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Partial Differential Equations (PDE) are fundamental to model different phenomena in science and engineering mathematically. Solving them is a crucial step towards a precise knowledge of the behaviour of natural and engineered systems. In…
This paper investigates the initial-boundary value problem for weakly coupled systems of time-fractional subdiffusion equations with spatially and temporally varying coupling coefficients. By combining the energy method with the coercivity…
This project introduces a groundbreaking approach to address the challenge of periodic signal compression. By proposing a novel adaptive coding method, coupled with hardware-assisted data compression, we have developed a new architecture…
Propagation characteristics of a wave are defined by the dispersion relationship, from which the governing partial differential equation (PDE) can be recovered. PDEs are commonly solved numerically using the finite-difference (FD) method,…
In this paper we consider the Schr\"odinger equation on a network formed by a tree with the last generation of edges formed by infinite strips. We give an explicit description of the solution of the linear Schr\"odinger equation with…
In wave propagation problems, finite difference methods implemented on staggered grids are commonly used to avoid checkerboard patterns and to improve accuracy in the approximation of short-wavelength components of the solutions. In this…
Coherence is a widely used measure to assess linear relationships between time series. However, it fails to capture nonlinear dependencies. To overcome this limitation, this paper introduces the notion of residual spectral density as a…
We investigate the spectrum of differentiation matrices for certain operators on the sphere that are generated from collocation at a set of scattered points $X$ with positive definite and conditionally positive definite kernels. We focus on…
In this paper we solve numerically a degenerate parabolic equation with dynamical boundary conditions of zero-coupon bond pricing. First, we discuss some properties of the differential equation. Then, starting from the divergent form of the…
This article presents a three-step framework for learning and solving partial differential equations (PDEs) using kernel methods. Given a training set consisting of pairs of noisy PDE solutions and source/boundary terms on a mesh, kernel…
Photonic computing has recently become an interesting paradigm for high-speed calculation of computing processes using light-matter interactions. Here, we propose and study an electromagnetic wave-based structure with the ability to…
In this paper, a class of finite difference numerical techniques is presented to solve the second-order linear inhomogeneous damped wave equation. The consistency, stability, and convergences of these numerical schemes are discussed. The…
The difference-of-convex algorithm (DCA) and its variants are the most popular methods to solve the difference-of-convex optimization problem. Each iteration of them is reduced to a convex optimization problem, which generally needs to be…
Coupled wave equations are popular tool for investigating longitudinal dynamical effects in semiconductor lasers, for example, sensitivity to delayed optical feedback. We study a model that consists of a hyperbolic linear system of partial…
Compact finite-difference (FD) schemes specify derivative approximations implicitly, thus to achieve parallelism with domain-decomposition suitable partitioning of linear systems is required. Consistent order of accuracy, dispersion, and…
We prove first-order convergence of the semi-explicit Euler scheme combined with a finite element discretization in space for elliptic-parabolic problems which are weakly coupled. This setting includes poroelasticity, thermoelasticity, as…
We introduce a method-of-lines formulation of the closest point method, a numerical technique for solving partial differential equations (PDEs) defined on surfaces. This is an embedding method, which uses an implicit representation of the…
The main goal of this thesis is to show the crucial role that plays the symbol in analysing the spectrum the sequence of matrices resulting from PDE approximation and in designing a fast method to solve the associated linear problem. In the…
Differential equations where the graph of some derivative of a function is composed of a finite number of similarity transformations of the graph of the function itself are defined. We call these self-similar differential equations (SSDEs)…
Active dendrites are the basis for biologically plausible neural networks possessing many desirable features of the biological brain including flexibility, dynamic adaptability, and energy efficiency. A formulation for active dendrites…