First-order Problem Research Articles

Numerical Study on Phase-Fitted and Amplification-Fitted Diagonally Implicit Two Derivative Runge-Kutta Method for Periodic IVPS

A fourth-order two stage Phase-fitted and Amplification-fitted Diagonally Implicit Two Derivative Runge-Kutta method (PFAFDITDRK) for the numerical integration of first-order Initial Value Problems (IVPs) which exhibits periodic solutions are constructed. The Phase-Fitted and Amplification-Fitted property are discussed thoroughly in this paper. The stability of the method proposed are also given herewith. Runge-Kutta (RK) methods of the similar property are chosen in the literature for the purpose of comparison by carrying out numerical experiments to justify the accuracy and the effectiveness of the derived method.

Water transfer to the deep mantle through hydrous, Al-rich silicates in subduction zones

AbstractConstraining deep-water recycling along subduction zones is a first-order problem to understand how Earth has maintained a hydrosphere over billions of years that created conditions for a habitable planet. The pressure-temperature stability of hydrous phases in conjunction with slab geotherms determines how much H2O leaves the slab or is transported to the deep mantle. Chlorite-rich, metasomatic rocks that form at the slab-mantle interface at 50–100 km depth represent an unaccounted, H2O-rich reservoir in subduction processes. Through a series of high-pressure experiments, we investigated the fate of such chlorite-rich rocks at the most critical conditions for subduction water recycling (5–6.2 GPa, 620–800 °C) using two different natural ultramafic compositions. Up to 5.7 GPa, 740 °C, chlorite breaks down to an anhydrous peridotite assemblage, and H2O is released. However, at higher pressures and lower temperatures, a hydrous Al-rich silicate (11.5 Å phase) is an important carrier to enable water transfer to the deep mantle for cold subduction zones. Based on the new phase diagrams, it is suggested that the deep-water cycle might not be in secular equilibrium.

Non-Linear First-Order Differential Boundary Problems with Multipoint and Integral Conditions

This paper considers boundary value problem (BVP) for nonlinear first-order differential problems with multipoint and integral boundary conditions. A suitable Green function was constructed for the first time in order to reduce this problem into a corresponding integral equation. So that by using the Banach contraction mapping principle (BCMP) and Schaefer’s fixed point theorem (SFPT) on the integral equation, we can show that the solution of the multipoint problem exists and it is unique.

Regularizing effect of homogeneous evolution equations with perturbation

Since the pioneering works by Aronson and Bénilan (1979), and Bénilan and Crandall (1981) it is well-known that first-order evolution problems governed by a nonlinear but homogeneous operator admit the smoothing effect that every corresponding mild solution is Lipschitz continuous at every positive time. Moreover, if the underlying Banach space has the Radon–Nikodým property, then this mild solution is a.e. differentiable, and the time-derivative satisfies global and point-wise bounds.In this paper, we show that these results remain true if the homogeneous operator is perturbed by a Lipschitz continuous mapping. More precisely, we establish global L1Aronson–Bénilan type estimates and point-wise Aronson–Bénilan type estimates. We apply our theory to derive global Lq-L∞-estimates on the time-derivative of the perturbed diffusion problem governed by the Dirichlet-to-Neumann operator associated with the p-Laplace–Beltrami operator and lower-order terms on a compact Riemannian manifold with a Lipschitz boundary.

Geometric and Viscosity Solutions for The Cauchy Problem of First Order

Geometric and Viscosity Solutions for The Cauchy Problem of First Order

Application of the regular perturbation method for the solution of first-order initial value problems

Several analytical and numerical methods are globally used in the solution of linear and non-linear ordinary differential equations. In recent times, an analytical method called the perturbation method was proposed by JiHuan He. This technique involves the use of the perturbation parameter, ε. The types of perturbation method which are usually used in Mathematics are the regular perturbation method and the singular perturbation method. In this research, various numerical examples of first-order IVPs are solved using the regular perturbation method. This method gives an approximate analytical solution to the first order IVPs and are discovered to be efficient and easy to adopt.

