Arbitrary Equation Research Articles

On nonoscillation of fractional order functional differential equations with forcing term and distributed delays

Abstract The paper deals with the nonoscillatory solutions of arbitrary noninteger-order neutral equations with distributed delays. Through the use of the LFD (Liouville Fractional Derivative) of order α ≥ 0 on the half-axis and BCP (Banach Contraction Principle), we are able to get the nonoscillation criteria. The obtained results are emphasized with some appropriate examples.

Simulating Non-Markovian Quantum Dynamics on NISQ Computers Using the Hierarchical Equations of Motion.

Quantum computing offers promising new avenues for tackling the long-standing challenge of simulating the quantum dynamics of complex chemical systems, particularly open quantum systems coupled to external baths. However, simulating such nonunitary dynamics on quantum computers is challenging since quantum circuits are specifically designed to carry out unitary transformations. Furthermore, chemical systems are often strongly coupled to the surrounding environment, rendering the dynamics non-Markovian and beyond the scope of Markovian quantum master equations like Lindblad or Redfield. In this work, we introduce a quantum algorithm designed to simulate non-Markovian dynamics of open quantum systems. Our approach enables the implementation of arbitrary quantum master equations on noisy intermediate-scale quantum (NISQ) computers. We illustrate the method as applied in conjunction with the numerically exact hierarchical equations of motion (HEOM) method. The effectiveness of the resulting quantum HEOM algorithm is demonstrated as applied to simulations of the non-Lindbladian electronic energy and charge transfer dynamics in models of the carotenoid-porphyrin-C60 molecular triad dissolved in tetrahydrofuran and the Fenna-Matthews-Olson complex.

A method for constructing a complete bifurcation picture of a boundary value problem for nonlinear partial differential equations: application of the Kolmogorov-Arnold theorem

The purpose of this study is to develop a numerical method for bifurcation analysis of nonlinear partial differential equations, based on the reduction of partial differential equations to ordinary ones, using the Kolmogorov-Arnold theorem. Methods. This paper describes a method for reducing partial differential equations to ordinary ones using the Kolmogorov-Arnold theorem, as well as methods for the bifurcation analysis of nonlinear boundary value problems for ordinary differential equations. Results. The paper presents a new method for solving and bifurcation analysis of nonlinear boundary value problems for partial differential equations, which allow variational formulation. The method was applied to a nonlinear two-dimensional Bratu problem with Dirichlettype boundary conditions. Conclusion. A new method of bifurcation analysis for nonlinear partial differential equations has been developed. Specifically, a method has been proposed for reducing partial different equations to ordinary equations, which allows the use of the developed apparatus of bifurcation analysis for boundary value problems of ordinary differential equations. This method allows conducting bifurcation analysis for arbitrary nonlinear partial differential equations.

Remarkable Localized Integral Identities for 3D Compressible Euler Flow and the Double-Null Framework

We derive new, localized geometric integral identities for solutions to the 3D compressible Euler equations under an arbitrary equation of state when the sound speed is positive. The integral identities are coercive in the derivatives of the specific vorticity (defined to be vorticity divided by density) and the derivatives of the entropy gradient vectorfield, and the error terms exhibit remarkable regularity and null structures. Our framework plays a fundamental role in our companion works (Abbrescia L, Speck J. The emergence of the singular boundary from the crease in 3D compressible Euler flow, 2022; Abbrescia and Speck, The emergence of the Cauchy horizon from the crease in 3D compressible Euler flow (in preparation)) on the structure of the maximal classical development for shock-forming solutions. It allows one to simultaneously unleash the full power of the geometric vectorfield method for both the wave- and transport- parts of the flow on compact regions, and our approach reveals fundamental new coordinate-invariant structural features of the flow. In particular, the integral identities yield localized control over one additional derivative of the vorticity and entropy compared to standard results, assuming that the initial data enjoy the same gain. Similar results hold for the solution’s higher derivatives. We derive the identities in detail for two classes of spacetime regions that frequently arise in PDE applications: (i) compact spacetime regions that are globally hyperbolic with respect to the acoustical metric, where the top and bottom boundaries are acoustically spacelike—but not necessarily equal to portions of constant Cartesian-time hypersurfaces; and (ii) compact regions covered by double-acoustically null foliations. Our results have implications for the geometry and regularity of solutions, the formation of shocks, the structure of the maximal classical development of the data, and for controlling solutions whose state along a pair of intersecting characteristic hypersurfaces is known. Our analysis relies on a recent new formulation of the compressible Euler equations that splits the flow into a geometric wave-part coupled to a div-curl-transport part. The main new contribution of the present article is our analysis of the spacelike, co-dimension one and two boundary integrals that arise in the div-curl identities. By exploiting interplay between the elliptic and hyperbolic parts of the new formulation and using careful geometric decompositions, we observe several crucial cancellations, which in total show that after a further integration with respect to an acoustical time function, the boundary integrals have a good sign, up to error terms that can be controlled due to their good null structure and regularity properties.

