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Novel Numerical Solutions of Fractional PDEs

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A special issue of Fractal and Fractional (ISSN 2504-3110). This special issue belongs to the section "Numerical and Computational Methods".

Deadline for manuscript submissions: closed (31 July 2022) | Viewed by 33346

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Special Issue Editors

Prof. Dr. Xiaoping Xie

E-Mail Website
Guest Editor

School of Mathematics, Sichuan University, Chengdu 610064, China
Interests: numerical analysis; finite element method; solver; partial differential equation; fractional partial differential equation

Dr. Xian-Ming Gu

E-Mail Website
Guest Editor

1. School of Mathematics, Southwestern University of Finance and Economics, Chengdu 611130, China
2. Mathematical Institute, Utrecht University, 3584 Utrecht, The Netherlands
Interests: numerical linear algebra; numerical (fractional) PDEs; parallel-in-time methods; Krylov subspace solvers
Special Issues, Collections and Topics in MDPI journals

Dr. Maohua Ran

E-Mail Website
Guest Editor

School of Mathematical Sciences, Sichuan Normal University, Chengdu 610068, China
Interests: finite difference method; numerical methods for fractional

Special Issue Information

Dear Colleagues,

During the past few decades, fractional partial differential equations (PDEs) have been widely used in biology, materials science, molecular dynamics and many other fields. In particular, time fractional PDEs, which are able to accurately describe the state or evolution with historical memory, have attracted much research interest in both theoretical and numerical aspects. Due to the non-local property of fractional derivatives, the development of numerical algorithms for the fractional PDEs faces new challenges and opportunities:

(1) Analysis and numerical treatment of the weak singularity of solutions;
(2) Fast and parallel algorithms;
(3) Applications and simulations for real-word models.

In this Special Issue, of particular interest are the following subtopics:

Fractional ordinary differential equations;
Fractional PDEs;
Nonlocal modeling and computation;
Integro-differential equations;
Fast and parallel methods;
Convolution quadrature;
Finite element methods;
Spectral and collocation methods;
Finite difference methods;
Convergence analysis;
Software and package for numerical (fractional) PDEs;
Modeling and simulations involving (fractional) PDEs.

Prof. Dr. Xiaoping Xie
Dr. Xian-Ming Gu
Dr. Maohua Ran
Guest Editors

Manuscript Submission Information

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Keywords

Fractional PDE
Fractional ODE
Integro-differential equation
Numerical method
Fast algorithm
Modeling and simulation

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Published Papers (14 papers)

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Research

32 pages, 2234 KiB

Open AccessArticle

Numerical Investigation of a Class of Nonlinear Time-Dependent Delay PDEs Based on Gaussian Process Regression

by Wei Gu, Wenbo Zhang and Yaling Han

Fractal Fract. 2022, 6(10), 606; https://doi.org/10.3390/fractalfract6100606 - 17 Oct 2022

Cited by 1 | Viewed by 1588

Abstract

Probabilistic machine learning and data-driven methods gradually show their high efficiency in solving the forward and inverse problems of partial differential equations (PDEs). This paper will focus on investigating the forward problem of solving time-dependent nonlinear delay PDEs with multi-delays based on multi-prior numerical Gaussian processes (MP-NGPs), which are constructed by us to solve complex PDEs that may involve fractional operators, multi-delays and different types of boundary conditions. We also quantify the uncertainty of the prediction solution by the posterior distribution of the predicted solution. The core of MP-NGPs is to discretize time firstly, then a Gaussian process regression based on multi-priors is considered at each time step to obtain the solution of the next time step, and this procedure is repeated until the last time step. Different types of boundary conditions are studied in this paper, which include Dirichlet, Neumann and mixed boundary conditions. Several numerical tests are provided to show that the methods considered in this paper work well in solving nonlinear time-dependent PDEs with delay, where delay partial differential equations, delay partial integro-differential equations and delay fractional partial differential equations are considered. Furthermore, in order to improve the accuracy of the algorithm, we construct Runge–Kutta methods under the frame of multi-prior numerical Gaussian processes. The results of the numerical experiments prove that the prediction accuracy of the algorithm is obviously improved when the Runge–Kutta methods are employed. Full article

(This article belongs to the Special Issue Novel Numerical Solutions of Fractional PDEs)

