Solutions of convolution integral and integral equations via double general orthogonal polynomials

Double general orthogonal polynomials are developed in this work to approximate the solutions of convolution integrals, Volterra integral equations, and Fredholm integral equations. The proposed method reduces the computations of integral equations to the successive solution of a set of linear algebraic equations in matrix form; thus, the computational complexity is considerably simplified. Furthermore, the solutions obtained by the general orthogonal polynomials include as special cases solutions by Chebyshev polynomials, Legendre polynomials, Laguerre polynomials, or Jacobi polynomials. A comparison of the results obtained via several different classical orthogonal polynomial approximations is also presented.     

Two general integral inequalities and their applications to stability analysis for systems with time‐varying delay

Integral inequalities have been widely used in stability analysis for systems with time‐varying delay because they directly produce bounds for integral terms with respect to quadratic functions. This paper presents two general integral inequalities from which almost all of the existing integral inequalities can be obtained, such as Jensen inequality, the Wirtinger‐based inequality, the Bessel–Legendre inequality, the Wirtinger‐based double integral inequality, and the auxiliary function‐based integral inequalities. Based on orthogonal polynomials defined in different inner spaces, various concrete single/multiple integral inequalities are obtained. They can produce more accurate bounds with more orthogonal polynomials considered. To show the effectiveness of the new inequalities, their applications to stability analysis for systems with time‐varying delay are demonstrated with two numerical examples. Copyright © 2016 John Wiley & Sons, Ltd.     

Direct and indirect boundary element methods for solving the heat conduction problem

The boundary element method is used to solve the stationary heat conduction problem as a Dirichlet, a Neumann or as a mixed boundary value problem. Using singularities which are interpreted physically, a number of Fredholm integral equations of the first or second kind is derived by the indirect method. With the aid of Green's third identity and Kupradze's functional equation further direct integral equations are obtained for the given problem. Finally a numerical method is described for solving the integral equations using Hermitian polynomials for the boundary elements and constant, linear, quadratic or cubic polynomials for the unknown functions.     

A collocation approach for solving systems of linear Volterra integral equations with variable coefficients

In this paper, a numerical method is introduced to solve a system of linear Volterra integral equations (VIEs). By using the Bessel polynomials and the collocation points, this method transforms the system of linear Volterra integral equations into the matrix equation. The matrix equation corresponds to a system of linear equations with the unknown Bessel coefficients. This method gives an analytic solution when the exact solutions are polynomials. Numerical examples are included to demonstrate the validity and applicability of the technique and comparisons are made with existing results. All of the numerical computations have been performed on computer using a program written in MATLAB v7.6.0 (R2008a).     

Some Integrals of the products of laguerre polynomials

The evaluation of an integral of the product of Laguerre polynomials was discussed recently in this Journal by Mavromatis [12] (1990) and Lee [9] (1997) [see also Ong and Lee [14] (2000)]. The main object of the present sequel to these earlier works is to consider a family of such integrals of the products of Laguerre, Hermite, and other classical orthogonal polynomials in a systematic and unified manner. Relevant connections of some of these integral formulas with various known integrals, as well as the computational and numerical aspects of the results presented here, are also pointed out.     

Approximate solution of dual integral equations using Chebyshev polynomials

The aim of the present work is to introduce solution of special dual integral equations by the orthogonal polynomials. We consider a system of dual integral equations with trigonometric kernels which appear in formulation of the potential distribution of an electrified plate with mixed boundary conditions and convert them to Cauchy-type singular integral equations. We use the Chebyshev orthogonal polynomials to construct approximate solution for Cauchy-type singular integral equations which will solve the main dual integral equations. Numerical results demonstrate effectiveness of this method.     

Solution of integral equations via generalized orthogonal polynomials

A new approximation method using a generalized orthogonal polynomial (GOP) is employed for solving integral equations. The integration operational matrix of the GOP, which can represent all kinds of individual orthogonal polynomial, is developed. The dependent variables in the integral equation are assumed to be expressed by a GOP series. A set of algebraic equations is obtained from the integral equation. The calculation of coefficients is straightforward and easy. Examples are given, and the results obtained from individual orthogonal polynomial approximations are compared with each other. It is found that nearly all individual orthogonal polynomials, except Hermite polynomials, offer excellent results.     

