Asymptotic Initial-Value Method for Singularly-Perturbed Boundary-Value Problems for Second-Order Ordinary Differential Equations

A computational method is presented to solve a class of nonturning-point singularly-perturbed two-point boundary-value problems for second-order ordinary differential equations with a small parameter multiplying the highest derivative, subject to Dirichlet-type boundary conditions. In this method, first we construct a zeroth order asymptotic expansion for the solution of the given boundary-value problem. Then, this problem is integrated to get an equivalent initial-value problem for first-order ordinary differential equations. This initial-value problem is solved by either a classical method or a fitted operator method after approximating some of the terms in the differential equations by using the zeroth order asymptotic expansion. This method is effective and easy to implement. An error estimate is derived for the numerical solution. Examples are given to illustrate the method.     

A Taylor polynomial approach for solving the most general linear Fredholm integro‐differential‐difference equations

In this study, a matrix method is developed to solve approximately the most general higher order linear Fredholm integro‐differential‐difference equations with variable coefficients under the mixed conditions in terms of Taylor polynomials. This technique reduces the problem into the linear algebraic system. The method is valid for any combination of differential, difference and integral equations. An initial value problem and a boundary value problem are also presented to illustrate the accuracy and efficiency of the method. Copyright © 2012 John Wiley & Sons, Ltd.     

An initial value theory for linear causal boundary value problems

A new formalism in the theory of linear boundary value problems involving causal functional differential equations is presented. The approach depends on the construction of a differentiable family of boundary problems into which the original boundary value problem is imbedded. The formalism then generates an initial value problem which is equivalent to the family of imbedded problems. An important aspect of the method is that the equations in the initial value algorithm are ordinary differential equations rather than functional differential equations, although nonlinear and of higher dimension. Applications of the theory to differential-delay and difference equations are given.     

Finding the Periodic Solution of Differential Equation via Solving Optimization Problem

In this paper, we propose a new method to find the periodic solutions of differential equations. The key technique is to convert the problem of finding periodic solutions of differential equations into an optimization problem. Then by solving the corresponding optimization problem, we can find the periodic solutions of differential equations. Finally, some numerical results are presented to illustrate the utility of the technique.     

Numerical solution of retarded functional differential equations as abstract Cauchy problems

An approach for the numerical solution of linear delay differential equations, different from the classical step-by-step integration, was presented in (Numer. Math. 84 (2000) 351). The problem is restated as an abstract Cauchy problem (or as the advection equation with a particular nonstandard boundary condition) and then, by using a scheme of order one, it is discretized as a system of ordinary differential equations by the method of lines. In this paper we introduce a class of related schemes of arbitrarily high order and we then extend the approach to general retarded functional differential equations. An analysis of convergence, and of asymptotic stability when the numerical schemes are applied to the complex scalar equation y′(t)=ay(t)+by(t−1), is provided.     

Convergence of the ghost fluid method for elliptic equations with interfaces

This paper proves the convergence of the ghost fluid method for second order elliptic partial differential equations with interfacial jumps. A weak formulation of the problem is first presented, which then yields the existence and uniqueness of a solution to the problem by classical methods. It is shown that the application of the ghost fluid method by Fedkiw, Kang, and Liu to this problem can be obtained in a natural way through discretization of the weak formulation. An abstract framework is given for proving the convergence of finite difference methods derived from a weak problem, and as a consequence, the ghost fluid method is proved to be convergent.

     

An inverse problem of Bilayer textile thickness determination in dynamic heat and moisture transfer

An inverse problem of bilayer textile thickness determination in dynamic heat and moisture transfer is presented satisfying the heat–moisture comfort level of human body. Heat and mass transfer law in bilayer textiles is displayed by proving the existence and uniqueness of solution to the coupled partial differential equations with initial-boundary value conditions. The finite difference method is employed to derive the numerical solution to partial differential equations. The regularized solution of the inverse problem is reformulated into solving function minimum problem through the Tikhonov regularization method. The golden section method is applied to solve the direct search problem and achieve the optimal solution to the inverse problem. Numerical algorithm and its numerical results provide theoretical explanation for textile materials research and development.     

Approximate controllability of differential inclusions in Hilbert spaces

In this paper, controllability for the system originating from semilinear functional differential equations in Hilbert spaces is studied. We consider the problem of approximate controllability of semilinear differential inclusion assuming that semigroup, generated by the linear part of the inclusion, is compact and under the assumption that the corresponding linear system is approximately controllable. By using resolvent of controllability Gramian operator and fixed point theorem, sufficient conditions have been formulated and proved. An example is presented to illustrate the utility and applicability of the proposed method.     

The scattering of acoustic waves by a cylinder with a non-uniform elastic coating

The problem of the scattering of a plane acoustic wave by a solid cylinder with a radially non-uniform elastic coating is considered. An analytical expression describing the scattered acoustic field is obtained. The equations of motion of the non-uniform elastic cylindrical layer are reduced to a system of ordinary differential equations, the boundary-value problem for which is solved by the power-series method. The results of calculations of the directional pattern of the scattered field are presented.     

