Fast Hankel Transforms Algorithm Based on Kernel Function Interpolation with Exponential Functions

The Pravin method for Hankel transforms is based on the decomposition of kernel function with exponential function. The defect of such method is the difficulty in its parameters determination and lack of adaptability to kernel function especially using monotonically decreasing functions to approximate the convex ones. This thesis proposed an improved scheme by adding new base function in interpolation procedure. The improved method maintains the merit of Pravin method which can convert the Hankel integral to algebraic calculation. The simulation results reveal that the improved method has high precision, high efficiency, and good adaptability to kernel function. It can be applied to zero-order and first-order Hankel transforms

Approximate Analytical Solutions of Space-Fractional Telegraph Equations by Sumudu Adomian Decomposition Method

The main goal in this work is to establish a new and efficient analytical scheme for space fractional telegraph equation (FTE) by means of fractional Sumudu decomposition method (SDM). The fractional SDM gives us an approximate convergent series solution. The stability of the analytical scheme is also studied. The approximate solutions obtained by SDM show that the approach is easy to implement and computationally very much attractive. Further, some numerical examples are presented to illustrate the accuracy and stability for linear and nonlinear cases

Wavelet Theory

The wavelet is a powerful mathematical tool that plays an important role in science and technology. This book looks at some of the most creative and popular applications of wavelets including biomedical signal processing, image processing, communication signal processing, Internet of Things (IoT), acoustical signal processing, financial market data analysis, energy and power management, and COVID-19 pandemic measurements and calculations. The editor’s personal interest is the application of wavelet transform to identify time domain changes on signals and corresponding frequency components and in improving power amplifier behavior

Broadband Multilevel Fast Multipole Methods

Numerical simulations of electromagnetic fields are very important for a plethora of modern applications like antenna design, wireless communication systems, optical systems, high-frequency circuits and so on. As a consequence, there is much interest in finding algorithms that make these simulations as computationally efficient as possible. One of the leading classes of algorithms consists of the so-called Fast Multipole Methods. These methods use a subdivision of the geometry into boxes on multiple levels, in combination with a decomposition of the Green function. For high frequency simulations, where the wavelength is smaller then the smallest features of the geometry, a propagating plane wave decomposition leads to a very efficient algorithm. Unfortunately, this decomposition fails when the geometry contains features smaller than the wavelength, which is the case for broadband simulations. Broadband simulations are becoming increasingly important, for example in the simulation of high frequency printed circuit boards and microwave circuits, metamaterials or the scattering of radar waves off complex shapes. Because of the failure of the propagating plane wave decomposition, performing broadband simulations requires the construction of a hybrid algorithm which uses the propagating plane wave decomposition when the boxes are large enough and some low frequency decomposition when they are not. However, the known low frequency decompositions are usually suboptimal compared to the theoretical performance of the propagating plane wave decomposition. In this work, the focus will be on these low frequency decompositions. First, an improvement over a known low frequency decomposition (the spectral decomposition) is presented. Among other techniques, the well-known Beltrami decomposition of electromagnetic fields is shown to significantly reduce the computational burden in this scheme. Secondly, entirely novel ways of decomposing the Green function are developed in both two and three dimensions. These decompositions use evanescent plane waves, so they can handle small boxes. Nevertheless, they have the same convergence characteristics as the propagating plane wave decomposition. Therefore, these decompositions are also very efficient. Finally, the novel techniques are applied in the full-wave homogenization of various metamaterials

Enriched finite elements for the solution of hyperbolic PDEs

This doctoral research endeavors to reduce the computational cost involved in the solution of initial boundary value problems for the hyperbolic partial differential equation, with special functions used to enrich the solution basis for highly oscillatory solutions. The motivation for enrichment functions is derived from the fact that the typical solutions of the hyperbolic partial differential equations are wave-like in nature. To this end, the nodal coefficients of the standard finite element method are decomposed into plane waves of variable amplitudes. These plane waves form the basis for the proposed enrichment method, that are used for interpolating the solution over the elements, and thus allow for a coarse computational mesh without jeopardizing the numerical accuracy. In this research, the time dependant wave problem is established into a semi-discrete finite element formulation. Both implicit as well as explicit discretization schemes are employed for temporal integration. In either approach, the assembled system matrix needs to be inverted only at the first time step. This inverted matrix is then reused in the subsequent time steps to update the numerical solution with evolution of time. The implicit approach provides unconditional stability, whereas the explicit scheme allows lumping the mass matrix into blocks that are cheaper to invert as opposed to the consistent mass matrix. These methods are validated with several numerical examples. A comparison of the performances of the implicit and the explicit schemes, in conjunction with the enriched finite element basis, is presented. Numerical results are also compared to gauge the performance of the enriched approach against the standard polynomial based finite element approaches. Industrially relevant numerical examples are also studied to illustrate the utility of the numerical methods developed through this research

Digital Filters and Signal Processing

Digital filters, together with signal processing, are being employed in the new technologies and information systems, and are implemented in different areas and applications. Digital filters and signal processing are used with no costs and they can be adapted to different cases with great flexibility and reliability. This book presents advanced developments in digital filters and signal process methods covering different cases studies. They present the main essence of the subject, with the principal approaches to the most recent mathematical models that are being employed worldwide

