An unstructured mesh control volume method for two-dimensional space fractional diffusion equations with variable coefficients on convex domains

In this paper, we propose a novel unstructured mesh control volume method to deal with the space fractional derivative on arbitrarily shaped convex domains, which to the best of our knowledge is a new contribution to the literature. Firstly, we present the finite volume scheme for the two-dimensional space fractional diffusion equation with variable coefficients and provide the full implementation details for the case where the background interpolation mesh is based on triangular elements. Secondly, we explore the property of the stiffness matrix generated by the integral of space fractional derivative. We find that the stiffness matrix is sparse and not regular. Therefore, we choose a suitable sparse storage format for the stiffness matrix and develop a fast iterative method to solve the linear system, which is more efficient than using the Gaussian elimination method. Finally, we present several examples to verify our method, in which we make a comparison of our method with the finite element method for solving a Riesz space fractional diffusion equation on a circular domain. The numerical results demonstrate that our method can reduce CPU time significantly while retaining the same accuracy and approximation property as the finite element method. The numerical results also illustrate that our method is effective and reliable and can be applied to problems on arbitrarily shaped convex domains.Comment: 18 pages, 5 figures, 9 table

Novel numerical analysis of multi-term time fractional viscoelastic non-Newtonian fluid models for simulating unsteady MHD Couette flow of a generalized Oldroyd-B fluid

In recent years, non-Newtonian fluids have received much attention due to their numerous applications, such as plastic manufacture and extrusion of polymer fluids. They are more complex than Newtonian fluids because the relationship between shear stress and shear rate is nonlinear. One particular subclass of non-Newtonian fluids is the generalized Oldroyd-B fluid, which is modelled using terms involving multi-term time fractional diffusion and reaction. In this paper, we consider the application of the finite difference method for this class of novel multi-term time fractional viscoelastic non-Newtonian fluid models. An important contribution of the work is that the new model not only has a multi-term time derivative, of which the fractional order indices range from 0 to 2, but also possesses a special time fractional operator on the spatial derivative that is challenging to approximate. There appears to be no literature reported on the numerical solution of this type of equation. We derive two new different finite difference schemes to approximate the model. Then we establish the stability and convergence analysis of these schemes based on the discrete $H^1$ norm and prove that their accuracy is of $O(\tau+h^2)$ and $O(\tau^{\min\{3-\gamma_s,2-\alpha_q,2-\beta\}}+h^2)$, respectively. Finally, we verify our methods using two numerical examples and apply the schemes to simulate an unsteady magnetohydrodynamic (MHD) Couette flow of a generalized Oldroyd-B fluid model. Our methods are effective and can be extended to solve other non-Newtonian fluid models such as the generalized Maxwell fluid model, the generalized second grade fluid model and the generalized Burgers fluid model.Comment: 19 pages, 8 figures, 3 table

Optimal convergence rates for semidiscrete finite element approximations of linear space-fractional partial differential equations under minimal regularity assumptions

We consider the optimal convergence rates of the semidiscrete finite element approximations for solving linear space-fractional partial differential equations by using the regularity results for the fractional elliptic problems obtained recently by Jin et al. \cite{jinlazpasrun} and Ervin et al. \cite{ervheuroo}. The error estimates are proved by using two approaches. One approach is to apply the duality argument in Johnson \cite{joh} for the heat equation to consider the error estimates for the linear space-fractional partial differential equations. This argument allows us to obtain the optimal convergence rates under the minimal regularity assumptions for the solution. Another approach is to use the approximate solution operators of the corresponding fractional elliptic problems. This argument can be extended to consider more general linear space-fractional partial differential equations. Numerical examples are given to show that the numerical results are consistent with the theoretical results

