scholarly journals A higher order numerical method for time fractional partial differential equations with nonsmooth data

2018 ◽  
Vol 357 ◽  
pp. 305-323 ◽  
Author(s):  
Yanyuan Xing ◽  
Yubin Yan
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Numerical Method ◽  
Higher Order ◽  
Fractional Partial Differential Equations ◽  
Partial Differential ◽  
Nonsmooth Data

scholarly journals An approach to construct higher order time discretisation schemes for time fractional partial differential equations with nonsmooth data

2017 ◽  
Vol 20 (5) ◽  
Author(s):  
Neville J. Ford ◽  
Yubin Yan
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Higher Order ◽  
Fractional Partial Differential Equations ◽  
Order Accuracy ◽  
Partial Differential ◽  
Nonsmooth Data ◽  
Standard Time ◽  
Time Discretisation ◽  
Theoretical Results

AbstractIn this paper, we shall review an approach by which we can seek higher order time discretisation schemes for solving time fractional partial differential equations with nonsmooth data. The low regularity of the solutions of time fractional partial differential equations implies standard time discretisation schemes only yield first order accuracy. To obtain higher order time discretisation schemes when the solutions of time fractional partial differential equations have low regularities, one may correct the starting steps of the standard time discretisation schemes to capture the singularities of the solutions. We will consider these corrections of some higher order time discretisation schemes obtained by using Lubich’s fractional multistep methods, L1 scheme and its modification, discontinuous Galerkin methods, etc. Numerical examples are given to show that the theoretical results are consistent with the numerical results.

scholarly journals Legendre Polynomials Operational Matrix Method for Solving Fractional Partial Differential Equations with Variable Coefficients

2015 ◽  
Vol 2015 ◽  
pp. 1-9 ◽  
Author(s):  
Yongqiang Yang ◽  
Yunpeng Ma ◽  
Lifeng Wang
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Numerical Method ◽  
Legendre Polynomials ◽  
Linear Equations ◽  
Operational Matrix ◽  
Variable Coefficients ◽  
Fractional Partial Differential Equations ◽  
Partial Differential ◽  
Operational Matrix Method

A numerical method for solving a class of fractional partial differential equations with variable coefficients based on Legendre polynomials is proposed. A fractional order operational matrix of Legendre polynomials is also derived. The initial equations are transformed into the products of several matrixes by using the operational matrix. A system of linear equations is obtained by dispersing the coefficients and the products of matrixes. Only a small number of Legendre polynomials are needed to acquire a satisfactory result. Results obtained using the scheme presented here show that the numerical method is very effective and convenient for solving fractional partial differential equations with variable coefficients.

scholarly journals Oscillation criteria of certain fractional partial differential equations

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Di Xu ◽  
Fanwei Meng
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Fractional Derivatives ◽  
Sufficient Conditions ◽  
Riccati Transformation ◽  
Fractional Partial Differential Equations ◽  
Partial Differential ◽  
Oscillation Criteria ◽  
Generalized Riccati Transformation

Abstract In this article, we regard the generalized Riccati transformation and Riemann–Liouville fractional derivatives as the principal instrument. In the proof, we take advantage of the fractional derivatives technique with the addition of interval segmentation techniques, which enlarge the manners to demonstrate the sufficient conditions for oscillation criteria of certain fractional partial differential equations.

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