Non-polynomial Cubic Spline Method for the Solution of Second-order Linear Hyperbolic Equation

Second-order linear hyperbolic equations are solved by using a new three level method based on nonpolynomial spline in the space direction and Taylor expansion in the time direction. Numerical results reveal that three level method based on non-polynomial spline is implemented and effective

A Brief Overview of Results about Uniqueness of the Quantization in Cosmology

The purpose of this review is to provide a brief overview of recent conceptual developments regarding possible criteria to guarantee the uniqueness of the quantization in a variety of situations that are found in cosmological systems. These criteria impose certain conditions on the representation of a group of physically relevant linear transformations. Generally, this group contains any existing symmetry of the spatial sections. These symmetries may or may not be sufficient for the purpose of uniqueness and may have to be complemented with other remaining symmetries that affect the time direction or with dynamical transformations that are, in fact, not symmetries. We discuss the extent to which a unitary implementation of the resulting group suffices to fix the quantization—a demand that can be seen as a weaker version of the requirement of invariance. In particular, a strict invariance under certain transformations may eliminate some physically interesting possibilities in the passage to the quantum theory. This is the first review in which this unified perspective is adopted to discuss otherwise different uniqueness criteria proposed either in homogeneous loop quantum cosmology or in the Fock quantization of inhomogeneous cosmologies.

Sample paths of white noise in spaces with dominating mixed smoothness

The sample paths of white noise are proved to be elements of certain Besov spaces with dominating mixed smoothness. Unlike in isotropic spaces, here the regularity does not get worse with increasing space dimension. Consequently, white noise is actually much smoother than the known sharp regularity results in isotropic spaces suggest. An application of our techniques yields new results for the regularity of solutions of Poisson and heat equation on the half space with boundary noise. The main novelty is the flexible treatment of the interplay between the singularity at the boundary and the smoothness in tangential, normal and time direction.

Computational Sinc-scheme for extracting analytical solution for the model Kuramoto-Sivashinsky equation

The present article is designed to supply two different numerical<br />solutions for solving Kuramoto-Sivashinsky equation. We have made<br />an attempt to develop a numerical solution via the use of<br />Sinc-Galerkin method for Kuramoto-Sivashinsky equation, Sinc<br />approximations to both derivatives and indefinite integrals reduce<br />the solution to an explicit system of algebraic equations. The fixed<br />point theory is used to prove the convergence of the proposed<br />methods. For comparison purposes, a combination of a Crank-Nicolson<br />formula in the time direction, with the Sinc-collocation in the<br />space direction is presented, where the derivatives in the space<br />variable are replaced by the necessary matrices to produce a system<br />of algebraic equations. In addition, we present numerical examples<br />and comparisons to support the validity of these proposed<br />methods.

BRIM: An Accurate Electricity Spot Price Prediction Scheme-Based Bidirectional Recurrent Neural Network and Integrated Market

For the benefit from accurate electricity price forecasting, not only can various electricity market stakeholders make proper decisions to gain profit in a competitive environment, but also power system stability can be improved. Nevertheless, because of the high volatility and uncertainty, it is an essential challenge to accurately forecast the electricity price. Considering that recurrent neural networks (RNNs) are suitable for processing time series data, in this paper, we propose a bidirectional long short-term memory (LSTM)-based forecasting model, BRIM, which splits the state neurons of a regular RNN into two parts: the forward states (using the historical electricity price information) are designed for processing the data in positive time direction and backward states (using the future price information available at inter-connected markets) for the data in negative time direction. Moreover, due to the fact that inter-connected power exchange markets show a common trend for other neighboring markets and can provide signaling information for each other, it is sensible to incorporate and exploit the impact of the neighboring markets on forecasting accuracy of electricity price. Specifically, future electricity prices of the interconnected market are utilized both as input features for forward LSTM and backward LSTM. By testing on day-ahead electricity prices in the European Power Exchange (EPEX), the experimental results show the superiority of the proposed method BRIM in enhancing predictive accuracy in comparison with the various benchmarks, and moreover Diebold-Mariano (DM) shows that the forecast accuracy of BRIM is not equal to other forecasting models, and thus indirectly demonstrates that BRIM statistically significantly outperforms other schemes.

Numerical simulations of two components ecological models with sinc collocation method

It is well known that reaction-diffusion systems can describe the interactions of different species in ecological systems. In this work, we propose a numerical method for solving a general two-species reaction-diffusion system. The method comprises a standard finite difference to discretize in time direction and Sinc collocation method in spatial direction. A series of numerical experiments demonstrate the accuracy and good performance of the algorithm. The biological significance of the numerical results was also discussed and presented in figures.