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32 pages, 664 KiB

Open AccessArticle

A Uniform Accuracy High-Order Finite Difference and FEM for Optimal Problem Governed by Time-Fractional Diffusion Equation

by Junying Cao, Zhongqing Wang and Ziqiang Wang

Fractal Fract. 2022, 6(9), 475; https://doi.org/10.3390/fractalfract6090475 - 28 Aug 2022

Cited by 3 | Viewed by 1488

Abstract

In this paper, the time fractional diffusion equations optimal control problem is solved by

3 - α

order with uniform accuracy scheme in time and finite element method (FEM) in space. For the state and adjoint state equation, the piecewise linear polynomials are [...] Read more.

In this paper, the time fractional diffusion equations optimal control problem is solved by

3 - α

order with uniform accuracy scheme in time and finite element method (FEM) in space. For the state and adjoint state equation, the piecewise linear polynomials are used to make the space variables discrete, and obtain the semidiscrete scheme of the state and adjoint state. The priori error estimates for the semidiscrete scheme for state and adjoint state equation are established. Furthermore, the

3 - α

order uniform accuracy scheme is used to make the time variable discrete in the semidiscrete scheme and construct the full discrete scheme for the control problems based on the first optimal condition and ‘first optimize, then discretize’ approach. The fully discrete scheme’s stability and truncation error are analyzed. Finally, two numerical examples are denoted to show that the theoretical analysis are correct. Full article

(This article belongs to the Special Issue Novel Numerical Solutions of Fractional PDEs)

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13 pages, 1184 KiB

Open AccessArticle

Parameter Estimation for Several Types of Linear Partial Differential Equations Based on Gaussian Processes

by Wenbo Zhang and Wei Gu

Fractal Fract. 2022, 6(8), 433; https://doi.org/10.3390/fractalfract6080433 - 8 Aug 2022

Cited by 2 | Viewed by 1737

Abstract

This paper mainly considers the parameter estimation problem for several types of differential equations controlled by linear operators, which may be partial differential, integro-differential and fractional order operators. Under the idea of data-driven methods, the algorithms based on Gaussian processes are constructed to solve the inverse problem, where we encode the distribution information of the data into the kernels and construct an efficient data learning machine. We then estimate the unknown parameters of the partial differential Equations (PDEs), which include high-order partial differential equations, partial integro-differential equations, fractional partial differential equations and a system of partial differential equations. Finally, several numerical tests are provided. The results of the numerical experiments prove that the data-driven methods based on Gaussian processes not only estimate the parameters of the considered PDEs with high accuracy but also approximate the latent solutions and the inhomogeneous terms of the PDEs simultaneously. Full article

(This article belongs to the Special Issue Novel Numerical Solutions of Fractional PDEs)

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18 pages, 2239 KiB

Open AccessArticle

Optimal H¹-Norm Estimation of Nonconforming FEM for Time-Fractional Diffusion Equation on Anisotropic Meshes

by Yabing Wei, Yanmin Zhao, Shujuan Lü, Fenling Wang and Yayun Fu

Fractal Fract. 2022, 6(7), 381; https://doi.org/10.3390/fractalfract6070381 - 4 Jul 2022

Cited by 2 | Viewed by 1808

Abstract

In this paper, based on the

L 2

1_{σ}

scheme and nonconforming

E Q_{1}^{r o t}

finite element method (FEM), a numerical approximation is presented for a class of two-dimensional time-fractional diffusion equations involving variable coefficients. A novel and [...] Read more.

In this paper, based on the

L 2

1_{σ}

scheme and nonconforming

E Q_{1}^{r o t}

finite element method (FEM), a numerical approximation is presented for a class of two-dimensional time-fractional diffusion equations involving variable coefficients. A novel and detailed analysis of the equations with an initial singularity is described on anisotropic meshes. The fully discrete scheme is shown to be unconditionally stable, and optimal second-order accuracy for convergence and superconvergence can be achieved in both time and space directions. Finally, the obtained numerical results are compared with the theoretical analysis, which verifies the accuracy of the proposed method. Full article

(This article belongs to the Special Issue Novel Numerical Solutions of Fractional PDEs)