First-order nonlinear dynamic initial value problems

First-order nonlinear dynamic initial value problems

A Class of Lagrangian–Eulerian Shock-Capturing Schemes for First-Order Hyperbolic Problems with Forcing Terms

In this work, we develop an improved shock-capturing and high-resolution Lagrangian–Eulerian method for hyperbolic systems and balance laws. This is a new method to deal with discontinuous flux and complicated source terms having concentrations for a wide range of applications science and engineering, namely, 1D shallow-water equations, sedimentation processes, Geophysical flows in 2D, N-Wave models, and Riccati-type problems with forcing terms. We also include numerical simulations of a 1D two-phase flow model in porous media, with gravity and a nontrivial singular $$\delta $$ -source term representing an injection point. Moreover, we present approximate solutions for 2D nonlinear systems (Compressible Euler Flows and Shallow-Water Equations) for distinct benchmark configurations available in the literature aiming to present convincing and robust numerical results. In addition, for the linear advection model in 1D and for a smooth solution of the nonlinear Burgers’ problem, second order approximations were obtained. We also present a high-resolution approximation of the nonlinear non-convex Buckley–Leverett problem. Based on the work of A. Harten, we derive a convergent Lagrangian–Eulerian scheme that is total variation diminishing and second-order accurate, away from local extrema and discontinuous data. Additionally, using a suitable Kružkov’s entropy definition, introduced by K. H. Karlsen and J. D. Towers, we can verify that our improved Lagrangian–Eulerian scheme converges to the unique entropy solution for conservation laws with a discontinuous space-time dependent flux. A key hallmark of our method is the dynamic tracking forward of the no-flow curves, which are locally conservative and preserve the natural setting of weak entropic solutions related to hyperbolic problems that are not reversible systems in general. In the end, we have a general procedure to construct a class of Lagrangian–Eulerian schemes to deal with hyperbolic problems with or without forcing terms. The proposed scheme is free of Riemann problem solutions and no adaptive space-time discretizations are needed. The numerical experiments verify the efficiency and accuracy of our new Lagrangian–Eulerian method.

First-order nonlinear dynamic initial value problems

We prove three existence theorems for solutions of first-order dynamic initial value problems, including corresponding continuous and discrete cases. The main tools are fixed point theorems and dyn...

Some results associated to first-order set-valued evolution problems with subdifferentials

Some results associated to first-order set-valued evolution problems with subdifferentials

一阶非线性常微分方程边值问题正解的存在性

一阶非线性常微分方程边值问题正解的存在性

Local Truncation Error-Informed Code Verification

Abstract The method of manufactured solutions (MMS) has become increasingly popular in conducting code verification studies on predictive codes, such as nuclear power system codes and computational fluid dynamic codes. The reason for the popularity of this approach is that it can be used when an analytical solution is not available. Using MMS, code developers are able to verify that their code is free of coding errors that impact the observed order of accuracy. While MMS is still an excellent tool for code verification, it does not identify coding errors that are of the same order as the numerical method. This paper presents a method that combines MMS with modified equation analysis (MEA), which calculates the local truncation error (LTE) to identify coding error up to and including the order of the numerical method. This method is referred to as modified equation analysis methd of manufactured solutions (MEAMMS). MEAMMS is then applied to a custom-built code, which solves the shallow water equations, to test the performance of the code verification method. MEAMMS is able to detect all coding errors that impact the implementation of the numerical scheme. To show how MEAMMS is different than MMS, they are both applied to the same first-order numerical method test problem with a first-order coding error. When there are first-order coding errors, only MEAMMS is able to identify them. This shows that MEAMMS is able to identify a larger set of coding errors while still being able to identify the coding errors MMS is able to identify.

A Stable Approach for Numerical Differentiation by Local Regularization Method with its Regularization Parameter Selection Strategies

The local regularization method for solving the first-order numerical differentiation problem is considered in this paper. The a-priori and a-posteriori selection strategy of the regularization parameter is introduced, and the convergence rate of local regularization solution under some assumption of the exact derivative is also given. Numerical comparison experiments show that the local regularization method can reflect sharp variations and oscillations of the exact derivative while suppress the noise of the given data effectively.

