Fractional multiwavelet methods for solving spatiotemporal fractional diffusion equations with non-smooth solutions

In this work, anisotropic bilinear finite element and second‐order temporal approximation are adopted to establish a fully discrete scheme for the fractional substantial diffusion equation with variable coefficient. We prove the proposed discrete scheme is unconditionally stable in the senses of ‐norm and ‐norm. By introducing a new projection, the optimal convergence error in ‐norm and the superclose property in ‐norm are obtained under the condition of , where is the exact solution of the problem. Through interpolation post processing technique, we arrive at the global superconvergence of the interpolation. Furthermore, an improved algorithm is proposed to solve the problem with nonsmooth solution. Finally, numerical tests are given to illustrate the validity and efficiency of our theoretical analysis.

A fully discrete scheme is proposed for the nonlinear fractional delay diffusion equations with smooth solutions, where the fractional derivative is described in Caputo sense with the order $\alpha$ ($0 \lt \alpha \lt 1$). The scheme is constructed by combining finite difference method in time and Legendre spectral approximation in space. Stability and convergence are proved rigorously. Moreover, a modified scheme is proposed for the equation with nonsmooth solutions by adding correction terms to the approximations of fractional derivative operator and nonlinear term. Numerical examples are carried out to support the theoretical analysis.

Finite difference scheme on graded meshes to the time-fractional neutron diffusion equation with non-smooth solutions

Fast algorithm based on TT-M FE system for space fractional Allen–Cahn equations with smooth and non-smooth solutions

Purpose This study aims to develop and analyze a numerical method to solve fractional diffusion equations (FDEs) of one dimension (1D) and two dimensions (2D) that incorporate the Caputo derivative with a generalized kernel (CDGK). Design/methodology/approach The collocation method is applied to compute the numerical solution, with a detailed discussion of the error and convergence properties. Findings The proposed CDGK is efficient, accurate and effective for solving 1D and 2D FDEs. The study demonstrates that the provided approach successfully handles both smooth and nonsmooth solutions. Furthermore, varying the scale function within the CDGK framework significantly influences, providing flexibility in modeling complex diffusion processes. Originality/value The novelty stems from applying the collocation method within the CDGK framework, providing a comprehensive investigation of error and convergence properties. Unlike existing methods, this study systematically explores the impact of varying the scale function on numerical solutions. It addresses both smooth and nonsmooth solutions under homogeneous and nonhomogeneous boundary conditions.

This study presents the development and evaluation process of a vibrating wire sensor readout named DUT Vibro, with both hardware and software entirely developed by Vietnamese researchers. The primary objective of this paper is to evaluate the efficacy of two methods to determine the resonant frequency of the vibrating wire to facilitate precise strain calculations. In this study, parabolic interpolation is investigated to improve the accuracy of the resonant frequency of a steel wire, addressing the limitation of Fast Fourier Transform (FFT) constrained by the storage capacity of the microcontroller. In addition, the determined resonant frequency of DUT Vibro is compared with a commercial DIGIANGLE (DAS). Furthermore, two stimulation signals—sine and square waves—were employed to compare their impact on measurement accuracy. The results indicate that the parabolic interpolation method yields the lowest standard deviation, closely aligning with the DAS readout, and demonstrates stability across both low and high load conditions. In contrast, the FFT method exhibits greater error variability, particularly in the medium load range, due to the influence of noise and non-linearities in the response signal. The sine wave stimulus combined with parabolic interpolation achieves the highest accuracy. The measurement system maintains high linearity, with linearity errors below 0.5% of full scale (FS), and the lowest linearity error is 0.129% FS when using a sine wave stimulus. Linear regression analysis reveals a slope coefficient of approximately 0.052, reflecting a linear relationship between load and measured strain. Based on these findings, the parabolic interpolation method has been integrated into the DUT Vibro readout, meeting stringent accuracy requirements for strain measurement applications.