Dynamical Systems in Spiking Neuromorphic Hardware

Dynamical systems are universal computers. They can perceive stimuli, remember, learn from feedback, plan sequences of actions, and coordinate complex behavioural responses. The Neural Engineering Framework (NEF) provides a general recipe to formulate models of such systems as coupled sets of nonlinear differential equations and compile them onto recurrently connected spiking neural networks – akin to a programming language for spiking models of computation. The Nengo software ecosystem supports the NEF and compiles such models onto neuromorphic hardware. In this thesis, we analyze the theory driving the success of the NEF, and expose several core principles underpinning its correctness, scalability, completeness, robustness, and extensibility. We also derive novel theoretical extensions to the framework that enable it to far more effectively leverage a wide variety of dynamics in digital hardware, and to exploit the device-level physics in analog hardware. At the same time, we propose a novel set of spiking algorithms that recruit an optimal nonlinear encoding of time, which we call the Delay Network (DN). Backpropagation across stacked layers of DNs dramatically outperforms stacked Long Short-Term Memory (LSTM) networks—a state-of-the-art deep recurrent architecture—in accuracy and training time, on a continuous-time memory task, and a chaotic time-series prediction benchmark. The basic component of this network is shown to function on state-of-the-art spiking neuromorphic hardware including Braindrop and Loihi. This implementation approaches the energy-efficiency of the human brain in the former case, and the precision of conventional computation in the latter case

Fractional Calculus and Special Functions with Applications

The study of fractional integrals and fractional derivatives has a long history, and they have many real-world applications because of their properties of interpolation between integer-order operators. This field includes classical fractional operators such as Riemann–Liouville, Weyl, Caputo, and Grunwald–Letnikov; nevertheless, especially in the last two decades, many new operators have also appeared that often define using integrals with special functions in the kernel, such as Atangana–Baleanu, Prabhakar, Marichev–Saigo–Maeda, and the tempered fractional equation, as well as their extended or multivariable forms. These have been intensively studied because they can also be useful in modelling and analysing real-world processes, due to their different properties and behaviours from those of the classical cases.Special functions, such as Mittag–Leffler functions, hypergeometric functions, Fox's H-functions, Wright functions, and Bessel and hyper-Bessel functions, also have important connections with fractional calculus. Some of them, such as the Mittag–Leffler function and its generalisations, appear naturally as solutions of fractional differential equations. Furthermore, many interesting relationships between different special functions are found by using the operators of fractional calculus. Certain special functions have also been applied to analyse the qualitative properties of fractional differential equations, e.g., the concept of Mittag–Leffler stability.The aim of this reprint is to explore and highlight the diverse connections between fractional calculus and special functions, and their associated applications

Higher-Order Tensors and Differential Topology in Diffusion MRI Modeling and Visualization

Diffusion Weighted Magnetic Resonance Imaging (DW-MRI) is a noninvasive method for creating three-dimensional scans of the human brain. It originated mostly in the 1970s and started its use in clinical applications in the 1980s. Due to its low risk and relatively high image quality it proved to be an indispensable tool for studying medical conditions as well as for general scientific research. For example, it allows to map fiber bundles, the major neuronal pathways through the brain. But all evaluation of scanned data depends on mathematical signal models that describe the raw signal output and map it to biologically more meaningful values. And here we find the most potential for improvement. In this thesis we first present a new multi-tensor kurtosis signal model for DW-MRI. That means it can detect multiple overlapping fiber bundles and map them to a set of tensors. Compared to other already widely used multi-tensor models, we also add higher order kurtosis terms to each fiber. This gives a more detailed quantification of fibers. These additional values can also be estimated by the Diffusion Kurtosis Imaging (DKI) method, but we show that these values are drastically affected by fiber crossings in DKI, whereas our model handles them as intrinsic properties of fiber bundles. This reduces the effects of fiber crossings and allows a more direct examination of fibers. Next, we take a closer look at spherical deconvolution. It can be seen as a generalization of multi-fiber signal models to a continuous distribution of fiber directions. To this approach we introduce a novel mathematical constraint. We show, that state-of-the-art methods for estimating the fiber distribution become more robust and gain accuracy when enforcing our constraint. Additionally, in the context of our own deconvolution scheme, it is algebraically equivalent to enforcing that the signal can be decomposed into fibers. This means, tractography and other methods that depend on identifying a discrete set of fiber directions greatly benefit from our constraint. Our third major contribution to DW-MRI deals with macroscopic structures of fiber bundle geometry. In recent years the question emerged, whether or not, crossing bundles form two-dimensional surfaces inside the brain. Although not completely obvious, there is a mathematical obstacle coming from differential topology, that prevents general tangential planes spanned by fiber directions at each point to be connected into consistent surfaces. Research into how well this constraint is fulfilled in our brain is hindered by the high precision and complexity needed by previous evaluation methods. This is why we present a drastically simpler method that negates the need for precisely finding fiber directions and instead only depends on the simple diffusion tensor method (DTI). We then use our new method to explore and improve streamsurface visualization.<br /