Numerical methods for time-fractional evolution equations with nonsmooth data: a concise overview

Over the past few decades, there has been substantial interest in evolution equations that involving a fractional-order derivative of order $\alpha\in(0,1)$ in time, due to their many successful applications in engineering, physics, biology and finance. Thus, it is of paramount importance to develop and to analyze efficient and accurate numerical methods for reliably simulating such models, and the literature on the topic is vast and fast growing. The present paper gives a concise overview on numerical schemes for the subdiffusion model with nonsmooth problem data, which are important for the numerical analysis of many problems arising in optimal control, inverse problems and stochastic analysis. We focus on the following aspects of the subdiffusion model: regularity theory, Galerkin finite element discretization in space, time-stepping schemes (including convolution quadrature and L1 type schemes), and space-time variational formulations, and compare the results with that for standard parabolic problems. Further, these aspects are showcased with illustrative numerical experiments and complemented with perspectives and pointers to relevant literature.Comment: 24 pages, 3 figure

The Magnus expansion and some of its applications

Approximate resolution of linear systems of differential equations with varying coefficients is a recurrent problem shared by a number of scientific and engineering areas, ranging from Quantum Mechanics to Control Theory. When formulated in operator or matrix form, the Magnus expansion furnishes an elegant setting to built up approximate exponential representations of the solution of the system. It provides a power series expansion for the corresponding exponent and is sometimes referred to as Time-Dependent Exponential Perturbation Theory. Every Magnus approximant corresponds in Perturbation Theory to a partial re-summation of infinite terms with the important additional property of preserving at any order certain symmetries of the exact solution. The goal of this review is threefold. First, to collect a number of developments scattered through half a century of scientific literature on Magnus expansion. They concern the methods for the generation of terms in the expansion, estimates of the radius of convergence of the series, generalizations and related non-perturbative expansions. Second, to provide a bridge with its implementation as generator of especial purpose numerical integration methods, a field of intense activity during the last decade. Third, to illustrate with examples the kind of results one can expect from Magnus expansion in comparison with those from both perturbative schemes and standard numerical integrators. We buttress this issue with a revision of the wide range of physical applications found by Magnus expansion in the literature.Comment: Report on the Magnus expansion for differential equations and its applications to several physical problem

International Conference on Mathematical Analysis and Applications in Science and Engineering – Book of Extended Abstracts

The present volume on Mathematical Analysis and Applications in Science and Engineering - Book of Extended Abstracts of the ICMASC’2022 collects the extended abstracts of the talks presented at the International Conference on Mathematical Analysis and Applications in Science and Engineering – ICMA2SC'22 that took place at the beautiful city of Porto, Portugal, in June 27th-June 29th 2022 (3 days). Its aim was to bring together researchers in every discipline of applied mathematics, science, engineering, industry, and technology, to discuss the development of new mathematical models, theories, and applications that contribute to the advancement of scientific knowledge and practice. Authors proposed research in topics including partial and ordinary differential equations, integer and fractional order equations, linear algebra, numerical analysis, operations research, discrete mathematics, optimization, control, probability, computational mathematics, amongst others. The conference was designed to maximize the involvement of all participants and will present the state-of- the-art research and the latest achievements.info:eu-repo/semantics/publishedVersio

A spectral representation solution for electromagnetic scattering from complex structures

