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Spherical Gauss-Laguerre (SGL) basis functions, i.e., normalized functions of the type $L_{n-l-1}^{(l + 1/2)} (r^2) r^{l} Y_{lm}(\vartheta,\varphi)$, $|m| \leq l < n \in \mathbb{N}$, $L_{n-l-1}^{(l + 1/2)}$ being a generalized Laguerre…
Due to the lack of corresponding analysis on appropriate mapping operator between two grids, high-order two-grid difference algorithms are rarely studied. In this paper, we firstly discuss the boundedness of a local bi-cubic Lagrange…
In this work, we propose an algorithm for a filter based on the Fast Fourier Transform (FFT), which, due to its characteristics, allows for an efficient computational implementation, ease of use, and minimizes amplitude variation in the…
A crucial part of successful wave propagation related inverse problems is an efficient and accurate numerical scheme for solving the seismic wave equations. In particular, the numerical solution to a multi-dimensional Helmholtz equation can…
The primary goal of this research is to propose a novel architecture for a deep neural network that can solve fractional differential equations accurately. A Gaussian integration rule and a $L_1$ discretization technique are used in the…
How well do multisymplectic discretisations preserve travelling wave solutions? To answer this question, the 5-point central difference scheme is applied to the semi-linear wave equation. A travelling wave ansatz leads to an ordinary…
In this work, an efficient approximation scheme has been proposed for getting accurate approximate solution of nonlinear partial differential equations with constant or variable coefficients satisfying initial conditions in a series of…
Partial differential equations (PDEs) are crucial for modeling various physical phenomena such as heat transfer, fluid flow, and electromagnetic waves. In computer-aided engineering (CAE), the ability to handle fine resolutions and large…
In this paper, we propose a new optimization framework, the layer separation (LySep) model, to improve the deep learning-based methods in solving partial differential equations. Due to the highly non-convex nature of the loss function in…
In this paper, we consider the problem of accelerating the numerical simulation of time dependent problems by time domain decomposition. The available algorithms enabling such decompositions present severe efficiency limitations and are an…
In this paper we consider the numerical solution of the Hamiltonian wave equation in two spatial dimension. We use the Mimetic Finite Difference (MFD) method to approximate the continuous problem combined with a symplectic integration in…
Fractional differential equations (FDEs) are an extension of the theory of fractional calculus. However, due to the difficulty in finding analytical solutions, there have not been extensive applications of FDEs until recent decades. With…
Fourier Neural Operators are deep learning models that learn mappings between function spaces and can be used to learn and solve partial differential equations (PDEs), in some cases significantly faster than traditional PDE solvers. Within…
Let $z_{1},z_{2},...,z_{N}$ be a sequence of distinct grid points. A finite difference formula approximates the $m$-th derivative $f^{(m)}(0)$ as $\sum w_{k}f(z_{k})$, with $w_{k}$ being the weights. We derive an algorithm for finding the…
We develop a new type of orthogonal polynomial, the modified discrete Laguerre (MDL) polynomials, designed to accelerate the computation of bosonic Matsubara sums in statistical physics. The MDL polynomials lead to a rapidly convergent…
We analyze and test using Fourier extensions that minimize a Hilbert space norm for the purpose of solving partial differential equations (PDEs) on surfaces. In particular, we prove that the approach is arbitrarily high-order and also show…
Solutions to fractional models inherently exhibit non-smooth behavior, which significantly deteriorates the accuracy and therefore efficiency of existing numerical methods. We develop a two-stage data-infused computational framework for…
Nonlinear differential equations are challenging to solve numerically and are important to understanding the dynamics of many physical systems. Deep neural networks have been applied to help alleviate the computational cost that is…
We present a finite-difference integration algorithm for solution of a system of differential equations containing a diffusion equation with nonlinear terms. The approach is based on Crank-Nicolson method with predictor-corrector algorithm…
In this article I present a fast and direct method for solving several types of linear finite difference equations (FDE) with constant coefficients. The method is based on a polynomial form of the translation operator and its inverse, and…