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27 pages, 535 KiB

Open AccessArticle

A Higher-Order Numerical Scheme for Two-Dimensional Nonlinear Fractional Volterra Integral Equations with Uniform Accuracy

by Zi-Qiang Wang, Qin Liu and Jun-Ying Cao

Fractal Fract. 2022, 6(6), 314; https://doi.org/10.3390/fractalfract6060314 - 2 Jun 2022

Cited by 6 | Viewed by 1636

Abstract

In this paper, based on the modified block-by-block method, we propose a higher-order numerical scheme for two-dimensional nonlinear fractional Volterra integral equations with uniform accuracy. This approach involves discretizing the domain into a large number of subdomains and using biquadratic Lagrangian interpolation on each subdomain. The convergence of the high-order numerical scheme is rigorously established. We prove that the numerical solution converges to the exact solution with the optimal convergence order

O (h_{x}^{4 - α} + h_{y}^{4 - β})

for

0 < α, β < 1

. Finally, experiments with four numerical examples are shown, to support the theoretical findings and to illustrate the efficiency of our proposed method. Full article

(This article belongs to the Special Issue Novel Numerical Solutions of Fractional PDEs)

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19 pages, 2053 KiB

Open AccessArticle

High-Order Dissipation-Preserving Methods for Nonlinear Fractional Generalized Wave Equations

by Yu Li, Wei Shan and Yanming Zhang

Fractal Fract. 2022, 6(5), 264; https://doi.org/10.3390/fractalfract6050264 - 10 May 2022

Cited by 2 | Viewed by 2008

Abstract

In this paper, we construct and analyze a class of high-order and dissipation-preserving schemes for the nonlinear space fractional generalized wave equations by the newly introduced scalar auxiliary variable (SAV) technique. The system is discretized by a fourth-order Riesz fractional difference operator in spatial discretization and the collocation methods in the temporal direction. Not only can the present method achieve fourth-order accuracy in the spatial direction and arbitrarily high-order accuracy in the temporal direction, but it also has long-time computing stability. Then, the unconditional discrete energy dissipation law of the present numerical schemes is proved. Finally, some numerical experiments are provided to certify the efficiency and the structure-preserving properties of the proposed schemes. Full article

(This article belongs to the Special Issue Novel Numerical Solutions of Fractional PDEs)

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15 pages, 528 KiB

Open AccessArticle

Linearly Implicit High-Order Exponential Integrators Conservative Runge–Kutta Schemes for the Fractional Schrödinger Equation

by Yayun Fu, Qianqian Zheng, Yanmin Zhao and Zhuangzhi Xu

Fractal Fract. 2022, 6(5), 243; https://doi.org/10.3390/fractalfract6050243 - 28 Apr 2022

Viewed by 1704

Abstract

In this paper, a family of high-order linearly implicit exponential integrators conservative schemes is constructed for solving the multi-dimensional nonlinear fractional Schrödinger equation. By virtue of the Lawson transformation and the generalized scalar auxiliary variable approach, the equation is first reformulated to an exponential equivalent system with a modified energy. Then, we construct a semi-discrete conservative scheme by using the Fourier pseudo-spectral method to discretize the exponential system in space direction. After that, linearly implicit energy-preserving schemes which have high accuracy are given by applying the Runge–Kutta method to approximate the semi-discrete system in temporal direction and using the extrapolation method to the nonlinear term. As expected, the constructed schemes can preserve the energy exactly and implement efficiently with a large time step. Numerical examples confirm the constructed schemes have high accuracy, energy-preserving, and effectiveness in long-time simulation. Full article

(This article belongs to the Special Issue Novel Numerical Solutions of Fractional PDEs)

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18 pages, 1793 KiB

Open AccessArticle

Second-Order Time Stepping Scheme Combined with a Mixed Element Method for a 2D Nonlinear Fourth-Order Fractional Integro-Differential Equations

by Deng Wang, Yang Liu, Hong Li and Zhichao Fang

Fractal Fract. 2022, 6(4), 201; https://doi.org/10.3390/fractalfract6040201 - 2 Apr 2022

Cited by 4 | Viewed by 2467

Abstract

In this article, we study a class of two-dimensional nonlinear fourth-order partial differential equation models with the Riemann–Liouville fractional integral term by using a mixed element method in space and the second-order backward difference formula (BDF2) with the weighted and shifted Grünwald integral (WSGI) formula in time. We introduce an auxiliary variable to transform the nonlinear fourth-order model into a low-order coupled system including two second-order equations and then discretize the resulting equations by the combined method between the BDF2 with the WSGI formula and the mixed finite element method. Further, we derive stability and error results for the fully discrete scheme. Finally, we develop two numerical examples to verify the theoretical results. Full article

