Solving boundary value problems via the Nyström method using spline Gauss rules

We propose to use spline Gauss quadrature rules for solving boundary value problems (BVPs) using the Nyström method. When solving BVPs, one converts the corresponding partial differential equation inside a domain into the Fredholm integral equation of the second kind on the boundary in the sense of boundary integral equation (BIE). The Fredholm integral equation is then solved using the Nyström method, which involves the use of a particular quadrature rule, thus, converting the BIE problem to a linear system. We demonstrate this concept on the 2D Laplace problem over domains with smooth boundary as well as domains containing corners. We validate our approach on benchmark examples and the results indicate that, for a fixed number of quadrature points (i.e., the same computational effort), the spline Gauss quadratures return an approximation that is by one to two orders of magnitude more accurate compared to the solution obtained by traditional polynomial Gauss counterparts

Solving Boundary Value Problems Via the Nyström Method Using Spline Gauss Rules

We propose to use spline Gauss quadrature rules for solving boundary value problems (BVPs) using the Nyström method. When solving BVPs, one converts the corresponding partial differential equation inside a domain into the Fredholm integral equation of the second kind on the boundary in the sense of boundary integral equation (BIE). The Fredholm integral equation is then solved using the Nyström method, which involves a use of a particular quadrature rule, thus, converting the BIE problem to a linear system. We demonstrate this concept on the 2D Laplace problem over domains with smooth boundary as well as domains containing corners. We validate our approach on benchmark examples and the results indicate that, for a fixed number of quadrature points (i.e., the same computational effort), the spline Gauss quadratures return an approximation that is by one to two orders of magnitude more accurate compared to the solution obtained by traditional polynomial Gauss counterparts

On numerical solution of Fredholm and Hammerstein integral equations via Nystr\"{o}m method and Gaussian quadrature rules for splines

Nystr\"{o}m method is a standard numerical technique to solve Fredholm integral equations of the second kind where the integration of the kernel is approximated using a quadrature formula. Traditionally, the quadrature rule used is the classical polynomial Gauss quadrature. Motivated by the observation that a given function can be better approximated by a spline function of a lower degree than a single polynomial piece of a higher degree, in this work, we investigate the use of Gaussian rules for splines in the Nystr\"{o}m method. We show that, for continuous kernels, the approximate solution of linear Fredholm integral equations computed using spline Gaussian quadrature rules converges to the exact solution for $m \rightarrow \infty$, $m$ being the number of quadrature points. Our numerical results also show that, when fixing the same number of quadrature points, the approximation is more accurate using spline Gaussian rules than using the classical polynomial Gauss rules. We also investigate the non-linear case, considering Hammerstein integral equations, and present some numerical tests.RYC-2017-2264

A numerical solution of fredholm integral equations of the second kind based on tight framelets generated by the oblique extension principle

© 2019 by the authors. In this paper, we present a new computational method for solving linear Fredholm integral equations of the second kind, which is based on the use of B-spline quasi-affine tight framelet systems generated by the unitary and oblique extension principles. We convert the integral equation to a system of linear equations. We provide an example of the construction of quasi-affine tight framelet systems. We also give some numerical evidence to illustrate our method. The numerical results confirm that the method is efficient, very effective and accurate

Aircraft turbulence and gust identification using simulated in-flight data

Gust and turbulence events are of primary importance for the analysis of flight incidents, for the design of gust load alleviation systems and for the calculation of loads in the airframe. Gust and turbulence events cannot be measured directly but they can be obtained through direct or optimisation-based methods. In the direct method the discretisation of the Fredholm Integral equation is associated with an ill conditioned matrix. In this work the effects of regularisation methods including Tikhonov regularisation, Truncated Single Value Decomposition (TSVD), Damped Single Value Decomposition (DSVD) and a recently proposed method using cubic B-spline functions are evaluated for aeroelastic gust identification using in flight measured data. The gust identification methods are tested in the detailed aeroelastic model of FFAST and an equivalent low-fidelity aeroelastic model developed by the authors. In addition, the accuracy required in the model for a reliable identification is discussed. Finally, the identification method based on B-spline functions is tested by simultaneously using both low-fidelity and FFAST aeroelastic models so that the response from the FFAST model is used as measurement data and the equivalent low-fidelity model is used in the identification process