Mathematical Physics Research Articles

Strong error bounds for Trotter and strang-splittings and their implications for quantum chemistry

Efficient error estimates for the Trotter product formula are central in quantum computing, mathematical physics, and numerical simulations. However, the Trotter error's dependency on the input state and its application to unbounded operators remains unclear. Here, we present a general theory for error estimation, including higher-order product formulas, with explicit input state dependency. Our approach overcomes two limitations of the existing operator-norm estimates in the literature. First, previous bounds are too pessimistic as they quantify the worst-case scenario. Second, previous bounds become trivial for unbounded operators and cannot be applied to a wide class of Trotter scenarios, including atomic and molecular Hamiltonians. Our method enables analytical treatment of Trotter errors in chemistry simulations, illustrated through a case study on the hydrogen atom. Our findings reveal the following: (i) for states with fat-tailed energy distribution, such as low-angular-momentum states of the hydrogen atom, the Trotter error scales worse than expected (sublinearly) in the number of Trotter steps; (ii) certain states do not admit an advantage in the scaling from higher-order Trotterization and, thus, the higher-order Trotter hierarchy breaks down for these states, including the hydrogen atom's ground state; (iii) the scaling of higher-order Trotter bounds might depend on the order of the Hamiltonians in the Trotter product for states with fat-tailed energy distribution. Physically, the enlarged Trotter error is caused by the atom's ionization due to the Trotter dynamics. Mathematically, we find that certain domain conditions are not satisfied by some states so higher moments of the potential and kinetic energies diverge. Our analytical error analysis agrees with numerical simulations, indicating that we can estimate the state-dependent Trotter error scaling genuinely. Published by the American Physical Society 2024

Propagation of lump and travelling wave solutions to the (4+1)-dimensional Fokas equation arise in mathematical physics

Propagation of lump and travelling wave solutions to the (4+1)-dimensional Fokas equation arise in mathematical physics

Lie symmetries, exact solution and conservation laws of (2 + 1)-dimensional time fractional Kadomtsev–Petviashvili system

Abstract In this paper, Lie symmetry analysis method is applied to the ( 2 + 1 ) (2+1) -dimensional time fractional Kadomtsev–Petviashvili (KP) system, which is an important model in mathematical physics. We obtain all the Lie symmetries admitted by the KP system and use them to reduce the ( 2 + 1 ) (2+1) -dimensional fractional partial differential equations with Riemann–Liouville fractional derivative to some ( 1 + 1 ) (1+1) -dimensional fractional partial differential equations with Erdélyi–Kober fractional derivative or Riemann–Liouville fractional derivative, thereby getting some exact solutions of the reduced equations. In addition, the new conservation theorem and the generalization of Noether operators are developed to construct the conservation laws for the system studied.

Speed controller design for turboshaft engine using a high-fidelity AeroThermal model

Abstract One of the main objectives of this research is to construct a high-fidelity aerothermal model for controlling the turboshaft engine power turbine rotational speed. Firstly, a high-fidelity aerothermal model of General Electric T700 is formed. The turboshaft engine aerothermal model is simulated and compared to engine design point data on MATLAB/Simulink environment. Thermodynamic equations and some algebraic equations are used. In predicting compressor mass flow rate, the adaptive neuro-fuzzy inference system method is preferred. A propeller model is also added to the turbo shaft engine aerothermal model and to keep the power turbine rotational speed at a constant value, a Proportional Integral Derivative controller is designed and applied. Open-loop and closed-loop simulations are conducted and successful outcomes are achieved. Results indicate that the differences between simulation and actual engine data are within acceptable limits. The suggested model can be readily updated and expanded to control additional parameters of the engine. PACS 2010 Classification: Control theory in mathematical physics, 02.30. Yy, Computer modeling and simulation, 07.05.Tp, 05.70.Ce Thermodynamic functions and equations of state.