(This article belongs to the Special Issue Novel Numerical Solutions of Fractional PDEs)

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24 pages, 766 KiB

Open AccessArticle

Initial Value Problems of Fuzzy Fractional Coupled Partial Differential Equations with Caputo gH-Type Derivatives

by Fan Zhang, Hai-Yang Xu and Heng-You Lan

Fractal Fract. 2022, 6(3), 132; https://doi.org/10.3390/fractalfract6030132 - 25 Feb 2022

Cited by 4 | Viewed by 2070

Abstract

The purpose of this paper is to investigate a class of initial value problems of fuzzy fractional coupled partial differential equations with Caputo

g H

-type derivatives. Firstly, using Banach fixed point theorem and the mathematical inductive method, we prove the existence and [...] Read more.

The purpose of this paper is to investigate a class of initial value problems of fuzzy fractional coupled partial differential equations with Caputo

g H

-type derivatives. Firstly, using Banach fixed point theorem and the mathematical inductive method, we prove the existence and uniqueness of two kinds of

g H

-weak solutions of the coupled system for fuzzy fractional partial differential equations under Lipschitz conditions. Then we give an example to illustrate the correctness of the existence and uniqueness results. Furthermore, because of the coupling in the initial value problems, we develop Gronwall inequality of the vector form, and creatively discuss continuous dependence of the solutions of the coupled system for fuzzy fractional partial differential equations on the initial values and

ε

-approximate solution of the coupled system. Finally, we propose some work for future research. Full article

(This article belongs to the Special Issue Novel Numerical Solutions of Fractional PDEs)

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14 pages, 846 KiB

Open AccessArticle

A Mixed Element Algorithm Based on the Modified L1 Crank–Nicolson Scheme for a Nonlinear Fourth-Order Fractional Diffusion-Wave Model

by Jinfeng Wang, Baoli Yin, Yang Liu, Hong Li and Zhichao Fang

Fractal Fract. 2021, 5(4), 274; https://doi.org/10.3390/fractalfract5040274 - 14 Dec 2021

Cited by 3 | Viewed by 2312

Abstract

In this article, a new mixed finite element (MFE) algorithm is presented and developed to find the numerical solution of a two-dimensional nonlinear fourth-order Riemann–Liouville fractional diffusion-wave equation. By introducing two auxiliary variables and using a particular technique, a new coupled system with three equations is constructed. Compared to the previous space–time high-order model, the derived system is a lower coupled equation with lower time derivatives and second-order space derivatives, which can be approximated by using many time discrete schemes. Here, the second-order Crank–Nicolson scheme with the modified

L 1

-formula is used to approximate the time direction, while the space direction is approximated by the new MFE method. Analyses of the stability and optimal

L^{2}

error estimates are performed and the feasibility is validated by the calculated data. Full article

(This article belongs to the Special Issue Novel Numerical Solutions of Fractional PDEs)

18 pages, 1263 KiB

Open AccessArticle

A Fast Preconditioned Semi-Implicit Difference Scheme for Strongly Nonlinear Space-Fractional Diffusion Equations

by Yu-Yun Huang, Xian-Ming Gu, Yi Gong, Hu Li, Yong-Liang Zhao and Bruno Carpentieri

Fractal Fract. 2021, 5(4), 230; https://doi.org/10.3390/fractalfract5040230 - 18 Nov 2021

Cited by 10 | Viewed by 2156

Abstract

In this paper, we propose a semi-implicit difference scheme for solving one-dimensional nonlinear space-fractional diffusion equations. The method is first-order accurate in time and second-order accurate in space. It uses a fractional central difference formula and the backward Euler method to approximate its space and time derivatives, respectively. Stability and convergence properties of the proposed scheme are proved with the help of a discrete Grönwall inequality. Moreover, we extend the method to the solution of two-dimensional nonlinear models. A fast matrix-free implementation based on preconditioned Krylov subspace methods is presented for solving the discretized linear systems. The resulting fast preconditioned semi-implicit difference scheme reduces the memory requirement of conventional semi-implicit difference schemes from

