Theoretical and Numerical Methods for Fractional and Nonlocal Partial Differential Equations

Dear Colleagues,

Fractional and nonlocal partial differential equations (PDEs) have emerged as powerful tools for modeling complex phenomena exhibiting memory, anomalous diffusion, long-range interactions, and spatial heterogeneity. These equations extend classical PDEs by incorporating fractional derivatives or nonlocal operators, which capture effects that local differential models fail to describe adequately. As a result, fractional and nonlocal PDEs now play central roles in diverse scientific and engineering domains, including physics, biology, finance, and materials science.

Despite their growing relevance, the analysis and computation of solutions to fractional and nonlocal PDEs present significant challenges due to their intrinsic nonlocality, singularities, and often limited regularity. This has motivated intense research efforts aimed at developing robust theoretical frameworks and efficient numerical methods to better understand and approximate these models.

This Special Issue is dedicated to advancing both the theoretical foundations and computational techniques for approaching fractional and nonlocal PDEs. We invite original research articles and high-quality reviews that address recent progress in areas such as existence and uniqueness theory, regularity analysis, numerical approximation schemes, error estimates, and the application of these equations in real-world systems.

Topics of interest include, but are not limited to, the following:

• Analytical methods for fractional and nonlocal PDEs;
• The existence, uniqueness, and regularity of solutions;
• Variational and functional analytic techniques;
• Finite element, finite difference, spectral, and meshfree methods;
• Stability and convergence analysis of numerical schemes;
• Fast algorithms for nonlocal operators and fractional Laplacians;
• Applications in physics, biology, engineering, and finance;
• Inverse problems and control theory involving fractional/nonlocal PDEs.

We welcome contributions that deepen our mathematical understandings of nonlocal models or introduce novel numerical frameworks with theoretical justification and demonstrated applicability.

Prof. Dr. Giuseppe Di Fazio
Prof. Dr. Giovanni Molica Bisci
Guest Editors

Manuscript Submission Information

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Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Fractal and Fractional is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2700 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

• nonlocal equations
• fractional models
• numerical methods
• variational methods
• regularity theory