Analytical solution of time-fractional Emden-Fowler equations arising in astronomy and mathematical physics

The time-fractional Emden-Fowler model is an interesting mathematical model because it is widely used and important in mathematics and applied sciences. The current investigation compiles the Elzaki transform homotopy perturbation method (ET-HPM) and the Laplace transform homotopy perturbation method (LT-HPM) for the analytical solution of the considered model. In their developments, we show that fractional orders are simple to compute, and the resulting series lead to accurate results after a few iterations. We consider the fractional orders in the context of the Caputo form. The primary aim of these approaches is to compare the analytical results of the models under consideration. These schemes do not impose any constraints on the parameters or make any assumptions that could complicate the process of obtaining the analytical results. We show that the derived outcomes are in convergent sequence with a rapid iterative formula. The iterative series and the graphical illustrations in various fractional orders confirm the efficiency of the proposed approaches. The derived results are also compared to the precise solutions, which show that our suggested approaches are simple, effective, and accurate.

Martínez–Kaabar Fractal–Fractional Laplace Transformation with Applications to Integral Equations

This paper addresses the extension of Martinez–Kaabar (MK) fractal–fractional calculus (for simplicity, in this research work, it is referred to as MK calculus) to the field of integral transformations, with applications to some solutions to integral equations. A new notion of Laplace transformation, named MK Laplace transformation, is proposed, which incorporates the MK α,γ-integral operator into classical Laplace transformation. Laplace transformation is very applicable in mathematical physics problems, especially symmetrical problems in physics, which are frequently seen in quantum mechanics. Symmetrical systems and properties can be helpful in applications of Laplace transformations, which can help in providing an effective computational tool for solving such problems. The main properties and results of this transformation are discussed. In addition, the MK Laplace transformation method is constructed and applied to the non-integer-order first- and second-kind Volterra integral equations, which exhibit a fractal effect. Finally, the MK Abel integral equation’s solution is also investigated via this technique.

Hybrid genetic algorithm with Wiener process for multi-scale colored balanced traveling salesman problem

Colored traveling salesman problem (CTSP) can be applied to Multi-machine Engineering Systems (MES) in industry, colored balanced traveling salesman problem (CBTSP) is a variant of CTSP, which can be used to model the optimization problems with partially overlapped workspace such as the planning optimization (For example, process planning, assembly planning, productions scheduling). The traditional algorithms have been used to solve CBTSP, however, they are limited both in solution quality and solving speed, and the scale of CBTSP is also restricted. Moreover, the traditional algorithms still have the problems such as lacking theoretical support of mathematical physics. In order to improve these, this paper proposes a novel hybrid genetic algorithm (NHGA) based on Wiener process (ITÖ process) and generating neighborhood solution (GNS) to solve multi-scale CBTSP problem. NHGA firstly uses dual-chromosome coding to construct the solutions of CBTSP, then they are updated by the crossover operator, mutation operator and GNS. The crossover length of the crossover operator and the city number of the mutation operator are controlled by activity intensity based on ITÖ process, while the city keeping probability of GNS can be learned or obtained by Wiener process. The experiments show that NHGA can demonstrate an improvement over the state-of-art algorithms for multi-scale CBTSP in term of solution quality.

Mathematical Models for Removal of Pharmaceutical Pollutants in Rehabilitated Treatment Plants

This paper aims to investigate appropriate mathematical models devoted to the optimization of some cleaning processes related to pharmaceutical contaminant removal. In our recent works, we found the rehabilitation of the existing cleaning plants as a viable solution for the removal of this type of micropollutants from waters by introducing efficient techniques such as adsorption on granulated active carbon filters and micro-, nano-, or ultrafiltration. To have these processes under better control and to assure the transfer from small- to large-scale treatment stations, specific mathematical models are necessary. Starting from Navier–Stokes equations and imposing proper boundary conditions, some mathematical physics problems are obtained for which adequate solving methods via variational methods and surjectivity results are proposed. The importance of these solution characterizations consists in their continuation in adequate numerical methods and the possibility to visualize the result by using a CFD program.

