A new computational approach for solving optimal control of linear PDEs problem

In this paper, we present a new computational approach for solving an internal optimal control problem, which is governed by a linear parabolic partial differential equation. Our approach is to approximate the PDE problem by a nonhomogeneous ordinary differential equation system in higher dimension. Then, the homogeneous part of ODES is solved using semigroup theory. In the next step, the convergence of this approach is verified by means of Toeplitz matrix. In the rest of the paper, the optimal control problem is solved by utilizing the solution of homogeneous part. Finally, a numerical example is given.     

Numerical piecewise approximate solution of Fredholm integro-differential equations by the Tau method

A general form of numerical piecewise approximate solution of linear integro-differential equations of Fredholm type is discussed. It is formulated for using the operational Tau method to convert the differential part of a given integro-differential equation, or IDE for short, to its matrix representation. This formulation of the Tau method can be useful for such problems over long intervals and also can be used as a good and simple alternative algorithm for other piecewise approximations such as splines or collocation. A Tau error estimator is also adapted for piecewise application of the Tau method. Some numerical examples are considered to demonstrate the implementation and general effect of application of this (segmented) piecewise Chebyshev Tau method.     

Identification of Parameters through the Approximate Periodic Solutions of a Parabolic System

This work is concerned with the identification problem for the perturbation term or error term in a parabolic partial differential equation through its approximate periodic solutions. The observation is made over a subregion of the physical domain. The existence and uniqueness problem of the approximate periodic solutions is studied in the first part of the paper. A solution to the identification problem is given in the second part of the paper. The main ingredients to be used include the classical Galerkin method and the unique continuation property for a parabolic system. This work was supported by the National Natural Science Foundation of China Grant 10571161.     

Computational methods of solving the boundary value problems for the loaded differential and Fredholm integro‐differential equations

The article presents a new general solution to a loaded differential equation and describes its properties. Solving a linear boundary value problem for loaded differential equation is reduced to the solving a system of linear algebraic equations with respect to the arbitrary vectors of general solution introduced. The system's coefficients and right sides are computed by solving the Cauchy problems for ordinary differential equations. Algorithms of constructing a new general solution and solving a linear boundary value problem for loaded differential equation are offered. Linear boundary value problem for the Fredholm integro‐differential equation is approximated by the linear boundary value problem for loaded differential equation. A mutual relationship between the qualitative properties of original and approximate problems is obtained, and the estimates for differences between their solutions are given. The paper proposes numerical and approximate methods of solving a linear boundary value problem for the Fredholm integro‐differential equation and examines their convergence, stability, and accuracy.     

Reconstruction of the Linear Ordinary Differential System Based on Discrete Points

In this paper, we discuss an inverse problem, i.e., the reconstruction of a linear differential dynamic system from the given discrete data of the solution. We propose a model and a corresponding algorithm to recover the coefficient matrix of the differential system based on the normal vectors from the given discrete points, in order to avoid the problem of parameterization in curve fitting and approximation. We also give some theoretical analysis on our algorithm. When the data points are taken from the solution curve and the set composed of these data points is not degenerate, the coefficient matrix $A$ reconstructed by our algorithm is unique from the given discrete and noisefree data. We discuss the error bounds for the approximate coefficient matrix and the solution which are reconstructed by our algorithm. Numerical examples demonstrate the effectiveness of the algorithm.     

An adaptive numerical method for solving linear Fredholm integral equations of the first kind

A general procedure is presented for numerically solving linear Fredholm integral equations of the first kind. The approximate solution is expressed as a continuous piecewise linear (spline) function. The method involves collocation followed by the solution of an appropriately scaled stabilized linear algebraic system. The procedure may be used iteratively to improve the accuracy of the approximate solution. Several numerical examples are given.Supported in part by the Office of Naval Research under Contract No. NR 044-457.Supported in part by the National Science Foundation under Grant No. GJ-31827.     