Richtmyer–Meshkov instability when a shock is reflected for fluids with arbitrary equation of state

First predicted by Richtmyer in 1960 and experimentally confirmed by Meshkov in 1969, the Richtmyer–Meshkov instability (RMI) is crucial in fields such as physics, astrophysics, inertial confinement fusion and high-energy-density physics. These disciplines often deal with strong shocks moving through condensed materials or high-pressure plasmas that exhibit non-ideal equations of state (EoS), thus requiring theoretical models with realistic fluid EoS for accurate RMI simulations. Approximate formulae for asymptotic growth rates, like those proposed by Richtmyer, are helpful but rely on heuristic prescriptions for compressible materials. These prescriptions can sometimes approximate the RMI growth rate well, but their accuracy remains uncertain without exact solutions, as the fully compressible RMI growth rate is influenced by both vorticity deposited during shock refraction and multiple sonic wave refractions. This study advances previous work by presenting an analytic, fully compressible theory of RMI for reflected shocks with arbitrary EoS. It compares theoretical predictions with heuristic prescriptions using ideal gas, van der Waals gas and three-term constitutive equations for simple metals, the latter being analysed with detailed and simplified ideal-gas-like EoS. We additionally offer an alternative explicit approximate formula for the asymptotic growth rate. The comprehensive model also incorporates the effects of constant-amplitude acoustic waves at the interface, associated with the D'yakov–Kontorovich instability in shocks.

Analysis of multi-term arbitrary order implicit differential equations with variable type delay

Due to their capacity to simulate intricate dynamic systems containing memory effects and non-local interactions, fractional differential equations have attracted a great deal of attention lately. This study examines multi-term fractional differential equations with variable type delay with the goal of illuminating their complex dynamics and analytical characteristics. The introduction to fractional calculus and the justification for its use in many scientific and technical domains sets the stage for the remainder of the essay. It describes the importance of including variable type delay in differential equations and then applying it to model more sophisticated and realistic behaviours of real-world phenomena. The research study then presents the mathematical formulation of variable type delay and multi-term fractional differential equations. The system’s novelty stems from its unique combination of variable delay, generalized multi terms fractional differential operators (n and m), and integral implicit parameters, and studying the stability of the the newly formulated system as compared to the work in the existing literature. While the variable type delay is introduced as a function of time to describe instances where the delay is not constant, the fractional order derivatives are generated using the Caputo approach. The existence, uniqueness, and stability of solutions are the main topics of the theoretical analysis of the suggested differential equations. In order to establish important mathematical features, the inquiry makes use of spectral techniques, and fixed-point theorems. The study finishes by summarizing the major discoveries and outlining potential future research avenues in this developing field. It highlights the potential contribution of multi-term fractional differential equations with variable type delay to improving the control and design of complex systems. Overall, this study adds to the growing body of knowledge in the field of fractional calculus and provides insightful information about the investigation of multi-term fractional differential equations with variable type delay, making it pertinent for academics and practitioners from a variety of fields.

Improving the convergence of an iterative algorithm for solving arbitrary linear equation systems using classical or quantum binary optimization

Improving the convergence of an iterative algorithm for solving arbitrary linear equation systems using classical or quantum binary optimization

Reconfigurable metamaterial processing units that solve arbitrary linear calculus equations

Calculus equations serve as fundamental frameworks in mathematics, enabling describing an extensive range of natural phenomena and scientific principles, such as thermodynamics and electromagnetics. Analog computing with electromagnetic waves presents an intriguing opportunity to solve calculus equations with unparalleled speed, while facing an inevitable tradeoff in computing density and equation reconfigurability. Here, we propose a reconfigurable metamaterial processing unit (MPU) that solves arbitrary linear calculus equations at a very fast speed. Subwavelength kernels based on inverse-designed pixel metamaterials are used to perform calculus operations on time-domain signals. In addition, feedback mechanisms and reconfigurable components are used to formulate and solve calculus equations with different orders and coefficients. A prototype of this MPU with a compact planar size of 0.93λ0×0.93λ0 (λ0 is the free-space wavelength) is constructed and evaluated in microwave frequencies. Experimental results demonstrate the MPU’s ability to successfully solve arbitrary linear calculus equations. With the merits of compactness, easy integration, reconfigurability, and reusability, the proposed MPU provides a potential route for integrated analog computing with high speed of signal processing.

