High-Order Numerical Solution of Second-Order One-Dimensional Hyperbolic Telegraph Equation Using a Shifted Gegenbauer Pseudospectral Method

We present a high-order shifted Gegenbauer pseudospectral method (SGPM) to solve numerically the second-order one-dimensional hyperbolic telegraph equation provided with some initial and Dirichlet boundary conditions. The framework of the numerical scheme involves the recast of the problem into its integral formulation followed by its discretization into a system of well-conditioned linear algebraic equations. The integral operators are numerically approximated using some novel shifted Gegenbauer operational matrices of integration. We derive the error formula of the associated numerical quadratures. We also present a method to optimize the constructed operational matrix of integration by minimizing the associated quadrature error in some optimality sense. We study the error bounds and convergence of the optimal shifted Gegenbauer operational matrix of integration. Moreover, we construct the relation between the operational matrices of integration of the shifted Gegenbauer polynomials and standard Gegenbauer polynomials. We derive the global collocation matrix of the SGPM, and construct an efficient computational algorithm for the solution of the collocation equations. We present a study on the computational cost of the developed computational algorithm, and a rigorous convergence and error analysis of the introduced method. Four numerical test examples have been carried out in order to verify the effectiveness, the accuracy, and the exponential convergence of the method. The SGPM is a robust technique, which can be extended to solve a wide range of problems arising in numerous applications.Comment: 36 pages, articl

High-Order, Stable, And Efficient Pseudospectral Method Using Barycentric Gegenbauer Quadratures

The work reported in this article presents a high-order, stable, and efficient Gegenbauer pseudospectral method to solve numerically a wide variety of mathematical models. The proposed numerical scheme exploits the stability and the well-conditioning of the numerical integration operators to produce well-conditioned systems of algebraic equations, which can be solved easily using standard algebraic system solvers. The core of the work lies in the derivation of novel and stable Gegenbauer quadratures based on the stable barycentric representation of Lagrange interpolating polynomials and the explicit barycentric weights for the Gegenbauer-Gauss (GG) points. A rigorous error and convergence analysis of the proposed quadratures is presented along with a detailed set of pseudocodes for the established computational algorithms. The proposed numerical scheme leads to a reduction in the computational cost and time complexity required for computing the numerical quadrature while sharing the same exponential order of accuracy achieved by Elgindy and Smith-Miles (2013). The bulk of the work includes three numerical test examples to assess the efficiency and accuracy of the numerical scheme. The present method provides a strong addition to the arsenal of numerical pseudospectral methods, and can be extended to solve a wide range of problems arising in numerous applications.Comment: 30 pages, 10 figures, 1 tabl

Dynamic reliability analysis using the extended support vector regression (X-SVR)

© 2019 Elsevier Ltd For engineering applications, the dynamic system responses can be significantly affected by uncertainties in the system parameters including material and geometric properties as well as by uncertainties in the excitations. The reliability of dynamic systems is widely evaluated based on the first-passage theory. To improve the computational efficiency, surrogate models are widely used to approximate the relationship between the system inputs and outputs. In this paper, a new machine learning based metamodel, namely the extended support vector regression (X-SVR), is proposed for the reliability analysis of dynamic systems via utilizing the first-passage theory. Furthermore, the capability of X-SVR is enhanced by a new kernel function developed from the vectorized Gegenbauer polynomial, especially for solving complex engineering problems. Through the proposed approach, the relationship between the extremum of the dynamic responses and the input uncertain parameters is approximated by training the X-SVR model such that the probability of failure can be efficiently predicted without using other computational tools for numerical analysis, such as the finite element analysis (FEM). The feasibility and performance of the proposed surrogate model in dynamic reliability analysis is investigated by comparing it with the conventional ε-insensitive support vector regression (ε-SVR) with Gaussian kernel and Monte Carlo simulation (MSC). Four numerical examples are adopted to evidently demonstrate the practicability and efficiency of the proposed X-SVR method