Solving a parabolic PDE with nonlocal boundary conditions using the Sinc method

In this paper, the problem of solving the parabolic partial differential equations subject to given initial and nonlocal boundary conditions is considered. We change the problem to a system of Volterra integral equations of convolution type. By using Sinc-collocation method, the resulting integral equations are replaced by a system of linear algebraic equations. The convergence analysis is included, and it is shown that the error in the approximate solution is bounded in the infinity norm by the condition number of the coefficient matrix multiplied by a factor that decays exponentially with the size of the system. Some examples are considered to illustrate the ability of this method.     

Free vibration of non-uniform Euler–Bernoulli beam under various supporting conditions using Chebyshev wavelet collocation method

This paper proposes operational matrix of rth integration of Chebyshev wavelets. A general procedure of this matrix is given. Operational matrix of rth integration is taken as rth power of operational matrix of first integration in literature. But, this study removes this disadvantage of Chebyshev wavelets method. Free vibration problems of non-uniform Euler–Bernoulli beam under various supporting conditions are investigated by using Chebyshev Wavelet Collocation Method. The proposed method is based on the approximation by the truncated Chebyshev wavelet series. A homogeneous system of linear algebraic equations has been obtained by using the Chebyshev collocation points. The determinant of coefficients matrix is equated to the zero for nontrivial solution of homogeneous system of linear algebraic equations. Hence, we can obtain ith natural frequencies of the beam and the coefficients of the approximate solution of Chebyshev wavelet series that satisfied differential equation and boundary conditions. Mode shapes functions corresponding to the natural frequencies can be obtained by normalizing of approximate solutions. The computed results well fit with the analytical and numerical results as in the literature. These calculations demonstrate that the accuracy of the Chebyshev wavelet collocation method is quite good even for small number of grid points.     

A Shadowing Theorem for ordinary differential equations

A new notion of shadowing of a pseudo orbit, an approximate solution, of an autonomous system of ordinary differential equations by an associated nearby true orbit is introduced. Then a general theorem which guarantees the existence of shadowing of pseudo orbits in compact hyperbolic sets is proved.Supported in part by the Air Force.Supported in part by NSF grant DMS 9201951.     

N-S方程一般近似惯性流形构造和逼近

1．引言为了用有限维常微分方程来研究Navier七lobes（N七）方程的长时间动力学行为,Foias,Sell和Temaml］引入了耗散系统惯性流形的概念．但惯性流形存在的一个相当苛刻的条件一谱间隔条件一是包括N习方程在内的很多耗散系统无法满足的．因此,Foias,Manlea和TemamZ］随后又提出了近似惯性流形的概念．近似惯性流形也是一个光滑的Lipschitz流形,所有原方程的解在时间充分大时,将被吸引进入该流形的一个三邻域中．因其存在性不需要谱间隔条件来保证,从而可证明包括N－S方程在内的一大类耗散系统存在近似惯性流形．利用近似惯性…     

A computational method for large‐scale differential symmetric Stein equation

We propose a numerical method for solving large‐scale differential symmetric Stein equations having low‐rank right constant term. Our approach is based on projection the given problem onto a Krylov subspace then solving the low dimensional matrix problem by using an integration method, and the original problem solution is built by using obtained low‐rank approximate solution. Using the extended block Arnoldi process and backward differentiation formula (BDF), we give statements of the approximate solution and corresponding residual. Some numerical results are given to show the efficiency of the proposed method.     

混合广义Jacobi和Chebyshev谱配置法求解时间分数阶对流扩散方程

研究时间Caputo分数阶对流扩散方程的高效高阶数值方法.对于给定的时间分数阶偏微分方程,在时间和空间方向分别采用基于移位广义Jacobi函数为基底和移位Chebyshev多项式运算矩阵的谱配置法进行数值求解.这样得到的数值解可以很好地逼近一类在时间方向非光滑的方程解.最后利用一些数值例子来说明该数值方法的有效性和准确性.     