Convergence analysis of the anisotropic FEM for 2D time fractional variable coefficient diffusion equations on graded meshes

A Method for Threshold Setting and False Alarm Probability Evaluation for Radar Detectors

A direct digital frequency synthesizer (DDFS) based on a non-equal division parabolic polynomial interpolation method is proposed in this paper. To attain high spurious free dynamic range (SDRF) and reduce area cost, a parabolic polynomial interpolation method is adopted in the proposed design to replace conventional ROM-based phase-to-sine mapper methods. Particularly, the left 1/4 of the phase range is approximated using a low-curvature parabolic curve. The proposed design is manufactured using a standard 0.18 μm CMOS technology. The maximum output frequency is 50 MHz, the core area is 1.4528 mm2, and the spurious free dynamic range (SFDR) is 68.67 dBc. The proposed DDFS outperforms prior works' SFDR and energy efficiency.

Radial basis functions method for solving the fractional diffusion equations

Hybrid Parabolic Interpolation – Artificial Neural Network Method (HPI-ANNM) for long-term extreme response estimation of steel risers

In this paper, we introduce a new fractional basis function based on Lagrange polynomials. We define the new interpolation formula for approximation of the solutions of the second kind weakly singular Volterra integral equations. The product integration method is used for the numerical solution of these equations based on Jacobi polynomials. It is known that the weakly singular Volterra integral equations typically have solutions whose derivatives are unbounded at the left end-point of the interval of integration. We use the suitable transformations to overcome this non-smooth behaviour. An upper error bound of the proposed method is determined and the convergence analysis is discussed. Finally, some numerical examples with non-smooth solutions are prepared to test the efficiency and accuracy of the method.

We develop a numerical scheme for finding the approximate solution for one- and two-dimensional multi-term time fractional diffusion and diffusion-wave equations considering smooth and nonsmooth solutions. The concept of multi-term time fractional derivatives is conventionally defined in the Caputo view point. In the current research, the convergence analysis of Legendre collocation spectral method was carried out. Spectral collocation method is consequently tested on several benchmark examples, to verify the accuracy and to confirm effectiveness of proposed method. The main advantage of the method is that only a small number of shifted Legendre polynomials are required to obtain accurate and efficient results. The numerical results are provided to demonstrate the reliability of our method and also to compare with other previously reported methods in the literature survey.

Abstract: Fractional diffusion equations serve as fundamental tools for addressing the non-local properites and long range memory effects that observed in diffusion processes within complex media. This works focuses on solving non-integer order (fractional) diffusion equations by employing the natural decomposition approach which gives the solution in series form. Some numerical examples of one dimensional and two dimensional fractional order diffusion equations are presented to demonstrate its application and obtained solutions are interpreted with the help of the computational software. Compared to other analytical and numerical techniques, the fractional natural decomposition method demonstrates advantages such as reduced computational complexity and faster convergence. Additionally, it can also be readily applied to address linear as well as non-linear problems. The application of natural decomposition approach to solve non-integer order (fractional) diffusion equations provides the most comprehensive understanding of the anomalous diffusion process occurring within complex media, as the fractional model accurately captures the non-local properties and long-range memory effects associated with such processes. To support the technique, we have taken into account a few problems and analyzed their solution by fractional natural decomposition method (FNDM) with solutions for the classical diffusion equations.

This paper presents a novel architecture for direct digital frequency synthesizer (DDFS) based on a modified parabolic polynomial interpolation method. A 16-segment parabolic polynomial interpolation is adopted to replace conventional ROM-based phase-to-amplitude conversion methods. Besides, the proposed parabolic polynomial interpolation is realized in a multiplier-less fashion such that the speed can be significantly improved. The proposed DDFS is implemented in a standard 0.13 μm cell-based technology. The maximum clock rate is 227 MHz, and the core area is 0.25 mm <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> . The simulation result shows that the spurious free dynamic range (SFDR) is 117 dBc.