Scattering of water waves by rectangular thick barriers in presence of surface tension

The influence of surface tension over an oblique incident waves in presence of thick rectangular barriers present in water of uniform finite depth is discussed here. Three different structures of a bottom-standing submerged barrier, submerged rectangular block not extending down to the bottom and fully submerged block extending down to the bottom with a finite gap are considered. An appropriate multi-term Galekin approximation technique involving ultraspherical Gegenbauer polynomial is employed for solving the integral equations arising in the mathematical analysis. The reflection and transmission coefficients of the progressive waves for two-dimensional time har- monic motion are evaluated by utilizing linearized potential theory. The theoretical result is validated numerically and explained graphically in a number of figures. The present result will almost match analytically and graphically with those results already available in the literature without considering the effect of surface tension. From the graphical representation, it is clearly visible that the amplitude of reflection coefficient decreases with increasing values of surface tension. It is also seen that the presence of surface tension, the change of width, and the height of the thick barriers affect the nature of the reflection coefficients significantly

Structural Reliability Analysis via the Multivariate Gegenbauer Polynomial-Based Sparse Surrogate Model

Structural reliability analysis is usually realized based on a multivariate performance function that depicts failure mechanisms of a structural system. The intensively computational cost of the brutal-force Monte-Carlo simulation motivates proposing a Gegenbauer polynomial-based surrogate model for effective structural reliability analysis in this paper. By utilizing the orthogonal matching pursuit algorithm to detect significant explanatory variables at first, a small number of samples are used to determine a reliable approximation result of the structural performance function. Several numerical examples in the literature are presented to demonstrate potential applications of the Gegenbauer polynomial-based sparse surrogate model. Accurate results have justified the effectiveness of the proposed approach in dealing with various structural reliability problems.

On Decoupling the Integrals of Cosmological Perturbation Theory

Perturbation theory (PT) is often used to model statistical observables capturing the translation and rotation-invariant information in cosmological density fields. PT produces higher-order corrections by integration over linear statistics of the density fields weighted by kernels resulting from recursive solution of the fluid equations. These integrals quickly become high-dimensional and naively require increasing computational resources the higher the order of the corrections. Here we show how to decouple the integrands that often produce this issue, enabling PT corrections to be computed as a sum of products of independent 1-D integrals. Our approach is related to a commonly used method for calculating multi-loop Feynman integrals in Quantum Field Theory, the Gegenbauer Polynomial x-Space Technique (GPxT). We explicitly reduce the three terms entering the 2-loop power spectrum, formally requiring 9-D integrations, to sums over successive 1-D radial integrals. These 1-D integrals can further be performed as convolutions, rendering the scaling of this method Nglog Ng with Ng the number of grid points used for each Fast Fourier Transform. This method should be highly enabling for upcoming large-scale structure redshift surveys where model predictions at an enormous number of cosmological parameter combinations will be required by Monte Carlo Markov Chain searches for the best-fit values.

Admixed recurrent Gegenbauer polynomials neural network with mended particle swarm optimization control system for synchronous reluctance motor driving continuously variable transmission system

To cut down influence of nonlinear time-varying uncertainty action in a synchronous reluctance motor driving continuously variable transmission system, an admixed recurrent Gegenbauer polynomials neural network with mended particle swarm optimization control system is posed for improving control performance. The admixed recurrent Gegenbauer polynomials neural network with mended particle swarm optimization control system involves an observer control, a recurrent Gegenbauer polynomial neural network control and a remunerated control. The weights of recurrent Gegenbauer polynomials neural network controller are regulated by using the adaptive law and the gradient descent technology. The remunerated control with a reckoned law is derived and computed by means of the Lyapunov stability theorem so as to pledge stability of the control system. Likewise, to speedup convergence of weights in the recurrent Gegenbauer polynomial neural network, the mended particle swarm optimization algorithm is used for regulating two kinds of learning rates. At last, three kinds of experimental results are demonstrated to confirm the usefulness of the put forward control system with comparative studies.

On certain integral formulas involving the product of Bessel function and Jacobi polynomial

In the present paper, we establish some interesting integrals involving the product of Bessel function of the first kind with Jacobi polynomial, which are expressed in terms of Kampe de Feriet and Srivastava and Daoust functions. Some other integrals involving the product of Bessel (sine and cosine) function with ultraspherical polynomial, Gegenbauer polynomial, Tchebicheff polynomial, and Legendre polynomial are also established as special cases of our main results. Further, we derive an interesting connection between Kampe de Feriet and Srivastava and Daoust functions.