O (N_{s}^{2})

O (N_{s})

and the computational complexity from

O (N_{s}^{3})

O (N_{s} log N_{s})

in each iterative step, where

N_{s}

is the number of space grid points. Experiments with two numerical examples are shown to support the theoretical findings and to illustrate the efficiency of our proposed method. Full article

(This article belongs to the Special Issue Novel Numerical Solutions of Fractional PDEs)

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40 pages, 2339 KiB

Open AccessArticle

Novel Numerical Investigations of Fuzzy Cauchy Reaction–Diffusion Models via Generalized Fuzzy Fractional Derivative Operators

by Manar A. Alqudah, Rehana Ashraf, Saima Rashid, Jagdev Singh, Zakia Hammouch and Thabet Abdeljawad

Fractal Fract. 2021, 5(4), 151; https://doi.org/10.3390/fractalfract5040151 - 3 Oct 2021

Cited by 25 | Viewed by 2375

Abstract

The present research correlates with a fuzzy hybrid approach merged with a homotopy perturbation transform method known as the fuzzy Shehu homotopy perturbation transform method (SHPTM). With the aid of Caputo and Atangana–Baleanu under generalized Hukuhara differentiability, we illustrate the reliability of this scheme by obtaining fuzzy fractional Cauchy reaction–diffusion equations (CRDEs) with fuzzy initial conditions (ICs). Fractional CRDEs play a vital role in diffusion and instabilities may develop spatial phenomena such as pattern formation. By considering the fuzzy set theory, the proposed method enables the solution of the fuzzy linear CRDEs to be evaluated as a series of expressions in which the components can be efficiently identified and generating a pair of approximate solutions with the uncertainty parameter

λ \in [0, 1] .

To demonstrate the usefulness and capabilities of the suggested methodology, several numerical examples are examined to validate convergence outcomes for the supplied problem. The simulation results reveal that the fuzzy SHPTM is a viable strategy for precisely and accurately analyzing the behavior of a proposed model. Full article

(This article belongs to the Special Issue Novel Numerical Solutions of Fractional PDEs)

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21 pages, 803 KiB

Open AccessEditor’s ChoiceArticle

Numerical Solutions for Systems of Fractional and Classical Integro-Differential Equations via Finite Integration Method Based on Shifted Chebyshev Polynomials

by Ampol Duangpan, Ratinan Boonklurb and Matinee Juytai

Fractal Fract. 2021, 5(3), 103; https://doi.org/10.3390/fractalfract5030103 - 25 Aug 2021

Cited by 11 | Viewed by 2682

Abstract

In this paper, the finite integration method and the operational matrix of fractional integration are implemented based on the shifted Chebyshev polynomial. They are utilized to devise two numerical procedures for solving the systems of fractional and classical integro-differential equations. The fractional derivatives are described in the Caputo sense. The devised procedure can be successfully applied to solve the stiff system of ODEs. To demonstrate the efficiency, accuracy and numerical convergence order of these procedures, several experimental examples are given. As a consequence, the numerical computations illustrate that our presented procedures achieve significant improvement in terms of accuracy with less computational cost. Full article

(This article belongs to the Special Issue Novel Numerical Solutions of Fractional PDEs)

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15 pages, 1846 KiB

Open AccessArticle

A Numerical Study of Nonlinear Fractional Order Partial Integro-Differential Equation with a Weakly Singular Kernel

by Tayyaba Akram, Zeeshan Ali, Faranak Rabiei, Kamal Shah and Poom Kumam

Fractal Fract. 2021, 5(3), 85; https://doi.org/10.3390/fractalfract5030085 - 2 Aug 2021

Cited by 7 | Viewed by 3296

Abstract

Fractional differential equations can present the physical pathways with the storage and inherited properties due to the memory factor of fractional order. The purpose of this work is to interpret the collocation approach for tackling the fractional partial integro-differential equation (FPIDE) by employing the extended cubic B-spline (ECBS). To determine the time approximation, we utilize the Caputo approach. The stability and convergence analysis have also been analyzed. The efficiency and reliability of the suggested technique are demonstrated by two numerical applications, which support the theoretical results and the effectiveness of the implemented algorithm. Full article

(This article belongs to the Special Issue Novel Numerical Solutions of Fractional PDEs)

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