An explicit primitive conservative solver for the Euler equations with arbitrary equation of state

An explicit primitive conservative solver for the Euler equations with arbitrary equation of state

Verification and validation of detonation-shock-dynamics relations for explosives described by general equation of state and chemical reaction models

Detonation shock dynamics is a powerful method to model the behaviour of High Explosives (HE). However in order to use this method, the underlying relationship between the local radius of curvature and the detonation speed must be known. Previous work has developed methods to calculate this effect using simple, single-step Arrhenius and polytropic gas, models for the chemical reaction and the equation of state, respectively. In recent years, more complex models for both reaction rates and equations of state have been developed which show better agreement with experimental data than these simple models, especially when considering condensed phase explosives.. This work presents the governing equations for solving these problems in a way that is generalised to use arbitrary equations of state as well as reaction models which may have more than a single step and multiple product species. This implementation is verified against exact solutions, demonstrating that the equations were implemented properly. The verified algorithm is then validated against experimental data and high fidelity simulations, showing that it is able to make accurate predictions in a regime where the underlying assumptions of the governing equations are valid. This approach has many applications: from creating equivalent detonation shock dynamics models for existing reactive burn calibrations for HE; to developing new functional forms and calibrations of reactive burn models for condensed phase high explosives.

Hyers-Ulam stability and continuous dependence of the solution of a nonlocal stochastic-integral problem of an arbitrary (fractional) orders stochastic differential equation

Stochastic problems play a huge role in many applications including biology, chemistry, physics, economics, finance, mechanics, and several areas. In this paper, we are concerned with the nonlocal stochastic-integral problem of the arbitrary (fractional) orders stochastic differential equation where B is any Brownian motion, W is a standard Brownian motion, and X 0 is a second order random variable. The Hyers -Ulam stability of the problem will be studied. The existence of solution and its continuous dependence on the Brownian motion B will be proved. The three spatial cases Brownian bridge process, the Brownian motion with drift and the Brownian motion started at A will be considered.

On generalized discrete Ricker map

<p>In recent years, conventional Ricker maps have enjoyed widespread applications across crucial domains such as modeling and security. However, their limitation to a single changeable parameter poses constraints on their adaptability. This paper introduces a generalized form of the Ricker map, incorporating arbitrary powers, thus offering enhanced versatility compared to the traditional Ricker map. By introducing an additional parameter (arbitrary power), the map gains increased degrees of freedom, thereby accommodating a broader spectrum of applications. Consequently, the conventional Ricker map emerges as merely a special case within each proposed framework. This newfound parameter enhances system flexibility and elucidates the conventional system's performance across diverse contexts. Through numerous illustrations, we meticulously investigate the impact of the arbitrary power and equation parameters on equilibrium points, their positions, basin of attraction, stability conditions, and bifurcation diagrams, including the emergence of chaotic behavior.</p>

Об уравнениях в свободных группах с коммутантными ограничениями на решения

A polynomial algorithm has been constructed that allows, given an arbitrary equation of the form w(x1, . . . , xn) = [a, b], resolved with respect to unknowns, where w(x1, . . . , xn) is a group word in the alphabet of unknowns and [a, b] is the commutator of free generators a and b of the free group F2, to determine whether there is a solution to this equation that satisfies the condition x1 . . . , xn ∈ F (1) 2 , where F (1) 2 is the commutator of group F2. The existence of a polynomial algorithm has been established that allows, given an arbitrary equation of the form w(x1, . . . , xn) = g(a, b), where g(a, b) is an element of length less than 4 of the free group F2, to determine whether a solution to this equation exists, that satisfies the condition x1, . . . , xt ∈ F (1) 2 , where t is an arbitrary fixed number between 1 and n. The algorithmic solvability of a similar problem has been proven for the equations w(x1, a, b) = 1 with one variable x1.

