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In this article, an advanced differential quadrature (DQ) approach is proposed for the high-dimensional multi-term time-space-fractional partial differential equations (TSFPDEs) on convex domains. Firstly, a family of high-order difference…
A radial basis function (RBF) method based on matrix-valued kernels is presented and analyzed for computing two types of vector decompositions on bounded domains: one where the normal component of the divergence-free part of the field is…
Accurate interpolation of functions and derivatives is crucial in solving partial differential equations (PDEs). The Radial Basis Function (RBF) method has become an extremely popular and robust approach for interpolation on scattered data.…
In this paper, we approximate the solution and also discuss the periodic behavior termed as eventual periodicity of solutions of (IBVPs) for some dispersive wave equations on a bounded domain corresponding to periodic forcing. The…
We present an efficient method for propagating the time-dependent Kohn-Sham equations in free space, based on the recently introduced Fourier contour deformation (FCD) approach. For potentials which are constant outside a bounded domain,…
We propose a new method of calculating electronically excited states that combines a density functional theory (DFT) based ground state calculation with a linear response treatment that employs approximations used in the time-dependent…
Divergence-free (div-free) and curl-free vector fields are pervasive in many areas of science and engineering, from fluid dynamics to electromagnetism. A common problem that arises in applications is that of constructing smooth approximants…
We present an efficient and systematically convergent approach to all-electron real-time time-dependent density functional theory (TDDFT) calculations using a mixed basis, termed as enriched finite element (EFE) basis. The EFE basis…
The closest point method (Ruuth and Merriman, J. Comput. Phys. 227(3):1943-1961, [2008]) is an embedding method developed to solve a variety of partial differential equations (PDEs) on smooth surfaces, using a closest point representation…
In this chapter, we demonstrate a general formulation of the Finite Element Method allowing to calculate the diffraction efficiencies from the electromagnetic field diffracted by arbitrarily shaped gratings embedded in a multilayered stack…
Most problems in electrodynamics do not have an analytical solution so much effort has been put in the development of numerical schemes, such as the finite-difference method, volume element methods, boundary element methods, and related…
The increasing complexity and scale of photonic and electromagnetic devices demand efficient and accurate numerical solvers. In this work, we develop a parallel overlapping domain decomposition method (DDM) based on the finite-difference…
Advances in deep learning have enabled physics-informed neural networks to solve partial differential equations. Numerical differentiation using the finite-difference (FD) method is efficient in physics-constrained designs, even in…
Density functional theory (DFT), the most widely adopted method in modern computational chemistry, fails to describe accurately the electronic structure of strongly correlated systems. Here we show that DFT can be formally and practically…
We demonstrate that the finite difference grid method (FDM) can be simply modified to satisfy the variational principle and enable calculations of both real and complex poles of the scattering matrix. These complex poles are known as…
The thesis deals with applications of fractional calculus to fractals. It introduces the notion of local fractional derivative (LFD). Fractal and multifractal functions have been studied in the thesis using LFD. New kind of equations are…
Approximating differential operators defined on two-dimensional surfaces is an important problem that arises in many areas of science and engineering. Over the past ten years, localized meshfree methods based on generalized moving least…
An exact arithmetic, memory efficient direct solution method for finite element method (FEM) computations is outlined. Unlike conventional black-box or low-rank direct solvers that are opaque to the underlying physical problem, the proposed…
Basis Function (BF) expansions are a cornerstone of any engineer's toolbox for computational function approximation which shares connections with both neural networks and Gaussian processes. Even though BF expansions are an intuitive and…
In this work, we use the finite differences in time domain (FDTD) numerical method to compute and assess the validity of Hopf solutions, or hopfions, for the electromagnetic field equations. In these solutions, field lines form closed loops…