Study of the parametric effect of the wave profiles of the time-space fractional soliton neuron model equation arising in the topic of neuroscience

Time-space fractional nonlinear problems (T-SFNLPs) play a crucial role in the study of nonlinear wave propagation. Time-space nonlinearity is prevalent across various fields of applied science, nonlinear dynamics, mathematical physics, and engineering, including biosciences, neurosciences, plasma physics, geochemistry, and fluid mechanics. In this context, we examine the time-space fractional soliton neuron model (TSFSNM), which holds significant importance in neuroscience. This model explains how action potentials are initiated and propagated by axons, based on a thermodynamic theory of nerve pulse transmission. The signals passing through the cell membrane (CM) are proposed to take the form of solitary sound pulses, which can be represented as solitons. To investigate these soliton solutions, nonlinear fractional differential equations (NLFDEs) are transformed into corresponding partial differential equations (PDEs) using a fractional complex transform (FCT). The Kudryashov method is then applied to determine the wave profiles for the TSFSNM equation. We present 3D, 2D, contour, and density plots of the TSFSNM equation, and further analyze how fractional and time-space parameters influence these wave profiles through additional graphical representations. Kink, singular kink and different types of soliton solutions are successfully recovered through the Kudryashov method. The outcomes of various studies show that the applied method is highly efficient and well-suited for tackling problems in applied sciences and mathematical physics. Graphical representations, coupled with numerical data, reinforce the validity and accuracy of the technique. The proposed method is a convenient and powerful tool for handling the solution of nonlinear equations, making it particularly effective in exploring complex wave phenomena in diverse scientific fields.

Una breve storia dei concetti di matematica e fisica

It is shown that there is no clear boundary between mathematics and physics, and a brief history of the concepts of mathematics and physics is outlined. Not only the scientists of antiquity, but also Copernicus and Galileo referred to all the exact sciences by the term mathematics and used the word physics for the philosophy of nature. In Newton’s time, the unity of the exact sciences broke down, and from the combination of part of mathematics with the philosophy of nature modern physics was born. The article then describes how mathematical physics, which began as part of physics, was pushed back into mathematics by the birth of theoretical physics. Finally, the epistemological status of the current branches of the exact sciences is discussed.

Soliton solutions to the (3+1)-dimensional KP and BA models using advanced exp⁡(-ϕ(ξ))-expansion scheme in mathematical physics

Soliton solutions to the (3+1)-dimensional KP and BA models using advanced exp⁡(-ϕ(ξ))-expansion scheme in mathematical physics

Dynamics of optical solitons and sensitivity analysis in fiber optics

The nonlinear Schrödinger equation (NLSE) and its various forms have significant applications in the field of soliton theory. The Fokas-Lenells (FL) equation stands as a cornerstone in deepening our understanding of nonlinear wave dynamics within optical systems, particularly concerning the behavior of ultrashort pulses across different media. Its significance lies in providing a comprehensive framework to study and analyze complex phenomena, ultimately contributing to advancements in optical technology and applications. The FL equation is an integrable extension of the NLSE that provides a description of the nonlinear propagation of pulses in optical fiber. This paper seeks to discover optical soliton solutions for the FL equation by employing a modified sub-equation method. Additionally, the sensitivity analysis is described by using the various initial conditions. The main novelty of this paper lies in conducting a sensitivity analysis of the FL equation by examining the effects of various initial conditions, providing deeper insights into how these conditions influence the behavior of soliton solutions. For the physical behavior of the models, some solutions are graphically shown in 2D, 3D, and contour graphs by assigning specific values to the parameters under the provided situation at each solution. As a result, we discovered several new families of exact traveling wave solutions, such as bright solitons, dark solitons, and combined bright and dark solitons. This research opens numerous avenues for further exploration in the field of nonlinear wave dynamics and optical soliton theory. The discovery of exact soliton solutions for the FL equation through a modified sub-equation method paves the way for deeper investigations for newcomer researchers. The results of this study will contribute further to the field of mathematical physics, particularly in enhancing the understanding of nonlinear wave propagation and soliton theory in optical and other physical systems.

