator. The equivalence between eigenvalues research and critical lengths calculation of a differential system is demonstrated. In this context, we present a case of optimal control law calculation of a linear system. The efficiency of this preliminary convergence calculation is illustrated by a numerical example.     

带不等式路径约束最优控制问题的惩罚函数法

控制变量参数化（Control variable parameterization,CVP）方法是目前求解流程工业中最优操作问题的主流数值方法,但如果问题中包含路径约束,特别是不等式路径约束时,CVP方法则需要考虑专门的处理手段.为了克服该缺点,本文提出一种基于L1精确惩罚函数的方法,能够有效处理关于控制变量、状态变量、甚至控制变量/状态变量复杂耦合形式下的不等式路径约束.此外,为了能使用基于梯度的成熟优化算法,本文还引进了最新出现的光滑化技巧对非光滑的惩罚项进行磨光.最终得到了能高效处理不等式路径约束的改进型CVP架构,并给出相应数值算法.经典的带不等式路径约束最优控制问题上的测试结果及与国外文献报道的比较研究表明：本文所提出的改进型CVP 架构及相应算法在精度和效率上兼有良好表现.     

求解最优控制问题的混合变量变分方法及其航天控制应用

针对非线性最优控制导出的Hamiltonian系统两点边值问题,提出一种以离散区段右端状态和左端协态为混合独立变量的数值求解方法, 将非线性Hamiltonian系统两点边值问题的求解通过混合独立变量变分原理转化为非线性方程组求解.所提出的算法综合了求解最优控制 的"直接法"和"间接法"的特征,既满足最优控制理论的一阶必要条件,又不需要对协态初值的准确猜测,避免了求解大规模非线性规划问题. 通过两个航天控制算例讨论了本文算法的精度和效率等问题.与近年来在航空航天控制中备受关注的高斯伪谱方法相比较,本文算法无论是在 精度还是效率上都具有明显的优势.     

Quintic nonpolynomial spline method for the solution of a second-order boundary-value problem with engineering applications

Nonpolynomial quintic spline functions are used to develop a numerical algorithm for computing an approximation to the solution of a system of second order boundary value problems associated with heat transfer. We show that the approximate solutions obtained by our algorithm are better than those produced by other spline and domain decomposition methods. A comparison of our algorithm with nonpolynomial quadratic spline method is discussed with the help of two numerical examples.     

On the Convergence of a Spline Method for Singular Two Point Boundary Value Problems Arising in Physiology

The second order spline method developed by Iyengar et al. [11] for the problems $$eqalign{ x^{-alpha}(x^alpha y^prime)^prime & = f(x,y),quad 0