Newly Developments in Fractional Laplacian: Numerical Methods and Inverse Problems

Dear Colleagues,

During the past few decades, scientists have been exploring fractional calculus as a tool for developing more sophisticated but mathematically tractable models. The reason is that they can accurately describe complex physical phenomena, manifesting in long-range and nonlocal interactions, self-similar structures, sharp peaks, and memory effects. More and more people are accepting the fractional model as a promising remedy to the traditionally inaccurate integer-order model in many exciting applications. However, the efficient computation of this model on bounded domains is still challenging as highly accurate and efficient numerical methods are not yet available, which prevents its broader applications among the scientific and engineering community. This Special Issue will focus on recent developments in numerical methods.

Moreover, regularity plays a crucial role in developing numerical methods. As limited regularity results exist, this issue also welcomes the submission of PDE theory involving the new regularity results. Another exciting topic this Special Issue will focus on is extending the neural networks to inverse fractional problems on model learning via optimal control and machine learning techniques. Fractional models for materials and media are often hand-tuned or rely on engineering or scientific intuition. When limited data or a priori information are available, one can resort to solving an inverse problem to recover the unknown parameters and define a more accurate, data-driven mathematical model.

Dr. Zhao-peng Hao
Dr. Rui Du
Guest Editors

Manuscript Submission Information

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• Fractional Laplacian 
• Regularity 
• Numerical methods 
• Inverse problem 
• Machine learning