General Linear Equation Research Articles

Ulam stability of linear differential equations using Fourier transform

The purpose of this paper is to study the Hyers-Ulam stability and generalized Hyers-Ulam stability of general linear differential equations of $n$th order with constant coefficients by using the Fourier transform method. Moreover, the Hyers-Ulam stability constants are obtained for these differential equations.

Asymptotics for null-timelike boundary problems for general linear wave equations

We study the linear wave equation $\Box_{g}u=0$ in Bondi-Sachs coordinates, for an asymptotically flat Lorentz metric $g$. We consider the null-timelike boundary problem, where an initial value is given on the null surface $\tau=0$ and a boundary value on the timelike surface $r=r_{0}$. We obtain spacetime $H^{p}$-estimates of $ru$ for $r>r_0$ and derive an asymptotic exapnsion of $ru$ in terms of ${1}/{r}$ as $r\to\infty$.

Constraining values of bag constant for strange star candidates

We provide a strange star model under the framework of general relativity by using a general linear equation of state (EOS). The solution set thus obtained is employed on altogether 20 compact star candidates to constraint values of MIT bag model. No specific value of the bag constant ([Formula: see text]) a priori is assumed, rather possible range of values for bag constant is determined from observational data of the said set of compact stars. To do so, the Tolman–Oppenheimer–Volkoff (TOV) equation is solved by homotopy perturbation method (HPM) and hence we get a mass function for the stellar system. The solution to the Einstein field equations represents a nonsingular, causal and stable stellar structure which can be related to strange stars. Eventually, we get an interesting result on the range of the bag constant as [Formula: see text]. We have found the maximum surface redshift [Formula: see text] and shown that the central redshift ([Formula: see text]) cannot have value larger than [Formula: see text], where [Formula: see text]. Also, we provide a possible value of bag constant for neutron star with quark core using hadronic as well as quark EOS.

The general linear equation on open connected sets

Fix non-zero reals \(\alpha _1\), \(\ldots , \)\(\alpha _n\) with \(n\ge 2\) and let \(K\) be a non-empty open connected set in a topological vector space such that \(\sum _{i\le n}\alpha _iK\subseteq K\) (which holds, in particular, if \(K\) is an open convex cone and \(\alpha _1,\ldots ,\alpha _n>0\)). Let also \(Y\) be a vector space over \(\mathbb{F}\it :=\mathbb{Q} \it (\alpha _1,\ldots ,\alpha _n)\). We show, among others, that a function \(f : K\rightarrow Y\) satisfies the general linear equation $$\forall x_1,\ldots, x_n \in K, \quad f\big (\sum _{i\le n}\alpha _i x_i\big )=\sum _{i\le n}\alpha _i f(x_i)$$ if and only if there exist a unique \(\mathbb{F}\it \)-linear \(A X \rightarrow Y\) and unique \(b\in Y\) such that \(f(x)=A(x)+b\) for all \(x \in K\), with \(b=0\) if \(\sum _{i\le n}\alpha _i\ne 1\). The main tool of the proof is a general version of a result Rado and Baker on the existence and uniqueness of extension of the solution on the classical Pexider equation.

Equation—Domain Duality in the Dirichlet Problem for General Differential Equations in the Space L2

A development of an author’s observation that led to the creation of the equation—domain duality method is presented. This method is used in the study of the Dirichlet problem for a general partial differential equation in a semialgebraic domain. The exposition involves results of the general theory of boundary value problems and is aimed at extending these results to the generalized statements of such problems in L2(Ω). Results on the boundary properties of the L2-solution of a general linear partial differential equation in a domain are employed. It is demonstrated how the general construction under consideration is used in the study of the Dirichlet problem for specific equations with constant coefficients on the basis of the equation—domain duality method. It is also shown how one can extend to the generalized statement of the Dirichlet problem the earlier obtained necessary and sufficient conditions for the existence of a nontrivial smooth solution to the homogeneous Dirichlet problem for a general second-order equation with constant complex coefficients and a homogeneous symbol in a disk, as well as for an ultrahyperbolic equation in the n-dimensional ball.