릴의 방법과 지오지브라, 파이썬을 이용한 다항방정식 풀이

A major problem in algebra was to find a formula for the roots of polynomial equations of degree 5 or more, and it was proved by Abel and Ruffini that there was no formula for the roots. Lill developed the ‘graphical method’ by exploring the roots of polynomial equations using a geometric method rather than an algebraic method. To solve arbitrary polynomial equations, certain conditions of the ‘graphical method’ must be satisfied, but Lill did not provide a specific method for this. Beloch expressed quadratic and cubic equations using Lill's method and then discovered a method of satisfying specific conditions using origami. However, when the degree is 4 or more, several conditions must be satisfied simultaneously to solve it with origami, so there has been no research on solving the roots of arbitrary polynomial equations. Therefore, this study selected the following research questions. First, we develop a program that automatically solves quadratic and cubic equations using Lill's method, geogebra, and python. Second, we develop a program that automatically solves arbitrary polynomial equations of degree 4 or higher using Lill's method and python. As a result of the study, we were able to manually find the paths of quadratic and cubic equations using Geogebra and automatically find them using Python and develop a program to solve them using Lill's method. To find the path of an arbitrary nth degree polynomial equation of degree 4 or higher, we used object-oriented programming from the Pygame library to solve it. The significance of this research is that it visualizes the process of approximating the roots of an arbitrary polynomial equation, approaches it with a geometric method rather than an algebraic method, and develops a program to solve arbitrary polynomial equations in a generalized way.

Revisiting [formula omitted] cosmologies

Revisiting [formula omitted] cosmologies

Complex Matrix Equation Solver Based on Computational Metasurface

AbstractWith the increasing demands for computing powers, wave‐based computation has attracted researcher's attention due to its inherent high speed. Metasurfaces have flexible and powerful capabilities to manipulate electromagnetic waves and are good candidates for building a next‐generation of wave‐based computers. In this work, a computational‐metasurface‐based equation solver that yields real‐time solutions to arbitrary complex matrix equations (CME) in the form of K · x + c = 0 is proposed. Full‐wave simulations and experimental measurements are conducted to verify the functionality of the proposed solver. This work serves to provide a promising technique to overcome some shortcomings of the existing designs and lays the first stone of the bridge toward programmable wave‐space computers.

Qualitative Aspects of a Fractional-Order Integro-Differential Equation with a Quadratic Functional Integro-Differential Constraint

This manuscript investigates a constrained problem of an arbitrary (fractional) order quadratic functional integro-differential equation with a quadratic functional integro-differential constraint. We demonstrate that there is at least one solution x∈C[0,T] to the problem. Moreover, we outline the necessary demands for the solution’s uniqueness. In addition, the continuous dependence of the solution and the Hyers–Ulam stability of the problem are analyzed. In order to illustrate our results, we provide some particular cases and instances.

Experimenting with discrete dynamical systems

We demonstrate the power of Experimental Mathematics and Symbolic Computation to study intriguing problems on rational difference equations, studied extensively by Difference Equations giants, Saber Elaydi and Gerry Ladas (and their students and collaborators). In particular we rigorously prove some fascinating conjectures made by Amal Amleh and Gerry Ladas back in 2000. For other conjectures we are content with semi-rigorous proofs. We also extend the work of Emilie Purvine (formerly Hogan) and Doron Zeilberger for rigorously and semi-rigorously proving global asymptotic stability of arbitrary rational difference equations (with positive coefficients), and more generally rational transformations of the positive orthant of R k into itself.

Preparing a Fast Pauli Decomposition for Variational Quantum Solving Linear Equations

AbstractSolving linear equations is a basic and significant mathematical task, and it can be executed by variational quantum algorithm (VQA) with quantum advantages by leveraging near‐term quantum device and classical optimizer. In the above algorithm, the coefficient matrix should be decomposed first but slowly with the traditional method, for realizing an effective quantum circuit. In this paper, a general framework for preparing a fast Pauli decomposition for solving arbitrary linear equations using VQA is proposed. This method has a simpler form and reduces the complexity of matrix decomposition compared with the traditional one. Moreover, the concrete tables of two and three qubits cases are given for looking up quickly and the instances of Toeplitz matrix, Yule–Walker, and arbitrary equations are demonstrated. Finally, numerical simulations are given to verify this method. This work provides a more convenient and faster preparatory phase for solving linear equations using VQA.

Nontrivial solutions for arbitrary order discrete relaxation equations with periodic boundary conditions

Nontrivial solutions for arbitrary order discrete relaxation equations with periodic boundary conditions