Significant effort has been directed towards improving computational efficiency in calculating radiated or scattered fields from a complex structure over a broad frequency band. The formulation and solution of boundary integral equation methods in commercial and scientific software has seen considerable attention; methods presented in the literature are often abstract, “curve-fits” or lacking a sound foundation in the underlying physics of the problem. Anomalous results are often characterized incorrectly, or require user expertise for analysis, a clear disadvantage in a computer-aided design tool. This dissertation documents an investigation into the motivating theory, limitations and integration into SuperNEC of a technique for the analytical, continuous, wideband description of the response of a complex conducting body to an electromagnetic excitation. The method, referred to by the author as Transfer Function Estimation (TFE) has its foundations in the Singularity Expansion Method (SEM). For scattering and radiation from a perfect electric conductor, the Electric-Field Integral Equation (EFIE) and Magnetic-Field Integral Equation (MFIE) formulations in their Stratton-Chu form are used. Solution by spectral representation methods including the Singular Value Decomposition (SVD), the Singular Value Expansion (SVE), the Singular Function Method (SFM), Singularity Expansion Method (SEM), the Eigenmode Expansion Method (EEM) and Model-Based Parameter Estimation (MBPE) are evaluated for applicability to the perfect electric conductor. The relationships between them and applicability to the scattering problem are reviewed. A common theoretical basis is derived. The EFIE and MFIE are known to have challenges due to ill-posedness and uniqueness considerations. Known preconditioners present possible solutions. The Modified EFIE (MEFIE) and Modified Combined Integral Equation (MCFIE) preconditioner is shown to be consistent with the fundamental derivations of the SEM. Prony’s method applied to the SEM poleresidue approximation enables a flexible implementation of a reduced-order method to be defined, for integration into SuperNEC. The computational expense inherent to the calculation of the impedance matrix in SuperNEC is substantially reduced by a physically-motivated approximation based on the TFE method. iv Using an adaptive approach and relative error measures, SuperNEC iteratively calculates the best continuous-function approximation to the response of a conducting body over a frequency band of interest. The responses of structures with different degrees of resonant behaviour were evaluated: these included an attack helicopter, a log-periodic dipole array and a simple dipole. Remarkable agreement was achieved

Differential Equation Models in Applied Mathematics

The present book contains the articles published in the Special Issue “Differential Equation Models in Applied Mathematics: Theoretical and Numerical Challenges” of the MDPI journal Mathematics. The Special Issue aimed to highlight old and new challenges in the formulation, solution, understanding, and interpretation of models of differential equations (DEs) in different real world applications. The technical topics covered in the seven articles published in this book include: asymptotic properties of high order nonlinear DEs, analysis of backward bifurcation, and stability analysis of fractional-order differential systems. Models oriented to real applications consider the chemotactic between cell species, the mechanism of on-off intermittency in food chain models, and the occurrence of hysteresis in marketing. Numerical aspects deal with the preservation of mass and positivity and the efficient solution of Boundary Value Problems (BVPs) for optimal control problems. I hope that this collection will be useful for those working in the area of modelling real-word applications through differential equations and those who care about an accurate numerical approximation of their solutions. The reading is also addressed to those willing to become familiar with differential equations which, due to their predictive abilities, represent the main mathematical tool for applying scenario analysis to our changing world

Experimental and Novel Analytic Results for Couplings in Ordered Submicroscopic Systems: from Optomechanics to Thermomechanics

Theoretical modelling of challenging multiscale problems arising in complex (and sometimes bioinspired) solids are presented. Such activities are supported by analytical, numerical and experimental studies. For instance, this is the case for studying the response of hierarchical and nano-composites, nanostructured solid/semi-fluid membranes, polymeric nanocomposites, to electromagnetic, mechanical, thermal, and sometimes biological, electrical, and chemical agents. Such actions are notoriously important for sensors, polymeric films, artificial muscles, cell membranes, metamaterials, hierarchical composite interfaces and other novel class of materials. The main purpose of this project is to make significant advancements in the study of such composites, with a focus on the electromagnetic and mechanical performances of the mentioned structures, with particular regards to novel concept devices for sensing. These latter ones have been studied with different configuration, from 3D colloidal to 2D quasi-hemispherical micro voids elastomeric grating as strain sensors. Exhibited time-rate dependent behavior and structural phenomena induced by the nano/micro-structure and their adaptation to the applied actions, have been explored. Such, and similar, ordered submicroscopic systems undergoing thermal and mechanical stimuli often exhibit an anomalous response. Indeed, they neither follow Fourier’s law for heat transport nor their mechanical time-dependent behavior exhibiting classical hereditariness. Such features are known both for natural and artificial materials, such as bone, lipid membranes, metallic and polymeric “spongy” composites (like foams) and many others. Strong efforts have been made in the last years to scale-up the thermal, mechanical and micro-fluidic properties of such solids, to the extent of understanding their effective bulk and interface features. The analysis of the physical grounds highlighted above has led to findings that allow the describing of those materials’ effective characteristics through their fractional-order response. Fractional-order frameworks have also been employed in analyzing heat transfer to the extent of generalizing the classical Fourier and Cattaneo transport equations and also for studying consolidation phenomenon. Overall, the research outcomes have fulfilled all the research objectives of this thesis thanks to the strong interconnection between several disciplines, ranging from mechanics to physics, from structural health monitoring to chemistry, both from an analytical and numerical point of view to the experimental one