In this study, we present a numerical method for solving two--dimensional space--time fractional partial differential equations (FPDEs), where the solutions of the FPDEs are expanded in terms of the shifted Chebyshev polynomials. The numerical approximations are evaluated at the Chebyshev-Gauss-Lobatto points. In the case when the FPDE is nonlinear, we employ a Newton-Raphson approach to linearize the equation. Both the linear and nonlinear cases lead to a consistent system of linear algebraic equations. The scheme is tested on selected FPDEs and the numerical results show that the proposed numerical scheme is accurate and computationally efficient in terms of CPU times. To establish the accuracy of the method, we also present an error analysis which shows the convergence of the numerical method. These positive attributes make the proposed method a good approach for solving two--dimensional fractional partial differential equations with both space and time fractional orders.

We establish certain modular equations of degree three based on some results of Ramanujan's theta functions.

The present work concerns with presenting an explicit formula for new shifted wavelet (NSW) functions. Wavelet functions have many applications and advantages in both applied and theoretical fields. They are formulated with different orthogonal polynomials to construct new techniques for treating some problems in sciences, and engineering. A new important differentiation property of NSW in terms of NSW themselves is obtained and proved in this paper. Then it is utilized together with the state parameterization technique to find solution of optimal control problem (OCP) approximately. The suggested method converts the OCP into a quadratic programming problem, which can be easily determined on computer. As a result, the approximate solution closes with the exact solution even with a small number of NSW utilized in estimation. The error bound estimation for the proposed method is also discussed. Some test numerical examples are solved to demonstrate the applicability of the suggested method. For comparison, the exact known solutions against the obtained approximate results are listed in Tables.

This paper presents the design of a high frequency zero voltage switching (ZVS) full-bridge converter with a phase-shifted driving signal for photovoltaic applications. The resonant power converter can provide high-power capacity under high-frequency operation. The proposed power converter can also reduce the size of the transformer under the same power rating. The high-frequency transformer was developed by using the resonant and switching frequencies of the power converter to reduce the switching loss and to improve the system efficiency. Phase-shifted modulation was selected to drive the switches of a full-bridge power converter based on the switching loss minimization method. The desired output voltage was controlled using a closed-loop controller under a loop gain stability margin. The simulation results showed that the output voltage can be controlled to the desired constant when the input voltage changes from 30 VDC to 60 VDC. The desired output voltage of power converter is constant at 400 VDC. The power converter can transfer the DC supply to a 220 VAC household via grid-connected inverter. Therefore, the proposed study showed the effectiveness of the phase-shift ZVS full-bridge power converter with high-frequency transformer. This power converter can control the operation of the desired voltage system and has a small sizing of power converter system, low switching loss, and high system efficiency.

In this work, an improved collocation method based on the Bernoulli polynomials is presented to solve the Volterra integral equation (VIE) of the second kind. The main idea of the proposed method is to improve the results of the classic Bernoulli collocation method (BCM) by dividing the interval into some sub-intervals and considering the collocation points on each of them. Here, the zeros of the shifted Chebyshev polynomials (SCPs) are considered as collocation points. Then, BCM is applied step by step from the first sub-interval to the last one. By this process, a system of algebraic equations is attained for each sub-interval that could be easily solved. Convergence of the scheme is analyzed. For the purpose of demonstrating the validity, applicability, and efficiency of the suggested scheme several numerical examples are provided. Numerical results illustrate that the accuracy of the improved Bernoulli collocation method (IBCM) is more than BCM.

In recent years, the analysis of macro- and micro-expression has drawn the attention of researchers since they provide visual cues of an individual's emotions for a broad range of potential applications such as lie detection and criminal detection. In this paper, we address the challenge of spotting facial macro- and micro-expression from videos and present compelling results by using a deep learning approach to analyze the optical flow features. Different from other deep learning approaches that are mainly based on Convolutional Neural Networks (CNNs), we propose a Transformer-based deep learning approach that predicts a score indicating the probability of a frame being within an expression interval. In contrast to other Transformer-based models that achieve high performance by being pre-trained on large datasets, our deep learning model, called SL-Swin, which incorporates Shifted Patch Tokenization and Locality Self-Attention into the backbone network Swin Transformer, effectively spots macro- and micro-expressions by being trained from scratch on small-size expression datasets. Our evaluation outcomes surpass the MEGC 2022 spotting baseline result, obtaining an overall F1-score of 0.1366. Additionally, our approach also performs well in the MEGC 2021 spotting task with an overall F1-score of 0.1824 and 0.1357 on CAS(ME)^2 and SAMM Long Videos, respectively. The code is publicly available on GitHub (https://github.com/eddiehe99/pytorch-expression-spotting).