Control synthesis for a nonlinear nonstationary process

A system of nonlinear differential equations is considered. The relationship of the general solution of this system with the general solution of a certain linear system of differential equations is studied. The result leads to the solution of a control problem that produces a given optimal control synthesis.Kommunarnyi Mining and Metallurgical Institute. Translated from Vychislitel'naya i Prikladnaya Matematika, No. 67, 123–129, 1989.     

Pseudospectral approximation of eigenvalues of derivative operators with non-local boundary conditions

By taking as a “prototype problem” a one-delay linear autonomous system of delay differential equations we present the problem of computing the characteristic roots of a retarded functional differential equation as an eigenvalue problem for a derivative operator with non-local boundary conditions given by the particular system considered. This theory can be enlarged to more general classes of functional equations such as neutral delay equations, age-structured population models and mixed-type functional differential equations.It is thus relevant to have a numerical technique to approximate the eigenvalues of derivative operators under non-local boundary conditions. In this paper we propose to discretize such operators by pseudospectral techniques and turn the original eigenvalue problem into a matrix eigenvalue problem. This approach is shown to be particularly efficient due to the well-known “spectral accuracy” convergence of pseudospectral methods. Numerical examples are given.     

Reconstruction of solutions to a generalized Moisil-Theodorescu system in a spatial domain from their values on a part of the boundary

In this paper we consider the problem of reconstructing solutions to a generalized Moisil-Theodorescu system in a spatial domain from their values on a part of the domain boundary, i.e., the Cauchy problem. We construct an approximate solution to this problem with the help of the Carleman matrix method.     

Analysis of the second order matrix Riccati equations

This paper is concerned with periodic solutions of 2x2 autonomous matrix Riccati differential equations. The author had given a necessary and sufficient condition for periodicity of solutions of matrix Riccati differential equations of general type and some examples. However, it is not so simple to verify whether this condition is satisfied or not. So this paper simplifies the verification by restricting to special cases. In particular, we show that there may exist periodic solutions for any case where the coefficient matrix of the linear part of the equation has complex eigenvalues if we choose an initial value suitably. Many examples having a periodic solution are also shown by systematic analysis; such examples are seldom seen in the literature.     

On the solution of the non-local parabolic partial differential equations via radial basis functions

In this paper, the problem of solving the one-dimensional parabolic partial differential equation subject to given initial and non-local boundary conditions is considered. The approximate solution is found using the radial basis functions collocation method. There are some difficulties in computing the solution of the time dependent partial differential equations using radial basis functions. If time and space are discretized using radial basis functions, the resulted coefficient matrix will be very ill-conditioned and so the corresponding linear system cannot be solved easily. As an alternative method for solution, we can use finite-difference methods for discretization of time and radial basis functions for discretization of space. Although this method is easy to use but an accurate solution cannot be provided. In this work an efficient collocation method is proposed for solving non-local parabolic partial differential equations using radial basis functions. Numerical results are presented and are compared with some existing methods.     

Conditions for the existence of a solution of a system of differential equations close to an approximate solution

We consider nonautonomous systems of differential equations and state conditions for the existence of an exact solution in a neighborhood of an approximate one by analyzing the linear system of the first approximation in a neighborhood of the constructed approximate solution. We present conditions for the existence of a bounded solution of a linear inhomogeneous system of differential equations.     

Matrix displacement mappings in the numerical solution of functional and nonlinear differential equations with the tau method

The use of matrix displacement mappings reduces most matrix operations required in the construction of an approximate solution of a functional or differential equation by means of Ortiz' formulation of the Tau method to index shifts. The coefficient vector of the approximate solution is defined implicitly by a very sparse system of linear algebraic equations. The contributions of the differential or functional operator, and of the supplementary conditions of the problem (initial, boundary, or multipoint conditions) are treated with a single and versatile procedure of remarkable simplicity, which can be easily implemented in a computer. We give two nontrivial examples on the application of this approach: the first is a nonlinear boundary value problem with a continuous locus of singular points and multiple solutions, where stiffness is present, the second is a functional differential equation arising in analytic number theory. In both cases we obtain results of nigh accuracy.