C-Semigroups, subordination principle and the Lévy α-stable distribution on discrete time

In this paper, we introduce the notion of Lévy [Formula: see text]-stable distribution within the discrete setting. Using this notion, a subordination principle is proved, which relates a sequence of solution operators — given by a discrete [Formula: see text]-semigroup — for the abstract Cauchy problem of first order in discrete-time, with a sequence of solution operators for the abstract Cauchy problem of fractional order [Formula: see text] in discrete-time. As an application, we establish the explicit solution of the abstract Cauchy problem in discrete-time that involves the Hilfer fractional difference operator and prove that, in some cases, such solution converges to zero. Our findings give new insights on the theory, provide original concepts and extend as well as improve recent results of relevant references on the subject.

Effect of charge convection on gravitational settling of drop in uniform electric field

Dynamics of droplets in an electrified medium is largely dictated by an intricate interplay between interfacial charge convection and Ohmic conduction within the bulk. The extent of this interaction is quantified by the electric Reynolds number, ReE, delineating their relative strengths. The reported asymptotic theories consider vanishingly low values of ReE, i.e., negligible surface charge convection as compared to the bulk Ohmic conduction, which, in turn, enables decoupling of the contributions of drop deformation and charge convection. This, however, is grossly inaccurate toward establishing an appropriate inter-connection between surface charge convection and morpho-dynamic evolution of the drop beyond such limiting conditions. Circumventing these limits, here we present a theoretical approach that is capable of bringing out the underlying physics beyond low ReE limits. We realize this by incorporating nonlinear charge-convection effects in the leading-order and first-order problem. The present analytical model not only predicts the drop speed accurately but also shows noticeable improvement over the predictive capabilities of the existing asymptotic models. Our results demonstrate that convection of charges can lead to a substantial increase or decrease in gravitational settling speed, depending on the relative electrical properties of the droplet and the carrier. In sharp contrast to previously reported findings, we show that sufficiently strong charge convection can overwhelm the effect of deformation and hence can reverse the trends in the settling speed reported earlier. Comparison with results from full-scale numerical simulations justifies the accuracy of our analytical approach up to a fair level of high asymmetric deformation.

The European Commission's Sustainable Corporate Governance Report: A Critique

In July 2020, the European Commission published the “Study on directors’ duties and sustainable corporate governance” by EY. The Report purports to find evidence of debilitating short-termism in EU corporate governance and recommends many changes to support sustainable corporate governance. In this paper, we point out deep flaws in the Report’s evidence and analysis. We recently submitted the content of this paper in response to the European Commission’s call for feedback. First, the Report defines the corporate governance problem as one of pernicious short-termism that damages the environment, the climate, and stakeholders. But the Report mistakenly conflates time-horizon problems with externalities and distributional concerns. Cures for one are not cures for the others and a cure for one may well exacerbate the others. Second, the Report’s main ostensible evidence for an increase in corporate short-termism is rising gross payouts to shareholders (dividends and stock repurchases). However, the more relevant payout measure to assess corporations’ ability to fund long-term investment is net payouts (gross payouts minus equity issuances), which is much lower and has left plenty of funds available for long-term and short-term investment. Third, when the Report turns to other evidence for short-termism, it selectively picks academic studies that support its views on short-termism, while failing to engage substantial contrary literature. Significant studies fail to detect short-termism and some substantial studies show excessive long-termism. Conceptually, some short-termism is an unfortunate but an inevitable side effect of effective corporate governance and may not be a first-order problem warranting wholesale reform. Finally, the Report touts cures whose effectiveness has little evidentiary support and, for some, there is real evidence that the cures could be counterproductive and costly.

FORMULATION OF BLOCK SCHEMES WITH LINEAR MULTISTEP METHOD FOR THE APPROXIMATION OF FIRST-ORDER IVPS

ABSTRACT
 In this paper, formulation of an efficient numerical schemes for the approximation first-order initial value problems (IVPs) of ordinary differential equations (ODE) is presented. The method is a block scheme for some k-step linear multi-step methods (and) using the Hermite Polynomials a basis function. The continuous and discrete linear multi-step methods (LMM) are formulated through the technique of collocation and interpolation. Numerical examples of ODE have been examined and results obtained show that the proposed scheme can be efficient in solving initial value problems of first order ODE.