Involute-evolute curves with modified orthogonal frame in Galilean space G 3

UDC 514 We introduce and study an involute-evolute curve pair within a modified orthogonal frame in the context of 3-dimensional Galilean space G 3 . The proposed methodology involves the investigation of the aforementioned curve pair with respect to a modified orthogonal frame, specifically by analyzing their curvature and torsion properties. By using this approach, we derive various characterizations of these curves. The proposed findings contribute to the deeper understanding of the geometric properties and the behavior of involute-evolute curve pairs in the Galilean space, thereby offering potential applications in the differential geometry and mathematical physics.

Towards a probabilistic foundation of relativistic quantum theory: the one-body Born rule in curved spacetime

In this work, we establish a novel approach to the foundations of relativistic quantum theory, which is based on generalizing the quantum-mechanical Born rule for determining particle position probabilities to curved spacetime. A principal motivator for this research has been to overcome internal mathematical problems of relativistic quantum field theory (QFT) such as the ‘problem of infinities’ (renormalization), which axiomatic approaches to QFT have shown to be not only of mathematical but also of conceptual nature. The approach presented here is probabilistic by construction, can accommodate a wide array of dynamical models, does not rely on the symmetries of Minkowski spacetime, and respects the general principle of relativity. In the analytical part of this work, we consider the 1-body case under the assumption of smoothness of the mathematical quantities involved. This is identified as a special case of the theory of the general-relativistic continuity equation. While related approaches to the relativistic generalization of the Born rule assume the hypersurfaces of interest to be spacelike and the spacetime to be globally hyperbolic, we employ prior contributions by C. Eckart and J. Ehlers to show that the former condition is naturally replaced by a transversality condition and that the latter one is obsolete. We discuss two distinct formulations of the 1-body case, which, borrowing terminology from the non-relativistic analog, we term the Lagrangian and Eulerian pictures. We provide a comprehensive treatment of both. The main contribution of this work to the mathematical physics literature is the development of the Lagrangian picture. The Langrangian picture shows how one can address the ‘problem of time’ in this approach and, therefore, serves as a blueprint for the generalization to many bodies and the case that the number of bodies is not conserved. We also provide an example to illustrate how this approach can in principle be employed to model particle creation and annihilation.

Mathematical model for heat transfer during underground coal gasification process

Purpose. To develop a mathematical model of the “coal-gas” medium heat transfer during underground coal gasification to predict the combustion face advance velocity and the duration of gasification column mining. Methodology. To detect the temperature fields in the coal seam and gas, depending on the displacement length of the phase transition boundary, a boundary-value problem of mathematical physics has been developed. To solve this boundary-value problem, Boltzmann transformations, as well as methods for solving ordinary differential equations, are used. Newton-Raphson method, which has quadratic convergence, is used to find the roots of the transcendental equation. Findings. Tendencies in the formation of mathematical models when studying temperature fields around an underground gasifier have been analyzed, with highlighting their disadvantages. A mathematical model of heat transfer during underground gasification has been developed, taking into account the phase transition boundaries of the “coal-gas” medium. A computational experiment was conducted to determine the temperature of the phase transition boundary based on the length of the gasification column and the duration of the process. Originality. A mathematical model for heat transfer during coal gasification in the form of a boundary-value problem of mathematical physics, which consists of parabolic heat-transfer equations, the Stefan condition at the phase transition boundary, and the Dirichlet boundary conditions, has been constructed. As a result of solving the boundary-value problem, a self-similar solution has been obtained for the distribution of the coal seam and gas temperature fields, as well as the position of the phase transition boundary depending on the gasification duration and on the medium density, thermal conductivity coefficients, specific heat capacity of gas and coal, specific calorific value and temperature of coal combustion, initial coal temperature and constant temperature of the gasification process. The conducted analysis of numerical calculations provides a deeper understanding of the dynamics of underground coal gasification process and makes necessary corrections to achieve the maximum process efficiency. Practical value. A methodology for determining the displacement length of the phase transition boundary of the “coal-gas” medium has been developed, taking into account the change in the combustion face temperature along the gasification zone length depending on the duration of this process. Application of the methodology makes it possible to predict the time of mining the gasified coal column for drawing up a calendar plan for mining operations.

