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

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

Manuscript Submission Information

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