Chebyshev polynomial solution of nonlinear integral equations

In the current work, the Chebyshev collocation method is adopted to find an approximate solution for nonlinear integral equations. Properties of the Chebyshev polynomials and operational matrix are used in the integral equation of a system consisting of nonlinear algebraic equations with the unknown Chebyshev coefficients. Numerical examples are presented to illustrate the method and results are discussed.     

A numerical solution of a class of periodic coupled matrix equations

This paper studies the numerical solutions of a class of periodic coupled matrix equations. Based on the least square method, a finite iterative algorithm for a class of periodic coupled matrix equations is proposed, and the convergence of the algorithm is proved by theoretical derivation. For any initial value, the algorithm can converge to the solution in finite iterations. Since the equations considered in paper contain many variants, the proposed algorithm has a wide range of applications. Finally some numerical examples in practical systems are given to prove the effectiveness and efficiency of the algorithm.     

A mechanical method for graphical solution of polynomials

A mechanical synthesizer with thirty harmonic elements (fifteen sine components and fifteen cosine components) may be used to graph a polynomial in a complex plane. The sum of the sine components is recorded by a tracing point which moves vertically. The sum of the cosine components is recorded by horizontal motion of a drawing board.     

A comparative study of numerical methods for solving quadratic Riccati differential equations

In this paper, we use Legendre wavelet method for solving quadratic Riccati differential equations and perform a comparative study between the proposed method and other existing methods. Our results show that in comparison with other existing methods, the Legendre wavelet method provides a fast convergent series of easily computable components. The present study is illustrated by exploring two kinds of nonlinear Riccati differential equations that shows the pertinent features of the Legendre wavelet method.     

The numerical solution of physical problems modeled as a systems of differential-algebraic equations (DAEs)

In this paper, the numerical solution of differential-algebraic equations is considered by Padé approximation method. We applied this method to an example which is a physical model problem. Firstly the physical model problem has been converted to power series, then the numerical solution of physical model problem was put into Padé series form. Thus we obtained numerical solution of physical model problem.     

A collocation method using Hermite polynomials for approximate solution of pantograph equations

In this paper, a numerical method based on polynomial approximation, using Hermite polynomial basis, to obtain the approximate solution of generalized pantograph equations with variable coefficients is presented. The technique we have used is an improved collocation method. Some numerical examples, which consist of initial conditions, are given to illustrate the reality and efficiency of the method. In addition, some numerical examples are presented to show the properties of the given method; the present method has been compared with other methods and the results are discussed.     

Integraph solution of differential equations

In a previous article¹ a continuously recording integraph was described, by means of which differential equations, involving only one integration, could be solved. The present article describes a revision of this machine such that an equation involving two successive integrations, corresponding to practically any second-order total differential equation, with all terminal conditions included, can be solved. The need for a workable means of solving the differential equations involving empirical and discontinuous coefficients which occur repeatedly in electrical engineering and physics is recalled. In the machine described such solutions are effected by means of suitable interlinked integrating devices, the result being plotted continuously as a function of the independent variable. Tests and simple solutions show the over-all error to be approximately 1 or 2 per cent. The various sources of this error are discussed.     

Mechanical solution of algebraic equations

The paper describes a machine for determining the real and complex roots of higher-degree algebraic equations. The principle of operation is found in the correspondence between sine wave quantities and complex numbers. The particular machine is designed for equations of the eighth degree, and finds all the roots with engineering accuracy in the space of a few minutes.Though designed primarily with a view to determining the indicial admittance of electric networks, the machine should find utility in other fields of applied mathematics as well.     

微分方程数值解的优化格式

利用最优化思想，根据一阶常微分方程数值解的收敛性和稳定性，引进最优化技术，确定最优系数，得到强稳定的线性三步和四步公式以及三阶Runge-Kutta最优算法，经地实际计算，部分公式的结果优于目前已有的公式。     

A variational method for the numerical simulation of a boundary controllability problem for the linear and semilinear 1D wave equations

We pursue the numerical implementation of a boundary controllability problem for the 1D wave equation based on a recent variational approach to deal with such situations that consists in analyzing an error functional defined for feasible functions complying with appropriate initial, boundary, and final constraints. The nature of such scheme, as a minimization process for a certain error functional, leads to a natural numerical implementation by using typical descent algorithms. Our aim here is to explore the basic ingredients to set up such practical numerical approximations which allow us to address linear and semilinear equations with the same numerical scheme.     

A novel mesh discretization strategy for numerical solution of optimal control problems in aerospace engineering

An effective approach is proposed for optimal control problems in aerospace engineering. First, several interval lengths are treated as optimization variables directly to localize the switching points accurately. Second, the variable intervals are usually refined into more subintervals homogeneously to obtain the trajectories with high accuracy. To reduce the number of optimization variables and improve the efficiency, the control and the state vectors are parameterized using different meshes in this paper such that the control can be approximated asynchronously by fewer parameters where the trajectories change slowly. Then, the variables are departed as independent variables and dependent variables, the gradient formulae, based on the partial derivatives of dependent parameters with respect to independent parameters, are computed to solve nonlinear programming problems. Finally, the proposed approach is applied to the classic moon lander and hang glider problems. For the moon lander problem, the proposed approach is compared with CVP, Fast-CVP and GPM methods, respectively. For the hang glider problem, the proposed approach is compared with trapezoidal discretization and stopping criteria methods, respectively. The numerical results validate the effectiveness of the proposed approach.     

Solution of integral equations using a set of block pulse functions

A set of the block pulse functions is applied to solve the Fredholm's and the Volterra's integral equations of the second kind. An algebraic equation in matrix form which is equivalent to the solution of the integral equation is developed. The approximate results are easily obtained by a few computations. An accurate solution canbe evaluated in a digital computer by solving the algebraic equation. Two examples are given.     

Polynomial based differential quadrature method for numerical solution of nonlinear Burgers' equation

Nonlinear Burgers' equation is solved using polynomial based differential quadrature method (PDQ). Numerical simulations are studied for three well known test problems, namely shock-like solution, travelling wave and sinusoidal disturbance solutions of Burgers' equation. Obtained numerical results of the first and the third test problems are compared with some earlier numerical results. Discrete root mean square error norm and maximum error norm are computed for the first two test problems and a comparison with some earlier works is given.     

一类高振荡常微分方程数值解法的误差分析

以特殊的线性振荡方程y″ g(t)y=0(其中limt→∞g(t)= ∞)为例讨论了高振荡微分方程数值解的问题.分析了梯形格式的整体截断误差,并对梯形格式做了修改,讨论了修改后格式的局部截断误差对整体截断误差的影响,最后给出了数值结果.