Traffic Simulation-Based Approach for A Cradle-to-Grave Greenhouse Gases Emission Model

This paper presents a model to evaluate the life cycle greenhouse gases (GHG) emissions, expressed in terms of carbon dioxide equivalent (CO2eq), of a generic fleet composition as a function of the traffic simulation results. First we evaluated the complete life cycle of each category of the vehicles currently circulating; next, by defining a general linear equation, the traffic environmental performances of a real road network (city of Rome) were evaluated using a traffic simulation approach. Finally, the proposed methodology was applied to evaluate the GHG emission of a 100% penetration of battery electric vehicles (BEVs) and various electric and conventional vehicles composition scenarios. In terms of life cycle impacts, BEVs are the vehicles with the highest GHG emissions at the vehicle level (construction + maintenance + end-of-life processes) that are, on average, 20% higher than internal combustion engine vehicles, and 6.5% higher than hybrid electric vehicles (HEVs). Nevertheless, a 100% BEVs penetration scenario generates a reduction of the environmental impact at the mobility system level of about 65%.

The complexity of solution-free sets of integers for general linear equations

Given a linear equation L, a set A of integers is L-free if A does not contain any non-trivial solutions to L. Meeks and Treglown (2018) showed that for certain kinds of linear equations, it is NP-complete to decide if a given set of integers contains a solution-free subset of a given size. Also, for equations involving three variables, they showed that the problem of determining the size of the largest solution-free subset is APX-hard, and that for two such equations (representing sum-free and progression-free sets), the problem of deciding if there is a solution-free subset with at least a specified proportion of the elements is also NP-complete.We answer a number of questions posed by Meeks and Treglown, by extending the results above to all linear equations, and showing that the problems remain hard for sets of integers whose elements are polynomially bounded in the size of the set. For most of these results, the integers can all be positive as long as the coefficients do not all have the same sign.We also consider the problem of counting the number of solution-free subsets of a given set, and show that this problem is #P-complete for any linear equation in at least three variables.

Alienation of two general linear functional equations

We study the alienation problem for two general linear equations i.e. we compare the solutions of the system of equations ∑i=1nαif(pix+qiy)=0∑j=1mβjg(sjx+tjy)=0\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} \\left\\{ \\begin{array}{l}\\sum _{i=1}^n\\alpha _if(p_ix+q_iy)=0\\\\ \\sum _{j=1}^m\\beta _jg(s_jx+t_jy)=0\\end{array}\\right. \\end{aligned}$$\\end{document}with the solutions of the single equation ∑i=1nαif(pix+qiy)=∑j=1mβjg(sjx+tjy).\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} \\sum _{i=1}^n\\alpha _if(p_ix+q_iy)=\\sum _{j=1}^m\\beta _jg(s_jx+t_jy). \\end{aligned}$$\\end{document}To this end we introduce the notion of l-alienation—alienation in the class of monomial functions of order l. We use our results among others to study the alienation properties of two monomial functional equations.

Nitrogen controls the net primary production of an alpine Kobresia meadow in the northern Qinghai-Tibet Plateau.

