Non-constant Coefficients Research Articles

Difference Equations and Generating Functions for some Lattice Path Problems

An identity for generating functions is proved in this paper. A novel method to compute the number of restricted lattice paths is developed on the basis of this identity. The method employs a difference equation with non-constant coefficients. Dyck paths, Schr¨oder paths, Motzkins path and other paths are computed to illustrate this method

Semi-implicit Milstein approximation scheme for non-colliding particle systems

We introduce a semi-implicit Milstein approximation scheme for some class of non-colliding particle systems modeled by systems of stochastic differential equations with non-constant diffusion coefficients. We show that the scheme converges at the rate of order 1 in the mean-square sense.

Standing Waves for Nonautonomous Klein-Gordon-Maxwell Systems

We study a Klein-Gordon-Maxwell system, in a bounded spatial domain, under Neumann boundary conditions on the electric potential. We allow a nonconstant coupling coefficient. For sufficiently small data, we find infinitely many standing waves.

From structured data to evolution linear partial differential equations

This paper is devoted to the derivation of computational methods for constructing partial differential equations from data. Following some recent works [7,14,15,20], we propose a methodology based on symbolic calculus [8,9,13], pseudospectral methods [2,3] and stochastic processes [6], in order to determine non-constant coefficients of linear evolution Partial Differential Equations (PDEs), from a set of structured data constituted by solutions at given times and positions, of an unknown linear PDE.

Localised states and their capture characteristics in amorphous phase-change materials

As phase-change materials are poised to play a key role in next-generation computing systems, improving the current understanding of electrical transport in their amorphous phase can further strengthen their technological competitiveness. Even though the interaction of charge carriers with disorder-induced localised states largely affect the field-dependent conductivity, a clear link between electrical transport and specific features of the electronic density of states (DOS) could not be established yet due to a lack of knowledge of the capture characteristics of trap states. Here, we address this knowledge gap and employ modulated photocurrent spectroscopy (MPC) to investigate localised states in the frequently studied amorphous phase of Ge2Sb2Te5. Additionally, we present results on the DOS in the bandgap of amorphous AgIn-doped Sb2Te, which has not been subject to high-resolution DOS spectroscopy before. We find experimental evidence for clearly non-constant capture coefficients among a continuous spectrum of localised states in both studied materials. According to this observation especially in AgIn-doped Sb2Te, where no pronounced defect can be detected as main channel for carrier emission, we point out the necessity of modifying the current Poole-Frenkel-based transport modelling.

A fractional Fokker–Planck equation for non-singular kernel operators

Fractional diffusion equations imply non-Gaussian distributions that allow the characterisation of a wide variety of diffusive processes. Recent advances in fractional calculus lead to a class of new fractional operators defined by non-singular memory kernels. In this work we propose a generalisation of the Fokker–Planck equation in terms of a non-singular fractional temporal operator and considering a non-constant diffusion coefficient. We obtain analytical solutions for the Caputo–Fabrizio and the Atangana–Baleanu fractional kernel operators, from which non-Gaussian distributions emerge having unimodal or bimodal behavior according to whether the diffusion index ν is positive or negative respectively, where the diffusion coefficient of the power law type is considered. Thereby, we show that several types of diffusion can be characterised in the context of fractional kernel operators.

Numerical Analysis of a Semi-Infinite Solid with Temperature Dependent Thermal Conductivity using Truly Meshfree Method

This article presents Meshless Local Petrov-Galerkin (MLPG) method to obtain the numerical solution of linear and non-linear heat conduction in a semi-infinite solid object with specific heat flux. Moving least square approximants are used to approximate the unknown function of temperature T(x) with Th(x). These approximants are constructed by using a linear basis, a weight function and a set of non-constant coefficients. Essential boundary condition is imposed by the penalty function method. A predictor-corrector scheme based on direct substitution iteration has been applied to address the non-linearity and two-level  method for temporal discretization. The accuracy of MLPG method is verified by comparing the results for the simplified versions of the present model with the exact solutions. Once the accuracy of MLPG method is established, the method is further extended to investigate the effects of temperature-dependent properties.

Analytic solutions of fractional ODEs and PDEs

In this work, it is shown that the combined use of exponential operators and integral transforms provides a powerful tool to solve space fractional partial differential equation with non-constant coefficients in three dimensions. Also, a variety of Lamb–Bateman singular integral equation and a non-homogeneous time fractional Kd.V of order [Formula: see text] are solved. Constructive examples are also provided.