EFFICIENT FIFTH-ORDER CLASS FOR THE NUMERICAL SOLUTION OF FIRST ORDER ORDINARY DIFFERENTIAL EQUATIONS

This paper proposes continuous linear multistep methods for the numerical solution of first-order ordinary differential equations (ODEs) with step number k = 1 and k = 2. These methods are used to integrate some first-order initial value problems and the block method developed from the continuous method using interpolation and collocation approach adopting power series approximation as the basis function for the derivation of these methods. These methods are found to be consistent, zero stable, convergent, and accurate. It is noteworthy that the results generated from these methods are significantly accurate and efficient when compared with existing methods, which will be effective in solving first-order Ordinary Differential Equations.

Taking Rorty seriously: pragmatism, metaphilosophy, and truth

ABSTRACT Although Rorty’s work has become a touchstone for evaluating the metaphilosophical stance appropriate for pragmatism, the suspicion prevails that his ‘neopragmatism’ (NeoP) is undermined by a failure to take first-order philosophical problems seriously. We propose that this imputation is grounded in the assumption that he attempts to distinguish metaphilosophy from philosophy in order to insulate the former from the latter, and against it argue that pragmatism’s experimental attitude towards inquiry entails that there is and can be no such separation. We go on to suggest that philosophers such as Misak who define their ‘new pragmatist’ (NewP) position partly in opposition to Rorty’s, insist on this separation because they feel themselves answerable to some ‘transcendental’ urge. The conviction that to reject that urge is to reject the calling of philosophy itself is manifest in the debate between NeoP and NewP on the role of truth-talk. We argue that acceding to this urge prevents us from taking philosophy seriously, and by extension from taking Rorty’s contribution to philosophy in like fashion.

Frictionally Decaying Frontal Warm-Core Eddies

Purpose. The dynamics of nonstationary, nonlinear, axisymmetric, warm-core geophysical surface frontal vortices affected by Rayleigh friction is investigated semi-analytically using the nonlinear, nonstationary reduced-gravity shallow-water equations. The scope is to enlarge the number of known (semi)analytical solutions of nonstationary, nonlinear problems referring to geophysical problems and even to pave the way to their extension to broader geometries and/or velocity fields. Methods and Results. The used method to obtain the solutions is based on the decomposition of the original equations in a part expressing their prescribed spatial structure, so that they can be transformed into ordinary differential equations depending on time only. Based on that analytical procedure, the solutions are then found numerically. In this frame, it is found that vortices characterized by linear distributions of their radial velocity and arbitrary structures of their section and azimuthal velocity can be described exactly by a set of nonstationary, nonlinear coupled ordinary differential equations. The first-order problem (i. e., that describing vortices characterized by a linear azimuthal velocity field and a quadratic section) consists of a system of 4 differential equations, and each further order introduces in the system three additional ordinary differential equations and two algebraic equations. In order to illustrate the behavior of the nonstationary decaying vortices and to put them in the context of observed dynamics in the World Ocean, the system’s solution for the first-order and for the second-order problem is then obtained numerically using a Runge-Kutta method. The solutions demonstrate that inertial oscillations and an exponential attenuation dominate the vortex dynamics: expansions and shallowings, contractions and deepenings alternate during an exact inertial period while the vortex decays. The dependence of the vortex dissipation rate on its initial radius is found to be non-monotonic: it is higher for small and large radii. The possibility of solving (semi)analytically complex systems of differential equations representing observed physical phenomena is rare and very valuable. Conclusions. Our analysis adds realism to previous theoretical investigations on mesoscale vortices, represents an ideal tool for testing the accuracy of numerical models in simulating nonlinear, nonstationary frictional frontal phenomena in a rotating ocean, and paves the way to further extensions of (semi-) analytical solutions of hydrodynamical geophysical problems to more arbitrary forms and more complex density stratifications.