Explicit Formulas for Interpolating Splines of Degree 5 on the Triangle

Explicit formulas are derived for 21 Zlamal–Zenisek basis interpolating polynomials of degree 5 in each triangle of the triangulation. Their use significantly reduces the number of arithmetic operations in the FEM because otherwise 21 systems with 21 unknowns should be solved in each triangle to find all the 21 coefficients of each of the basis interpolating polynomials of degree 5. The formulas are also presented for interpolation operators with the use of these basis polynomials and for the integral representation of the remainder term of the approximation of differentiable functions by these operators.     

Double-shifted Chebyshev series for convolution integral and integral equations

Double-shifted Chebyshev polynomials are developed in this study to approximate the solutions of the convolution integral, Volterra integral equation, and Fredholm integral equation. This method simplifies the computations of integral equations to the successive solutions of a linear algebraic equation in matrix form. In addition, the computational complexity can be reduced remarkably. Three examples are illustrated. It is seen that the proposed approach is straightforward and convenient, and converges faster in finding approximations than other existing orthogonal function methods.     

3 种伪谱最优控制方法的积分形式及统一性证明

为了进一步提高伪谱最优控制方法的计算精度, 削弱微分形式伪谱法对状态变量近似误差的放大幅度, 研究基于积分形式的伪谱最优控制方法. 依次给出3 种伪谱法的积分伪谱离散形式, 证明当Lagrange 多项式对状态变量的近似误差等于零时, Gauss 伪谱法和Radau 伪谱法的积分形式与微分形式是等价的, 而Legendre 伪谱法的积分形式与微分形式是不等价的, 并分析了其不等价的原因.

     

Computation of two time-dependent coefficients in a parabolic partial differential equation subject to additional specifications

In this paper, we consider the problem of the simultaneous determination of time-dependent coefficients in a one-dimensional partial differential equation. The main aim is to apply the tau technique to determine unknown coefficients in a time-dependent partial differential equation. Our approach consists of reducing the problem to a set of algebraic equations by expanding the approximate solution in terms of shifted Legendre polynomials with unknown coefficients. The operational matrices of integral and derivative together with the tau method are then utilized to evaluate the unknown coefficients of shifted Legendre polynomials. Illustrative examples are included to demonstrate the validity and applicability of the presented technique.     

Bessel summation inequalities for stability analysis of discrete‐time systems with time‐varying delays

This paper is concerned with the stability analysis problems of discrete‐time systems with time‐varying delays using summation inequalities. In the literature focusing on the Lyapunov‐Krasovskii approach, the Jensen integral/summation inequalities have played important roles to develop less conservative stability criteria and thus have been widely studied. Recently, the Jensen integral inequality was successfully generalized to the Bessel‐Legendre inequalities constructed with arbitrary‐order Legendre polynomials. It was also shown that general inequality contributes to the less conservatism of stability criteria. In the case of discrete‐time systems, however, the Jensen summation inequality are hardly extensible to general ones since there have still not been general discrete orthogonal polynomials applicable to the developments of summation inequalities. Motivated by such observations, this paper proposes novel discrete orthogonal polynomials and then successfully derives general summation inequalities. The resulting summation inequalities are discrete‐time counterparts of the Bessel‐Legendre inequalities but are not based on the discrete Legendre polynomials. By developing hierarchical stability criteria based on the proposed summation inequalities, the effectiveness of the proposed approaches is demonstrated via three numerical examples for the stability analysis of discrete‐time systems with time‐varying delays.     

A Generalization of the Euler-Maclaurin Summation Formula: An Application to Numerical Computation of the Fermi-Dirac Integrals

The present paper deals with a generalization of the Euler-Maclaurin summation formula. The generalization is based on Bernoulli functions which are expressed in an integral form involving Bernoulli polynomials. Then the formula is used to numerical computation of the Fermi-Dirac integrals.     

