In this paper, we consider a variable-order fractional advection-diffusion equation with a nonlinear source term on a finite domain. Explicit and implicit Euler approximations for the equation are proposed. Stability and convergence of the methods are discussed. Moveover, we also present a fractional method of lines, a matrix transfer technique, and an extrapolation method for the equation. Some numerical examples are given, and the results demonstrate the effectiveness of theoretical analysis.

/pdf/numerical-methods-for-the-variable-order-fractional-3ezf7kytjf.pdf

Numerical Methods for the Variable-Order Fractional Advection-Diffusion Equation with a Nonlinear Source Term

In this paper, a new alternating direction implicit Galerkin-Legendre spectral method for the two-dimensional Riesz space fractional nonlinear reaction-diffusion equation is developed. The temporal component is discretized by the Crank-Nicolson method. The detailed implementation of the method is presented. The stability and convergence analysis is strictly proven, which shows that the derived method is stable and convergent of order 2 in time. An optimal error estimate in space is also obtained by introducing a new orthogonal projector. The present method is extended to solve the fractional FitzHugh-Nagumo model. Numerical results are provided to verify the theoretical analysis.

/pdf/a-crank-nicolson-adi-spectral-method-for-a-two-dimensional-eg5dxd8kw4.pdf

A Crank-Nicolson adi spectral method for a two-dimensional Riesz space fractional nonlinear reaction-diffusion equation

A direct O(Nlog2N) finite difference method for fractional diffusion equations

Abstract Variable-order (VO) fractional differential equations (FDEs) with a time (t), space (x) or other variables dependent order have been successfully applied to investigate time and/or space dependent dynamics. This study aims to provide a survey of the recent relevant literature and findings in primary definitions, models, numerical methods and their applications. This review first offers an overview over the existing definitions proposed from different physical and application backgrounds, and then reviews several widely used numerical schemes in simulation. Moreover, as a powerful mathematical tool, the VO-FDE models have been remarkably acknowledged as an alternative and precise approach in effectively describing real-world phenomena. Hereby, we also make a brief summary on different physical models and typical applications. This review is expected to help the readers for the selection of appropriate definition, model and numerical method to solve specific physical and engineering problems.

/pdf/a-review-on-variable-order-fractional-differential-equations-px13m3tdxp.pdf

A review on variable-order fractional differential equations: mathematical foundations, physical models, numerical methods and applications

We evaluated a time-domain wave equation for modeling acoustic wave propagation in attenuating media. The wave equation was derived from Kjartansson’s constant-Q constitutive stress-strain relation in combination with the mass and momentum conservation equations. Our wave equation, expressed by a second-order temporal derivative and two fractional Laplacian operators, described very nearly constant-Q attenuation and dispersion effects. The advantage of using our formulation of two fractional Laplacians over the traditional fractional time derivative approach was the avoidance of time history memory variables and thus it offered more economic computations. In numerical simulations, we formulated the first-order constitutive equations with the perfectly matched layer absorbing boundaries. The temporal derivative was calculated with a staggered-grid finite-difference approach. The fractional Laplacians are calculated in the spatial frequency domain using a Fourier pseudospectral implementation. We validated our numerical results through comparisons with theoretical constant-Q attenuation and dispersion solutions, field measurements from the Pierre Shale, and results from 2D viscoacoustic analytical modeling for the homogeneous Pierre Shale. We also evaluated different formulations to show separated amplitude loss and dispersion effects on wavefields. Furthermore, we generalized our rigorous formulation for homogeneous media to an approximate equation for viscoacoustic waves in heterogeneous media. We then investigated the accuracy of numerical modeling in attenuating media with different Q-values and its stability in largecontrast heterogeneous media. Finally, we tested the applicability of our time-domain formulation in a heterogeneous medium with high attenuation.

Modeling acoustic wave propagation in heterogeneous attenuating media using decoupled fractional Laplacians

http://eprints.qut.edu.au/29758/1/29758.pdf

Stability and convergence of a new explicit finite-difference approximation for the variable-order nonlinear fractional diffusion equation

In this paper, we investigate a delayed predator–prey model with a prey refuge where the predator population eats the prey according to the Beddington–DeAngelis type functional response. Firstly, we consider the existence of equilibrium points. By analyzing the corresponding characteristic equations, the local stability of the trivial equilibrium, the predator–extinction balance, and the coexistence equilibrium of the system are discussed, and the existence of Hopf bifurcations concerning both delays at the coexistence equilibrium are established. Then, in accordance with the standard form method and center manifold theorem, the explicit formulas which determine the direction of Hopf bifurcation and stability of bifurcating period solutions are derived. Finally, representative numerical simulations are performed to validate the theoretical analysis.

/pdf/qualitative-analysis-in-a-beddington-deangelis-type-predator-34y6ptbf.pdf

Qualitative Analysis in a Beddington-DeAngelis Type Predator-Prey Model with Two Time Delays

Handle everyday research tasks with reliable, citation-backed results

Your personal Research Agent to handle research tasks with citation-backed results

Popular Tasks used by Researchers

How can I help with your research?

Meet SciSpace

Get more enhanced response by uploading the PDFs you want me to reference.

No relevant PDFs in your library

SciSpace is the AI research assistant for academics. Run systematic literature reviews on 280M+ papers, and write papers with cited sources. Trusted by 1M+ students, PhDs & researchers.

SciSpace | AI for Research

Analyze PDFs

Code & Manuscripts

Funding & Grants

Literature & Patents

Medical & Clinical Data

Systematic Review

Visualize & Present

Web & Data

Build a Google Scholar-like website for your research.

Build a website

Create charts and images for your research

Create a Chart

Write a paper for submission to a journal

Draft a manuscript

Patent Search

Design eye-catching scientific posters in minutes.

Scientific Poster Generation

Systematic Literature Review

One task is running at the moment. Your messages will be shown right after.

Drag and drop or click here to browse

Loved by <highlight>1 million+</highlight> researchers

Extract a list of specific topics and their sources from unstructured text

Topics

Compare and analyze relevant papers that matches with your search

Papers

Get insights from PDFs and bookmarked papers from your library

My library

Recent searches

Try searching for:

Catch AI-generated content in scholarly and non-scholarly content

{ai} Detector

Ai Writer

Get PDF Summaries, highlighted text explanations 

Chat with PDF

Effortlessly create in-text citations and bibliographies in APA and 2,500 other formats

Citation generator

Get explanations, summaries, and answers on academic papers

Ease up your research workflow with {scispace}'s cohort of exciting AI tools

Elevate your academic writing skills and convey your ideas the way you want

Paraphraser

Explore our range of reading and writing tools

Your file is being prepared and should be ready in a few minutes. If it's a large file, it might take a bit longer. You can close this window, and we'll email you the file when it's done.

You have reached a maximum limit of <strong>{limit}</strong> columns in the table. Remove at least <strong>1</strong> column to add or create another one.

R. Lin

Author Tools

Chat about Author

Papers

Stability and convergence of a new explicit finite-difference approximation for the variable-order nonlinear fractional diffusion equation

Qualitative Analysis in a Beddington-DeAngelis Type Predator-Prey Model with Two Time Delays