New Optimal Periodic Control Policy for the Optimal Periodic Performance of a Chemostat Using a Fourier-Gegenbauer-Based Predictor-Corrector Method

In its simplest form, the chemostat consists of microorganisms or cells which grow continually in a specific phase of growth while competing for a single limiting nutrient. Under certain conditions on the cells' growth rate, substrate concentration, and dilution rate, the theory predicts and numerical experiments confirm that a periodically operated chemostat exhibits an "over-yielding" state in which the performance becomes higher than that at the steady-state operation. In this paper we show that an optimal control policy for maximizing the chemostat performance can be accurately and efficiently derived numerically using a novel class of integral-pseudospectral methods and adaptive h-integral-pseudospectral methods composed through a predictor-corrector algorithm. Some new formulas for the construction of Fourier pseudospectral integration matrices and barycentric shifted Gegenbauer quadratures are derived. A rigorous study of the errors and convergence rates of shifted Gegenbauer quadratures as well as the truncated Fourier series, interpolation operators, and integration operators for nonsmooth and generally T-periodic functions is presented. We introduce also a novel adaptive scheme for detecting jump discontinuities and reconstructing a discontinuous function from the pseudospectral data. An extensive set of numerical simulations is presented to support the derived theoretical foundations.Comment: 35 pages, 20 figure

Application of Zernike polynomials in solving certain first and second order partial differential equations

Integration operational matrix methods based on Zernike polynomials are used to determine approximate solutions of a class of non-homogeneous partial differential equations (PDEs) of first and second order. Due to the nature of the Zernike polynomials being described in the unit disk, this method is particularly effective in solving PDEs over a circular region. Further, the proposed method can solve PDEs with discontinuous Dirichlet and Neumann boundary conditions, and as these discontinuous functions cannot be defined at some of the Chebyshev or Gauss-Lobatto points, the much acclaimed pseudo-spectral methods are not directly applicable to such problems. Solving such PDEs is also a new application of Zernike polynomials as so far the main application of these polynomials seem to have been in the study of optical aberrations of circularly symmetric optical systems. In the present method, the given PDE is converted to a system of linear equations of the form Ax = b which may be solved by both l1 and l2 minimization methods among which the l1 method is found to be more accurate. Finally, in the expansion of a function in terms of Zernike polynomials, the rate of decay of the coefficients is given for certain classes of functions

Fast Spectral Collocation Method for Solving Nonlinear Time-Delayed Burgers-Type Equations with Positive Power Terms

Since the collocation method approximates ordinary differential equations, partial differential equations, and integral equations in physical space, it is very easy to implement and adapt to various problems, including variable coefficient and nonlinear differential equations. In this paper, we derive a Jacobi-Gauss-Lobatto collocation method (J-GL-C) to solve numerically nonlinear time-delayed Burgers-type equations. The proposed technique is implemented in two successive steps. In the first one, we apply nodes of the Jacobi-Gauss-Lobatto quadrature which depend upon the two general parameters , and the resulting equations together with the two-point boundary conditions constitute a system of ordinary differential equations (ODEs) in time. In the second step, the implicit Runge-Kutta method of fourth order is applied to solve a system of ODEs of second order in time. We present numerical results which illustrate the accuracy and flexibility of these algorithms

Radar Technology

In this book “Radar Technology”, the chapters are divided into four main topic areas: Topic area 1: “Radar Systems” consists of chapters which treat whole radar systems, environment and target functional chain. Topic area 2: “Radar Applications” shows various applications of radar systems, including meteorological radars, ground penetrating radars and glaciology. Topic area 3: “Radar Functional Chain and Signal Processing” describes several aspects of the radar signal processing. From parameter extraction, target detection over tracking and classification technologies. Topic area 4: “Radar Subsystems and Components” consists of design technology of radar subsystem components like antenna design or waveform design