Numerical Verification of Solutions of Nekrasov’s Integral Equation

This paper describes numerical verification of solutions of Nekrasov’s integral equation which is a mathematical model of two-dimensional water waves. This nonlinear and periodic integral equation includes a logarithmic singular kernel which is typically found in some two-dimensional potential problems. We propose the verification method using some properties of the singular integral for trigonometric polynomials and Schauder’s fixed point theorem in the periodic Sobolev space. A numerical example shows effectiveness of the present method.     

Integral and non-negativity preserving Bernstein-type polynomial approximations

In this paper, we consider the problem of approximating a function by Bernstein-type polynomials that preserve the integral and non-negativity of the original function on the interval [0, 1], obtaining the Kantorovich–Bernstein polynomials, but providing a novel approach with advantages in numerical analysis. We then develop a Markov finite approximation method based on piecewise Bernstein-type polynomials for the computation of stationary densities of Markov operators, providing numerical results for piecewise constant and piecewise linear algorithms.     

Generalization of generalized orthogonal polynomial operational matrices for fractional and operational calculus

A method is given for the approximation of generalized orthogonal polynomials (GOP) to solve the problems of fractional and operational calculus. A more rigorous derivation for the generalized orthogonal polynomial operational matrices is proposed. The Riemann-Liouville fractional integral for repeated fractional (and operational) integration is integrated exactly, then expanded in generalized orthogonal polynomials to yield the generalized orthogonal polynomial operational matrices. The generalized orthogonal polynomial operational matrices perform as s^α(α ≥ αε R) in the Laplace domain and as fractional (and operational) integrators in the time domain. Using these results, inversions of the Laplace transforms of irrational and rational transfer functions are solved in a simple way. Very accurate results are obtained.     

Operational matrices of Bernstein polynomials and their applications

The Bernstein polynomials (B-polynomials) operational matrices of integration P, differentiation D and product ? are derived. A general procedure of forming these matrices are given. These matrices can be used to solve problems such as calculus of variations, differential equations, optimal control and integral equations. Illustrative examples are included to demonstrate the validity and applicability of the operational matrices.     

Modified method for determining an approximate solution of the Fredholm–Volterra integral equations by Taylor’s expansion

A modified method for determining an approximate solution of the Fredholm–Volterra integral equations of the second kind is developed. Via Taylor’s expansion of the unknown function, the integral equation to be solved is approximately transformed into a system of linear equations for the unknown and its derivatives, which can be dealt with in an easy way. The obtained nth-order approximate solution is of high accuracy, and is exact for polynomials of degree n. In particular, an approximate solution with satisfactory accuracy of the weakly singular Volterra integral equation is also given. The efficiency of the method is illustrated by some numerical examples.     

Synthesis of PID controllers for integral processes with time delay

This article deals with the problem of determination of the stabilizing parameter sets of Proportional‐Integral‐Derivative (PID) controllers for first‐order and second‐order integral processes with time‐delay. First, the admissible stabilizing range of proportional‐gain is determined analytically in terms of a version of the Hermite–Biehler Theorem applicable to quasi‐polynomials. Then, based on a graphical stability condition developed in parameter space, the complete stabilizing regions in an integral‐derivative plane are drawn and identified graphically, not calculated mathematically, by sweeping over the admissible range of proportional‐gain. An actual algorithm for finding the stabilizing parameter sets of PID controllers is also proposed. Simulations show that the stabilizing regions in integral‐derivative space are either triangles or quadrilaterals. Copyright © 2009 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

A reliable computational approach for approximate solution of Hammerstein integral equations of mixed type

The aim of this article is to present an efficient analytical and numerical procedure for solving the nonlinear Hammerstein integral equations of mixed type. Our method mainly depends on a Taylor expansion approach. Also, we obtain the approximate solution of the nonlinear Volterra–Hammerstein integral equations of mixed type in terms of the Taylor polynomials. The reliability and efficiency of the proposed scheme are demonstrated by some numerical experiments.