Net primary production (NPP) is a fundamental property of natural ecosystems. Understanding the temporal variations of NPP could provide new insights into the responses of communities to environmental factors. However, few studies based on long‐term field biomass measurements have directly addressed this subject in the unique environment of the Qinghai‐Tibet plateau (QTP). We examined the interannual variations of NPP during 2008–2015 by monitoring both aboveground net primary productivity (ANPP) and belowground net primary productivity (BNPP), and identified their relationships with environmental factors with the general linear model (GLM) and structural equation model (SEM). In addition, the interannual variation of root turnover and its controls were also investigated. The results show that the ANPP and BNPP increased by rates of 15.01 and 143.09 g/m2 per year during 2008–2015, respectively. BNPP was mainly affected by growing season air temperature (GST) and growing season precipitation (GSP) rather than mean annual air temperature (MAT) or mean annual precipitation (MAP), while ANPP was only controlled by GST. In addition, available nitrogen (AN) was significantly positively associated with BNPP and ANPP. Root turnover rate averaged 30%/year, increased with soil depth, and was largely controlled by GST. Our results suggest that alpine Kobresia meadow was an N‐limited ecosystem, and the NPP on the QTP might increase further in the future in the context of global warming and nitrogen deposition.

Discrete linear evolution equations in a finite lattice

ABSTRACTWe present a general and effective method, known as the Fokas method, to solve an arbitrary discrete linear evolution equation posed in a finite lattice. The method is based on the simultaneous analysis of both parts of Lax pair, as well as the global relation that involves initial and boundary values. We show that, as in the continuum problems, the method can be applied effectively to solve general linear differential-difference equations in a finite lattice. In particular, we demonstrate the method by addressing a number of significant examples and we discuss the continuum limits of the solution and the boundary values.

Uniform continuity and delay robustness of an adaptive controller for Lagrangian systems

AbstractThis paper presents a method based on a continuity argument for analysing the delay robustness of nonlinear control systems with uncertainties. In particular, a delay-dependent stability condition is established in the form of a norm inequality for an adaptive control system with a time delay in the control input. The continuous dependence of the condition on the delay is derived via the uniform continuity in the delay of the fundamental solution of a general linear time-varying delay-differential equation. Since the stability condition holds when the delay is zero, the condition must be true for a certain non-zero positive delay due to continuity. Numerical simulations show the performance and robustness of the control system in the presence of an input delay. Remarkably, the results also illustrate a close agreement with the dependence on control parameters of the transient performance bounds predicted by the theoretical analysis in this paper.

Some notes on the Ulam stability of the general linear equation

We discuss the hyperstability outcome given by Aiemsomboon and Sintunavarat [1] and concerning the general linear equation. We give a simple proof of it via the hyperstability result for the Cauchy equation. Our proof is based on Brzdek’s fixed point theorem. Moreover, we use a weaker assumption.

A higher-order blended compact difference (BCD) method for solving the general 2D linear second-order partial differential equation

A higher-order blended compact difference (BCD) scheme is proposed to solve the general two-dimensional (2D) linear second-order partial differential equation. The distinguishing feature of the present method is that methodologies of explicit compact difference and implicit compact difference are blended together. Sixth-order accuracy approximations for the first- and second-order derivatives are employed, and the original equation is also discretized based on a 9-point stencil, which is different from the work of Lee et al. (J. Comput. Appl. Math. 264:23–37, 2014). A truncation error analysis is performed to show that the scheme is of sixth-order accuracy for the interior grid points. Simultaneously, sixth-order accuracy schemes are proposed to compute the grid points on the boundaries for the first- and second-order derivatives. Numerical experiments are conducted to demonstrate the accuracy and efficiency of the present method.

Non parametric estimation for fractional diffusion processes with random effects

ABSTRACTWe propose a nonparametric estimation for a class of fractional stochastic differential equations (FSDE) with random effects. We precisely consider general linear fractional stochastic differential equations with drift depending on random effects and non-random diffusion. We build ordinary kernel estimators and histogram estimators and study their risk (), when . Asymptotic results are evaluated as both and N tend to infinity.

The Fokas method for integrable evolution equations on a time-dependent interval

We demonstrate how to use the Fokas method to analyze initial-boundary value problems for integrable evolution equations posed on a time-dependent interval. In particular, we implement this method to a general dispersive linear evolution equation with spatial derivatives of arbitrary order and to the nonlinear Schrödinger equation, formulated in the time-dependent domain l1(t) < x < l2(t), 0 < t < T, where l1(t) and l2(t) are given, real, twice differentiable functions whose first derivatives are monotonic, and T is a positive fixed constant.

