Exploration and Analysis of Higher-Order Numerical Methods for Fractional Differential Equations

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: 31 December 2025 | Viewed by 1044

Special Issue Editors


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Guest Editor
School of Mathematics and Statistics, Guangxi Normal University, Guilin 541006, China
Interests: numerical and computational methods in fractional differential equations; higher-order numerical differential formulas for fractional derivatives

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Guest Editor
Department of Mathematics, Shanghai University, Shanghai 200444, China
Interests: fractional dynamics; numerical methods for fractional partial differential equations
Special Issues, Collections and Topics in MDPI journals

E-Mail Website
Guest Editor
Department of Mathematics, Shanghai University, Shanghai 200444, China
Interests: numerical methods for fractional partial differential equations

Special Issue Information

Dear Colleagues,

Fractional differential equations have significant applications in fields such as physics, chemistry, fluid mechanics, signal processing, and the social sciences. Unfortunately, due to the nonlocality of fractional-order derivatives, it is almost impossible to obtain analytical solutions for such equations in general. Even in special cases where analytical solutions are obtained, their expressions contain special functions, which also bring great difficulties to calculations. Therefore, in order to better analyze the dynamic behavior of fractional differential equations, we must resort to numerical methods.

Furthermore, when discretizing fractional derivatives, it is worth noting that the structural characteristics of the corresponding matrix after discretization are completely different from those of normal derivatives. Specifically, whether higher-order or lower-order numerical differentiation formulas are being used, the generated matrix is dense and requires the same amount of computation and storage. However, using the former can greatly improve computational efficiency. Therefore, constructing higher-order numerical differential formulas for fractional derivatives and higher-order numerical algorithms for fractional differential equations is very meaningful and is currently a research hotspot.

This Special Issue will be devoted to collating recent results from the numerical methods and applications of fractional differential equations. The topics encouraged for submissions include, but are not limited to, the following:

  • The modeling of fractional differential equations;
  • Finite difference methods for fractional differential equations;
  • Finite element methods for fractional differential equations;
  • Spectral methods for fractional differential equations;
  • The construction of higher-order numerical differential formulas for fractional derivatives.

Prof. Dr. Hengfei Ding
Prof. Dr. Changpin Li
Dr. Min Cai
Guest Editors

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Keywords

  • the modeling of fractional differential equations
  • finite difference methods for fractional differential equations
  • finite element methods for fractional differential equations
  • spectral methods for fractional differential equations
  • the construction of higher-order numerical differential formulas for fractional derivatives

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Published Papers (3 papers)

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Research

14 pages, 356 KiB  
Article
Pointwise Error Analysis of the Corrected L1 Scheme for the Multi-Term Subdiffusion Equation
by Qingzhao Li and Chaobao Huang
Fractal Fract. 2025, 9(8), 529; https://doi.org/10.3390/fractalfract9080529 - 14 Aug 2025
Viewed by 61
Abstract
This paper considers the multi-term subdiffusion equation with weakly singular solutions. In order to use sparser meshes near the initial time, a novel scheme (which we call the corrected L1 scheme) on graded meshes is constructed to estimate the multi-term Caputo fractional derivative [...] Read more.
This paper considers the multi-term subdiffusion equation with weakly singular solutions. In order to use sparser meshes near the initial time, a novel scheme (which we call the corrected L1 scheme) on graded meshes is constructed to estimate the multi-term Caputo fractional derivative by investigating a corrected term for the nonuniform L1 scheme. Combining this nonuniform corrected L1 scheme in the temporal direction and the finite element method (FEM) in the spatial direction, a fully discrete scheme for solving the multi-term subdiffusion equation is developed. The stability result of the developed scheme is given. Furthermore, the optimal pointwise-in-time error estimate of the developed scheme is derived. Finally, several numerical experiments are conducted to verify our theoretical findings. Full article
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20 pages, 506 KiB  
Article
Efficient Numerical Methods for Time-Fractional Diffusion Equations with Caputo-Type Erdélyi–Kober Operators
by Ruilian Du and Jianhua Tang
Fractal Fract. 2025, 9(8), 486; https://doi.org/10.3390/fractalfract9080486 - 24 Jul 2025
Viewed by 244
Abstract
This study proposes an L1 discretization scheme (an accurate second-order finite difference method) for time-fractional diffusion equations involving the Caputo-type Erdélyi–Kober operator, which models anomalous diffusion. Our key contributions include the following: (i) reformulation of the original problem into an equivalent fractional integral [...] Read more.
This study proposes an L1 discretization scheme (an accurate second-order finite difference method) for time-fractional diffusion equations involving the Caputo-type Erdélyi–Kober operator, which models anomalous diffusion. Our key contributions include the following: (i) reformulation of the original problem into an equivalent fractional integral equation to facilitate analysis; (ii) development of a novel L1 scheme for temporal discretization, which is rigorously proven to realize second-order accuracy in time; (iii) derivation of positive definiteness properties for discrete kernel coefficients; (iv) discretization of the spatial derivative using the classical second-order centered difference scheme, for which its second-order spatial convergence is rigorously verified through numerical experiments (this results in a fully discrete scheme, enabling second-order accuracy in both temporal and spatial dimensions); (v) a fast algorithm leveraging sum-of-exponential approximation, reducing the computational complexity from O(N2) to O(NlogN) and memory requirements from O(N) to O(logN), where N is the number of grid points on a time scale. Our numerical experiments demonstrate the stability of the scheme across diverse parameter regimes and quantify significant gains in computational efficiency. Compared to the direct method, the fast algorithm substantially reduces both memory requirements and CPU time for large-scale simulations. Although a rigorous stability analysis is deferred to subsequent research, the proven properties of the coefficients and numerical validation confirm the scheme’s reliability. Full article
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11 pages, 256 KiB  
Article
Improved High-Order Difference Scheme for the Conservation of Mass and Energy in the Two-Dimensional Spatial Fractional Schrödinger Equation
by Junhong Tian and Hengfei Ding
Fractal Fract. 2025, 9(5), 280; https://doi.org/10.3390/fractalfract9050280 - 25 Apr 2025
Cited by 1 | Viewed by 342
Abstract
In this paper, our primary objective is to develop a robust and efficient higher-order structure-preserving algorithm for the numerical solution of the two-dimensional nonlinear spatial fractional Schrödinger equation. This equation, which incorporates fractional derivatives, poses significant challenges due to its non-local nature and [...] Read more.
In this paper, our primary objective is to develop a robust and efficient higher-order structure-preserving algorithm for the numerical solution of the two-dimensional nonlinear spatial fractional Schrödinger equation. This equation, which incorporates fractional derivatives, poses significant challenges due to its non-local nature and nonlinearity, making it essential to design numerical methods that not only achieve high accuracy but also preserve the intrinsic physical and mathematical properties of the system. To address these challenges, we employ the scalar auxiliary variable (SAV) method, a powerful technique known for its ability to maintain energy stability and simplify the treatment of nonlinear terms. Combined with the composite Simpson’s formula for numerical integration, which ensures high precision in approximating integrals, and a fourth-order numerical differential formula for discretizing the Riesz derivative, we construct a highly effective finite difference scheme. This scheme is designed to balance computational efficiency with numerical accuracy, making it suitable for long-time simulations. Furthermore, we rigorously analyze the conserving properties of the numerical solution, including mass and energy conservation, which are critical for ensuring the physical relevance and stability of the results. Full article
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