Some exact solutions of a hyperbolic model of energy transmission in non-homogeneous media

In this note, we investigate the existence of exact solutions of a nonlinear partial differential equation with time-dependent coefficients that generalizes the well-known nonlinear wave model with damping. The model under consideration generalizes other classical models from physics, like the nonlinear Klein–Gordon equation, the ( 1 + 1 ) -dimensional ϕ 4 -theory, the Fisher–Kolmogorov equation from population dynamics and the Hodgkin–Huxley model used in the description of the propagation of electric signals through the nervous system. An extension of the trial equation method (also known as the direct integral method) for partial differential equations with non-constant coefficients is used in this work in order to derive traveling-wave solutions in exact form.

A General Method for the Ulam Stability of Linear Differential Equations

This paper deals with the Ulam stability of linear differential equations by using the method of variation of parameters, which provides a unified method to study the Ulam stability problem of linear differential equations of n-th order with constant and nonconstant coefficients. As an application of the main results, we also obtain the Hyers–Ulam stability of the Cauchy–Euler differential equations of second order, third order and n-th order. Our results make up to some deficiencies in the relevant literature.

Critical perspectives of the ‘new’ difference equation solution method of Rivera-Figueroa and Rivera-Rebolledo

ABSTRACTRecently, claims of a ‘new and straightforward’ method of solution to second-order linear difference equations have appeared in the mathematics education literature from Rivera-Figueroa and Rivera-Rebolledo. The claim of novelty is based on an assumption that ‘since the equation is worked in its canonical form’, the method within this context must be new. In addition, the assertion of straightforwardness is based on the position that ‘the solution comes naturally’ through this method, rather than artificially. In this article, we subject these claims and assumptions to closer scrutiny, examination and analysis. We note that the method has been published before, and we present the method in a more succinct form. We also discuss how the method can be extended to solve difference equations with non-constant coefficients, illustrating this via a discussion of an example.

Should future wind speed changes be taken into account in wind farm development?

Accurate wind resource assessments are crucial in the development of wind farm projects. However, it is common practice to estimate the wind yield over the next 20 years from short-term measurements and reanalysis data of the past 20 years, even though wind climatology is expected to change under the future climate. The present work examines future changes in wind power output over Europe using an ensemble of ESMs. The power output is calculated using the entire wind speed PDF and a non-constant power conversion coefficient. Based on this method, the ESM ensemble projects changes in near-future power outputs with a spatially varying magnitude between −12% and 8%. The most extreme changes occur over the Mediterranean region. For the first time, the sensitivity of these future change in power output to the type of wind turbine is also investigated. The analysis reveals that the projected wind power changes may vary in up to half of their magnitude, depending on the type of turbine and region of interest. As such, we recommend that wind industries fully account for projected near-future changes in wind power output by taking them into account as a well-defined loss/gain and uncertainty when estimating the yield of a future wind farm.

Solving a free boundary problem with nonconstant coefficients

The present article is concerned with the numerical solution of a free boundary problem for an elliptic state equation with nonconstant coefficients. We maximize the Dirichlet energy functional over all domains of fixed volume. The domain under consideration is represented by a level set function, which is driven by the objective's shape gradient. The state is computed by the finite element method where the underlying triangulation is constructed by means of a marching cubes algorithm. We show that the combination of these tools lead to an efficient solver for general shape optimization problems.

A note on the deformed Hermitian Yang-Mills PDE

ABSTRACTWe prove a priori estimates for a generalised Monge–Ampère PDE with ‘non-constant coefficients’ thus improving a result of Sun in the Kähler case. We apply this result to the deformed Hermitian Yang-Mills (dHYM) equation of Jacob–Yau to obtain an existence result and a priori estimates for some ranges of the phase angle assuming the existence of a subsolution. We then generalise a theorem of Collins–Szèkelyhidi on toric varieties and use it to address a conjecture of Collins–Jacob–Yau.