Quantum algorithm for simulating the wave equation

We present a quantum algorithm for simulating the wave equation under Dirichlet and Neumann boundary conditions. The algorithm uses Hamiltonian simulation and quantum linear system algorithms as subroutines. It relies on factorizations of discretized Laplacian operators to allow for improved scaling in truncation errors and improved scaling for state preparation relative to general purpose linear differential equation algorithms. We also consider using Hamiltonian simulation for Klein-Gordon equations and Maxwell's equations.

Brownian dynamics investigation of the Boltzmann superposition principle for orthogonal superposition rheology

The most general linear equation describing the stress response at time t to a time-dependent shearing perturbation may be written as the integral over the past history t' of a time dependent relaxation modulus, depending on t - t', multiplied by the perturbing shear rate at time t'. This is in agreement with the Boltzmann superposition principle, which says that the stress response of a system to a time dependent shearing deformation may be written as the sum of responses to a sequence of step-strain perturbations in the past. In equilibrium rheology, the Boltzmann superposition principle gives rise to the equality of the shear relaxation modulus, obtained from oscillatory experiments, and the stress relaxation modulus measured after a step-strain perturbation. In this paper, we describe the results of Brownian dynamics simulations of a simple soft matter system showing that the same conclusion does not hold when the system is steadily sheared in a direction perpendicular to the probing flows, and with a gradient parallel to that of the probing deformations, as in orthogonal superposition rheology. In fact, we find that the oscillatory relaxation modulus differs from the step-strain modulus even for the smallest orthogonal shear flows that we could simulate. We do find, however, that the initial or plateau levels of both methods agree and provide an equation relating the plateau value to the perturbation of the pair-function.

Time-Parallel Iterative Solvers for Parabolic Evolution Equations

We present original time-parallel algorithms for the solution of the implicit Euler discretization of general linear parabolic evolution equations with time-dependent self-adjoint spatial operators. Motivated by the inf-sup theory of parabolic problems, we show that the standard nonsymmetric time-global system can be equivalently reformulated as an original symmetric saddle-point system that remains inf-sup stable with respect to the same natural parabolic norms. We then propose and analyze an efficient and readily implementable parallel-in-time preconditioner to be used with an inexact Uzawa method. The proposed preconditioner is nonintrusive and easy to implement in practice. It also features the key theoretical advantage of robust spectral bounds, which lead to convergence rates that are independent of the number of time-steps, final time, and spatial mesh size. Finally, it has a theoretical parallel complexity that grows only logarithmically with respect to the number of time-steps. Numerical experiments with large-scale parallel computations show the effectiveness of the method, along with its good weak and strong scaling properties.

Computational uncertainty quantification for random non-autonomous second order linear differential equations via adapted gPC: a comparative case study with random Fröbenius method and Monte Carlo simulation

AbstractThis paper presents a methodology to quantify computationally the uncertainty in a class of differential equations often met in Mathematical Physics, namely random non-autonomous second-order linear differential equations, via adaptive generalized Polynomial Chaos (gPC) and the stochastic Galerkin projection technique. Unlike the random Fröbenius method, which can only deal with particular random linear differential equations and needs the random inputs (coefficients and forcing term) to be analytic, adaptive gPC allows approximating the expectation and covariance of the solution stochastic process to general random second-order linear differential equations. The random inputs are allowed to functionally depend on random variables that may be independent or dependent, both absolutely continuous or discrete with infinitely many point masses. These hypotheses include a wide variety of particular differential equations, which might not be solvable via the random Fröbenius method, in which the random input coefficients may be expressed via a Karhunen-Loève expansion.

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. (2015) and Ervin et al. (2018). The error estimates are proved by using two approaches. One approach is to apply the duality argument in Johnson (1987) 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.