An optimization method for solving some differential algebraic equations

In this paper, the problem of differential algebraic equations has been solved via Chebyshev integral method combined with an optimization method. Two approaches are used based on the index of the problem: in the first, the proposed method is applied on the original problem and in the second, the index of the problem is decreased and the modified problem is solved. An optimization technique is proposed to solve the resulting algebraic equations. Numerical results are included to confirm the efficiency and accuracy of the method.     

Approximate global convergence and quasireversibility for a coefficient inverse problem with backscattering data

A numerical method possessing the approximate global convergence property is developed for a 3-D coefficient inverse problem for hyperbolic partial differential equations with backscattering data resulting from a single measurement. An important part of this technique is the quasireversibility method. An approximate global convergence theorem is proved. Results of two numerical experiments are presented. Bibliography: 46 titles. Illustrations: 2 figures.     

Quaternion non-linear filter for estimation of rotating body attitude

A problem of optimal estimation of a rotating vehicle's attitude with both differential equations of motion and observations of stars by onboard equipment is considered. The problem is formulated like a determinate non-linear Kalman filter problem in quaternion terms. An exact analytical solution of the problem is also found in quaternion terms. Various types of solutions in accordance with a mutual arrangement of observed stars are presented. An example of only one observed star is demonstrated to support the method. The method is also suited for geodetic applications, e.g. geodetic positioning and navigation systems.     

Compact difference methods applied to initial-boundary value problems for mixed systems

Summary. An initial--boundary value problem to a system of nonlinear partial differential equations, which consists of a hyperbolic and a parabolic part, is taken into consideration. The problem is discretised by a compact finite difference method. An approximation of the numerical solution is constructed, at which the difference scheme is linearised. Nonlinear convergence is proved using the stability of the linearised scheme. Finally, a computational experiment for a noncompact scheme is presented. Received May 20, 1995     

Integrability and beyond

An algebraic approach to solving nonlinear functional equations in the Riemann theta functions is stated. By the inverse scattering method and some general methods of the theory of partial differential equations, the solution of the initial boundary value problem for the nonlinear Schrödinger equation is presented. Bibliography:17 titles.     

A coupled finite element-differential quadrature element method and its accuracy for moving load problem

In this paper a mixed method, which combines the finite element method and the differential quadrature element method (DQEM), is presented for solving the time dependent problems. In this study, the finite element method is first used to discretize the spatial domain. The DQEM is then employed as a step-by-step DQM in time domain to solve the resulting initial value problem. The resulting algebraic equations can be solved by either direct or iterative methods. Two general formulations using the DQM are also presented for solving a system of linear second-order ordinary differential equations in time. The application of the formulation is then shown by solving a sample moving load problem. Numerical results show that the present mixed method is very efficient and reliable.     

基于目标流线的曲线方向寻优

本文对于无约束问题提出了沿曲线寻优的思想,推导了确定寻优曲线的常微分方程组。对该常微分方程组研制了近似解析和数值的实用算法.借助对偶规划,这一方法由无约束问题推广到约束问题.     

又一类三阶中立型分布时滞微分方程的振动定理

本文研究三阶中立型分布时滞微分方程的振动问题.利用Riccati变换技巧和积分平均方法,获得了方程每一解振动或者收敛到零点的新准则,最后,给出了说明所得结果应用的例子.     

The method of lines for Poisson's equation with nonlinear or free boundary conditions

Summary The method of lines is used to solve Poisson's equation on an irregular domain with nonlinear or free boundary conditions. The partial differential equation is approximated by a system of second order ordinary differential equations subject to multi-point boundary conditions. The system is solved with an SOR iteration which employs invariant imbedding for each one dimensional problem. An application of the method to a boundary control problem and to a free surface problem arising in electrochemical machining is described. Finally, some theoretical convergence results are presented for a model problem with radiative boundary conditions on fixed boundaries.This work was supported by the U.S. Army Research Office under Grant DA-AG29-76-G-0261     

Solution of the laminar viscous flow in a semi-porous channel in the presence of a uniform magnetic field by using the homotopy analysis method

In this paper, the problem of laminar viscous flow in a semi-porous channel in the presence of a transverse magnetic field is presented and the homotopy analysis method (HAM) is employed to compute an approximation to the solution of the system of nonlinear differential equations governing the problem. It has been attempted to show the capabilities and wide-range applications of the homotopy analysis method in comparison with the numerical method in solving this problem. The obtained solutions, in comparison with the numeric solutions admit a remarkable accuracy. A clear conclusion can be drawn from the numerical method’s (NM) results that the HAM provides highly accurate solutions for nonlinear differential equations.     

The solution of high-order nonlinear ordinary differential equations by Chebyshev Series

By the use of the Chebyshev series, a direct computational method for solving the higher order nonlinear differential equations has been developed in this paper. This method transforms the nonlinear differential equation into the matrix equation, which corresponds to a system of nonlinear algebraic equations with unknown Chebyshev coefficients, via Chebyshev collocation points. The solution of this system yields the Chebyshev coefficients of the solution function. An algorithm for this nonlinear system is also proposed in this paper. The method is valid for both initial-value and boundary-value problems. Several examples are presented to illustrate the accuracy and effectiveness of the method.