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The computation time required by standard finite difference methods with fixed timesteps for solving fractional diffusion equations is usually very large because the number of operations required to find the solution scales as the square of…
Numerical solutions to fractional differential equations can be extremely computationally intensive due to the effect of non-local derivatives in which all previous time points contribute to the current iteration. In finite difference…
We consider finite approximations of a fractal generated by an iterated function system of affine transformations on $\mathbb{R}^d$ as a discrete set of data points. Considering a signal supported on this finite approximation, we propose a…
The numerical solution of implicit and stiff differential equations by implicit numerical integrators has been largely investigated and there exist many excellent efficient codes available in the scientific community, as Radau5 (based on a…
We develop a fast divided-and-conquer indirect collocation method for the homogeneous Dirichlet boundary value problem of variable-order space-fractional diffusion equations. Due to the impact of the space-dependent variable order, the…
It is well known that using high-order numerical algorithms to solve fractional differential equations leads to almost the same computational cost with low-order ones but the accuracy (or convergence order) is greatly improved, due to the…
In this paper, we consider a numerical method for the multi-term Caputo-Fabrizio time-fractional diffusion equations (with orders $\alpha_i\in(0,1)$, $i=1,2,\cdots,n$). The proposed method employs a fast finite difference scheme to…
We address a linear fractional differential equation and develop effective solution methods using algorithms for inversion of triangular Toeplitz matrices and the recently proposed QTT format. The inverses of such matrices can be computed…
Recent research in deep learning (DL) has investigated the use of the Fast Fourier Transform (FFT) to accelerate the computations involved in Convolutional Neural Networks (CNNs) by replacing spatial convolution with element-wise…
We give an algorithm to compute $N$ steps of a convolution quadrature approximation to a continuous temporal convolution using only $O(N \log N)$ multiplications and $O(\log N)$ active memory. The method does not require evaluations of the…
We investigate a local modification of a variable-order fractional wave equation, which describes the propagation of diffusive wave in viscoelastic media with evolving physical property. We incorporate an equivalent formulation to prove the…
An efficient, iterative semi-implicit (SI) numerical method for the time integration of stiff wave systems is presented. Physics-based assumptions are used to derive a convergent iterative formulation of the SI scheme which enables the…
Two quadrature-based algorithms for computing the matrix fractional power $A^\alpha$ are presented in this paper. These algorithms are based on the double exponential (DE) formula, which is well-known for its effectiveness in computing…
There are three main types of numerical computations for the Bessel function of the second kind: series expansion, continued fraction, and asymptotic expansion. In addition, they are combined in the appropriate domain for each. However,…
Computational methods for fractional differential equations exhibit essential instability. Even a minor modification of the coefficients or other entry data may switch good results to the divergent. The goal of this paper is to suggest the…
Fast Fourier transform algorithms are an arsenal of effective tools for solving various problems of analysis and high-speed processing of signals of various natures. Almost all of these algorithms are designed to process sequences of…
Discrete-velocity approximations represent a popular way for computing the Boltzmann collision operator. The direct numerical evaluation of such methods involve a prohibitive cost, typically $O(N^{2d+1})$ where $d$ is the dimension of the…
This paper develops a strong computational approach to simulate a three-dimensional nonlinear radiation-conduction model in optically thick media, subject to suitable initial and boundary conditions. The space derivatives are approximated…
In this work, we propose an efficient and robust multigrid method for solving the time-fractional heat equation. Due to the nonlocal property of fractional differential operators, numerical methods usually generate systems of equations for…
Different numerical approaches for the stray-field calculation in the context of micromagnetic simulations are investigated. We compare finite difference based fast Fourier transform methods, tensor grid methods and the finite-element…