Drag Coefficient Prediction of a Single Bubble Rising in Liquids

The motion of single bubbles rising in 2-octanol solutions was investigated experimentally. By using a high-speed video system to follow the rising bubble, the sequences of the recorded frames were digitized and analyzed using image analysis software. The periodical fluctuation of the bubble terminal velocity was observed, which is indicative of a nonconstant bubble drag coefficient. Then, the measured drag coefficient was compared with correlations available in the literature. The comparison shows that these correlations cannot give fully satisfactory results in predicting the fluctuated drag coefficient. Using the extensive experimental data, the authors proposed a new correlation to predict the drag coefficient calculated from the fluctuated motion with the added mass force and history force included. In virtue of nonlinear curve fitting, the drag coefficient of the single bubble is correlated as a function of the Re number, Eo number, We number, and Mo number based on the equivalent bubble diameter. T...

A Conditionally Integrable Bi-confluent Heun Potential Involving Inverse Square Root and Centrifugal Barrier Terms

Abstract We present a conditionally integrable potential, belonging to the bi-confluent Heun class, for which the Schrödinger equation is solved in terms of the confluent hypergeometric functions. The potential involves an attractive inverse square root term ~x −1/2 with arbitrary strength and a repulsive centrifugal barrier core ~x −2 with the strength fixed to a constant. This is a potential well defined on the half-axis. Each of the fundamental solutions composing the general solution of the Schrödinger equation is written as an irreducible linear combination, with non-constant coefficients, of two confluent hypergeometric functions. We present the explicit solution in terms of the non-integer order Hermite functions of scaled and shifted argument and discuss the bound states supported by the potential. We derive the exact equation for the energy spectrum and approximate that by a highly accurate transcendental equation involving trigonometric functions. Finally, we construct an accurate approximation for the bound-state energy levels.

A Modal Solution for Finite Length Rods with Non-Uniform Area

This paper derives a modal solution to the displacement field of a finite length rod whose area varies with respect to its length. This new method facilitates a solution to any problem where the area and derivative of the area can be represented as analytical functions. The problem begins by writing the longitudinal displacement of the non-uniform area rod as a series of indexed coefficients multiplied by the eigenfunctions of the uniform area rod. This series solution is inserted into the non-uniform area rod equation and multiplied by a single p-indexed eigenfunction. This equation is then integrated over the interval of the rod. Although the resultant expressions are not orthogonal, they can be written as a set of linear algebraic equations which can be solved to yield the unknown coefficients. Once these are known, the displacement of the system can be calculated. Five example problems are included: the first one has a non-uniform area that corresponds with a known analytical solution, the second has an area that can be represented by a Fourier series, the third and fourth have areas that do not have a known analytical solution and the fifth is a generic second order non-constant coefficient differential equation. Four of these problems are verified with other methods. Convergence of the series solution is discussed. It is shown that this new model is almost two orders of magnitude faster than corresponding finite element analysis.

Finite-Difference Time-Domain Modeling of Space–Time-Modulated Metasurfaces

A finite-difference time-domain modeling of finite-size zero thickness space–time-modulated Huygens’ metasurfaces based on generalized sheet transition conditions is proposed and numerically demonstrated. A typical all-dielectric Huygens’ unit cell is taken as an example and its material permittivity is modulated in both space and time, to emulate a traveling-type spatio-temporal perturbation on the metasurface. By mapping the permittivity variation onto the parameters of the equivalent Lorentzian electric and magnetic susceptibility densities, $\chi _{\text {ee}}$ and $\chi _{\text {mm}}$ , the problem is formulated into a set of second-order differential equations in time with nonconstant coefficients. The resulting field solutions are then conveniently solved using an explicit finite-difference technique and integrated with a Yee-cell-based propagation region to visualize the scattered fields taking into account the various diffractive effects from the metasurface of finite size. Several examples are shown for both linear and space–time varying metasurfaces which are excited with normally incident plane and Gaussian beams, showing detailed scattering field solutions. While the time-modulated metasurface leads to the generation of new collinearly propagating temporal harmonics, these harmonics are angularly separated in space, when an additional space modulation is introduced in the metasurface.

Dispersion-Dissipation Analysis for Advection Problems with Nonconstant Coefficients: Applications to Discontinuous Galerkin Formulations

This paper presents an extended version of von Neumann stability analysis to study dispersion and dissipation errors in nonconstant coefficient advection equations. This approach is used to analyze...

Exponential differential operators for singular integral equations and space fractional Fokker-Planck equation

In this article, it is shown that the combined use of exponential operators and integral transform provides a powerful tool to solve certain type of fractional PDEs. It is shown that exponential operators are powerful and effective method for solving certain differential equations and fractional Fokker-Planck equations with non-constant coefficients. Constructive examples are also provided.