Dynamical analysis and soliton solutions of a variety of quantum nonlinear Zakharov–Kuznetsov models via three analytical techniques

Some new types of truncated M-fractional exact soliton solutions of the two important quantum plasma physics models, extended quantum Zakharov–Kuznetsov and extended quantum nonlinear Zakharov–Kuznetsov, are successfully achieved by applying the expa function technique, the improved (G′/G)-expansion technique, and the Sardar sub-equation technique. These two models have many useful applications when explaining the waves in the quantum electron-positron-ion magnetoplasmas as well as weakly nonlinear ion-acoustic waves in plasma. The obtained results are in the form of dark, bright, periodic, and other soliton solutions. The results are verified and represented by two-dimensional, three-dimensional, and contour graphs. The results are newer than the existing results in the literature due to the use of fractional derivatives. Hence, the solutions will be fruitful in future studies on these models. The solutions obtained are useful in the areas of applied physics, applied mathematics, dynamical systems, and nonlinear waves in plasmas and in dense space plasma. The applied techniques are simple, fruitful, and reliable for solving other models in mathematical physics.

Investigating novel optical soliton solutions for a generalized (3+1)-dimensional q-deformed equation

In this study, we investigate the (3+1) q-deformed tanh-Gordon equation due to its importance in the context of mathematical physics. It describes solitonic solutions in quantum field theory; it can sometimes be used in condensed matter physics to describe interactions between particles in magnetic materials or superconductors; it can model light propagation in nonlinear optical fibers or photonic crystals where the refractive index has a q-deformed structure; and it also can be applied in studying shock waves, turbulence and rogue waves where the deformation introduces corrections to classical wave phenomena. Utilizing the (G′ωG′+G+r)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(\\frac{{\\mathfrak {G}}^{\\prime }}{\\omega {\\mathfrak {G}}^{\\prime }+{\\mathfrak {G}}+r})$$\\end{document}-expansion technique, we derive novel analytical solutions that enhance our understanding of the underlying dynamics. Additionally, we employ a finite element method (extended cubic B-spline method) to validate our analytical findings and explore the behavior of the q-deformed equation under different parameter regimes. Our results demonstrate the versatility of the q-deformed framework in generating rich optical phenomena, paving the way for future research in both theoretical and applied physics.

Interesting class of quasilinear systems

With this work, I want to draw attention to a class of differential equations that encompasses a number of important equations of mathematical physics and is convenient for constructing a mathematical theory.

The extended Adomian decomposition method and its application to the rotating shallow water system for the numerical pulsrodon solutions

The rotating shallow water system is an important physical model, which has been widely used in many scientific areas, such as fluids, hydrodynamics, geophysics, oceanic and atmospheric dynamics. In this paper, we extend the application of the Adomian decomposition method from the single equation to the coupled system to investigate the numerical solutions of the rotating shallow water system with an underlying circular paraboloidal basin. By introducing some special initial values, we obtain interesting approximate pulsrodon solutions corresponding to pulsating elliptic warm-core rings, which take the form of realistic series solutions. Numerical results reveal that the numerical pulsrodon solutions can quickly converge to the exact solutions derived by Rogers and An, which fully shows the efficiency and accuracy of the proposed method. Note that the method proposed can be effectively used to construct numerical solutions of many nonlinear mathematical physics equations. The results obtained provide some potential theoretical guidance for experts to study the related phenomena in geography, oceanic and atmospheric science.

Extracting traveling wave solutions for two nonlinear models of NPDEs in mathematical physics

In this work, we apply the Riccati-Bernoulli (RB) sub-ODE approach to provide some vital solitary wave solutions for the nonlinear dispersive modified Benjamin-Bona-Mahony (DMBBM) equation and the Klein-Gordan (KG) equation. The solutions that are provided here are helpful in describing several physical phenomena in inharmonic crystals, cold plasma, compressible fluids and quantum mechanics. The proposed approach is effective and easy, resulting in new generalised solitonic wave profiles. For suitable free parameter values, two-dimensional (2D) and three-dimensional (3D) graphs are depicted to show the shape of the obtained solutions. We also show the effect of the physical parameters on the behaviour of the solutions. Finally, the suggested approach may be extended to different equations appearing in mathematical physics.