### Functional PDEs for Conversation Laws and Beyond – (ARO MURI)

**Full Title: **Fractional PDEs for Conservation Laws and Beyond: Theory, Numerics, and Applications sponsored by the Army Research Office Multidisciplinary Research Program of the University Research Initiative (ARO MURI)

**Research Problem:** Despite significant progress over the last 50 years in simulating complex multiphysics problems using classical (integer order) partial differential equations (PDEs), many physical problems remain that cannot be adequately modeled using this approach. Examples include anomalous transport, non-Markovian behavior, and long-range interactions. Even well-known phenomena such as self-similarity, singular behavior, and decorrelation effects are not easily represented within the confines of standard calculus. We propose to break this deadlock by developing a new class of mathematical and computational tools based on fractional calculus, advancing both the field but also specific areas in computational mechanics. The fractional order may be a function of space-time or even a distribution, opening up great opportunities for modeling and simulation of multiscale and multiphysics phenomena based on a unified representation. Hence, we will construct data driven fractional differential operators that fit data from a particular experiment, including the effect of uncertainties, in which the fractional PDEs (FPDEs), are determined directly from the data!

**Technical Approach:** We shall address fundamental issues associated with the construction of fractional operators for conservation laws and related applications. An integrated framework will be developed that proceeds from the initial data-driven problem to ultimate engineering applications. This general methodology will allow us to develop new fractional physical models, test existing models and assess numerical methods in terms of accuracy and efficiency, which is very important in this early stage of fractional modeling. Our integrated framework is based on a dynamic integration of five areas: (1) Mathematical analysis of FPDEs; (2) Numerical approximation of FPDEs; (3) Development of fast solvers; (4) Fractional order estimation and validation, from data; and (5) Prototype application problems. Outcome: We will develop a new rigorous theoretical and computational framework enabling end-to-end fractional modeling of physical problems governed by conservation laws in large-scale simulations.

**Specific deliverables include:** Theory of well-posedness of FPDEs; Petrov-Galerkin methods for variable-order tempered FPDEs; numerical solvers for FPDEs; data-driven construction of fractional operators; benchmark examples, new models for turbulence and multiphase systems, and opensource code F-NEKTAR. Impact: Systematic and rigorous mathematical understanding of FPDEs will result in transformative changes to better model various physical processes of interest to DoD. The data-centric integrated framework we propose will lead to a paradigm shift, according to which we construct data-driven fractional operators to model a specific phenomenon instead of the current practice of tweaking free parameters that multiply pre-set integer order operators. Equally important, this integrated framework could potentially provide feedback to experiments – what needs to be measured and at what scale and fidelity – to allow true predictive modeling and rigorous validation.