Magnetic resonance imaging of muscle structure and function

Ziel dieser Arbeit ist die Implementierung und kombinierte Anwendung verschiedener MRT Techniken zur Untersuchung der Struktur und Funktion der humanen Skelettmuskulatur. Insbesondere steht deren Applikation an der Rückenmuskulatur im Vordergrund, um auf Basis dieser Untersuchungen einen Beitrag zur Ursachenforschung des unspezifischen - meist chronifizierten - Rückenschmerzes zu leisten. Vor diesem Hintergrund wurden in der vorliegenden Dissertation dezidierte MR-Pulssequenzen und Bildrekonstruktionsverfahren entwickelt, welche unter Verwendung der diffusionsgewichteten MR-Bildgebung (diffusion-weighted imaging, DWI) die 3D-Rekonstruktion der Muskelfaserarchitektur sowie die Quantifizierung der muskulären Vaskularität ermöglichen. Die Erfassung der Faserarchitektur basiert auf der Diffusionstensorbildgebung (diffusion tensor imaging, DTI) - einer Weiterentwicklung der DWI - und wurde am Tiermodell anhand sequentiell durchgeführter in vivo und post mortem Messungen validiert. Anschließend wurde diese Methode in einer Pilotstudie genutzt, um degenerative Veränderungen bei Patienten nach Wirbelsäulenoperation zu erfassen. Im zweiten Schritt dieser Arbeit, wurde ein Messprotokoll zur funktionellen MR-Untersuchung implementiert, welches Messungen vor, während und nach willkürlicher Muskelkontraktion beinhaltet. Dieses Protokoll sieht weiterhin die Applikation einer neuartigen perfusionsensitiven DWI-Sequenz sowie optimierten Sequenzen zur quantitativen T2-gewichteten MR-Bildgebung und ortaufgelösten 31P-MR-Spektroskopie vor, wobei die beiden letztgenannten Techniken es erlauben, komplexe funktionelle Vorgänge, wie beispielsweise die des Energiemetabolismus, unter Einfluss einer Belastungssituation zu untersuchen. Diese funktionellen MR-Messprinzipien werden in der Regel unter dem Begriff muscle functional MRI (mfMRI) subsumiert und ermöglichen die multi-parametrische Erfassung unterschiedlicher funktioneller und struktureller Eigenschaften der Muskelphysiologie. Dies wird in der vorliegenden Arbeit anhand einer gerontologischen Studie demonstriert, wobei die hierbei gewonnenen Ergebnisse Einblick in zahlreiche altersassoziierte Aspekte der Rückenphysiologie geben. Zusammenfassend werden in dieser Dissertation verschiedene Ansätze der MR-Bildgebung und MR-Spektroskopie vorgestellt, die einerseits für grundlagenwissenschaftliche Fragestellungen zum unspezifischen Rückenschmerz, andererseits aber auch in der klinischen Routine zur Untersuchung degenerativer Veränderungen der Skelettmuskulatur herangezogen werden können.The aim of this work is the implementation and combined application of different MRI-based methods, which facilitate the comprehensive assessment of the skeletal muscle structure and function. Especially, the application of these MRI techniques to human back muscles has been put into focus, which may provide deeper insights into the origin of non-specific - and in most cases chronic - back pain. To this end, dedicated MRI sequences and quantitative image reconstruction approaches were developed in this work, which are based on diffusion-weighted imaging (DWI) and enable the 3D reconstruction of the muscle fiber architecture as well as the assessment of a surrogate measure of the vascular capacity. The MRI-based reconstruction of the fiber architecture relies on diffusion tensor imaging (DTI) - an extension of DWI - and was validated by successive in vivo and post mortem measurements. Afterwards, this method was employed in a pre-clinical pilot study in order to assess surgeryrelated degenerative changes of the back muscles in patients after spinal surgery. In the second step of this work, functional measurements of the human skeletal muscles, which means quantitative measurements prior to, during and after muscular loading, were performed by using a novel perfusion-sensitive DWI sequence as well as optimized sequence protocols of quantitative T2-weighted MRI and spatially resolved 31P-MR spectroscopy. The latter two methods allow the quantification of complex physiological processes, such as of the high-energy metabolism, during the exercise of skeletal muscles, and are often subsumed under the term muscle functional MRI (mfMRI). Combined application of these mfMRI techniques provides multi-parametric evaluation of several structural and functional determinants and, thus, allows comprehensive characterization of the muscle physiology. In order to demonstrate the capabilities of the proposed multi-parametric mfMRI approach, a gerontological study is performed in this work, while the obtained results indicate several age-related aspects of the human back muscle physiology. Overall, this thesis introduces MR imaging and MR spectroscopy techniques, which may, on the one hand, contribute to research of low back pain and, on the other hand, serve as basis of clinical investigations in order to investigate degenerative processes of skeletal muscles

Bosonic Quantum Simulation in Circuit Quantum Electrodynamics

The development of controllable quantum machines is largely motivated by a desire to simulate quantum systems beyond the capabilities of classical computers. Investigating intrinsically multi-level model bosonic systems, using conventional quantum processors based on two-level qubits is inefficient and incurs a potentially prohibitive mapping overhead in the current near-term intermediate-scale quantum (NISQ) era. This motivates the development of hybrid quantum processors that contain multiple types of degrees of freedom, such that one can leverage an optimal one-to-one mapping between the model system and simulator. Circuit quantum electrodynamics (cQED) has emerged as a leading platform for quantum information processing owing to the immense flexibility of engineering high fidelity coherent interactions and measurements. In cQED, microwave photons act as bosonic particles confined within a nonlinear network of electromagnetic modes. Controlling these photons serves as the basis for a hardware efficient platform for simulation of naturally bosonic systems. In this thesis, we present two experiments that encapsulate this idea by simulating molecular dynamics in two different regimes of electronic-nuclear coupling: adiabatic and nonadiabatic. In the first experiment, we implement a boson sampling protocol for estimating Franck-Condon factors associated with photoelectron spectra. Importantly, we fulfill the scalability requirement by developing a novel single-shot number-resolved quantum non-demolition detector for microwave photons. In the second experiment, we develop and employ a model for simulating dissipative nonadiabatic dynamics through a conical intersection as a basis for modeling photochemical reactions. We directly observe branching of a coherent wave-packet upon passage through the conical intersection, revealing the competition between coherent evolution and dissipation in this system. The tools developed for the experiments in this thesis serve as a basis for implementing more complex bosonic simulations

Nonlinear Systems

The editors of this book have incorporated contributions from a diverse group of leading researchers in the field of nonlinear systems. To enrich the scope of the content, this book contains a valuable selection of works on fractional differential equations.The book aims to provide an overview of the current knowledge on nonlinear systems and some aspects of fractional calculus. The main subject areas are divided into two theoretical and applied sections. Nonlinear systems are useful for researchers in mathematics, applied mathematics, and physics, as well as graduate students who are studying these systems with reference to their theory and application. This book is also an ideal complement to the specific literature on engineering, biology, health science, and other applied science areas. The opportunity given by IntechOpen to offer this book under the open access system contributes to disseminating the field of nonlinear systems to a wide range of researchers

Non-Linear Lattice

The development of mathematical techniques, combined with new possibilities of computational simulation, have greatly broadened the study of non-linear lattices, a theme among the most refined and interdisciplinary-oriented in the field of mathematical physics. This Special Issue mainly focuses on state-of-the-art advancements concerning the many facets of non-linear lattices, from the theoretical ones to more applied ones. The non-linear and discrete systems play a key role in all ranges of physical experience, from macrophenomena to condensed matter, up to some models of space discrete space-time