Exact Analytical Solution Research Articles

Using machine learning to find exact analytic solutions to analytically posed physics problems

We investigate the use of machine learning for solving analytic problems in theoretical physics. In particular, symbolic regression (SR) is making rapid progress in recent years as a tool to fit data using functions whose overall form is not known in advance. Assuming that we have a mathematical problem that is posed analytically, e.g. through equations, but allows easy numerical evaluation of the solution for any given set of input variable values, one can generate data numerically and then use SR to identify the closed-form function that describes the data, assuming that such a function exists. In addition to providing a concise way to represent the solution of the problem, such an obtained function can play a key role in providing insight and allow us to find an intuitive explanation for the studied phenomenon. We use a state-of-the-art SR package to demonstrate how an exact solution can be found and make an attempt at solving an unsolved physics problem. We use the Landau-Zener problem and a few of its generalizations as examples to motivate our approach and illustrate how the calculations become increasingly complicated with increasing problem difficulty. Our results highlight the capabilities and limitations of the presently available SR packages, and they point to possible modifications of these packages to make them better suited for the purpose of finding exact solutions as opposed to good approximations. Our results also demonstrate the potential for machine learning to tackle analytically posed problems in theoretical physics.

Exact and explicit analytical solution for the Sakiadis boundary layer

It has recently been established [Naghshineh et al. IMA J. of Appl. Math., 88, 1 (2023)] that a convergent series solution may be obtained for the Sakiadis boundary layer problem once key parameters are determined iteratively using the series itself. Here, we provide exact and explicit analytical expressions for these parameters, including that associated with wall shear, thus completing the exact analytical solution. The resulting solution to the Sakiadis problem is summarized here for direct use.

An exact model for enhancing/suppressing primordial fluctuations

Abstract Enhancements of primordial curvature fluctuations in single field inflation often involve departures from attractor trajectories in the phase space. We study enhancement/suppression of primordial fluctuations in one of the simplest models with exact background solutions for arbitrary initial conditions: a single field inflationary model with a piecewise exponential potential. We then present close to exact analytical solutions for primordial fluctuations in a general transition between two slow-roll attractors, valid whether the first slow parameter increases or decreases. The main features in the primordial spectrum are determined by the ratio of exponents of the potential. We also discuss the imprint of such features in the induced GW spectrum. Lastly, we apply the δN formalism to discuss non-Gaussianities and the tail of the probability distribution. We find that while non-Gaussianities are at most 𝒪(1) in the case of enhancement, they can be very large in the case of suppression. Our work can be easily generalized to multiple piecewise exponential potentials.

Exact analytical solutions for micropolar magnetohydrodynamic flow: Insights into velocity slip and heat transfer characteristics

In this study, we investigate the laminar steady flow of a micropolar fluid along a permeable flexible surface subject to a uniform magnetic field acting normal to the surface, with wall transpiration and velocity slip at the surface. This analysis yields multiple exact solutions for the stretching/shrinking sheet, offering insights into the complex dynamics of the fluid–surface interaction. In addition, to enhance the realism of our model, we incorporate a power-law temperature distribution, providing a comprehensive understanding of the temperature profile in the context of the micropolar fluid flow. The multiplicity and existence of solutions in special cases are presented for both stretching and shrinking sheets. In particular, a unique solution is found for the stretching sheet in the cases of both suction and injection while dual solutions exist for the shrinking sheet over the range μ>μc in suction, and no solution exist for the injection. For the skin friction, dual solutions appear in the region 0≤a≤ac, but single solution exists for a<0. Furthermore, dual solutions are found for the heat transfer gradient in the regions −∞≤a≤ac and μc≤μ≤∞. Moreover, this research contributes to the broader comprehension of fluid dynamics on flexible surfaces and their intricate thermal behaviors.

Non-reflective Propagation of Kink Waves in Magnetic-Flux Tubes in the Solar Atmosphere

We study the non-reflective propagation of kink waves in inhomogeneous magnetic-flux tubes. We use the thin-tube and zero-beta plasma approximations. The wave equation with the variable velocity is reduced to the Euler–Poisson–Darboux equation. This equation contains one dimensionless parameter. There are two infinite sequences of this parameter, one monotonically increasing and the other monotonically decreasing, when exact analytical solutions for the Euler–Poisson–Darboux equation can be obtained. For the monotonically increasing sequences the Euler–Poisson–Darboux equation becomes the equation describing spherically symmetric waves in multi-dimensional spaces. The general results are applied to kink-wave propagation in coronal magnetic loops. We consider a coronal magnetic loop of a half-circular shape. We find that for a fixed loop height there is a one-parametric family of dependences of the loop cross-sectional radius on the coordinate along the loop corresponding to the non-reflective kink-wave propagation.

Dynamic behavior of pin-free beam with torsional spring and hub at pinned end

This paper described the free vibration of a Euler-Bernoulli beam and discovered the first five dimensionless natural frequencies and mode shapes of motion for the pin-free beam model with a helical spring and rotating mass on the pinned side of the beam. The effect of changing the dimensionless torsional stiffness ratios was investigated, as was the effect of changing the dimensionless rotational inertia torque ratios Concerning the vibration properties of the beam. The mathematical equations that regulate the pin-free beam model were constructed using the exact analytical solution, establishing the Admissible Function and the natural frequency equation for the basic pin-free beam system. The approximate Rayleigh-Ritz approach was then used to determine the first five dimensionless natural frequencies, as well as the movement pattern for the pin-free beam model with a helical spring and a rotating mass on the side of the hinge. The results demonstrated that modifying the beam's natural frequencies is dependent on the ratios of the additional torsional spring parameters as well as the ratios rotational moment of inertia. The threshold system frequencies increase when the spring parameters are the major factor, and decrease when the inertia moment parameters are the main factor. Both play a role in influencing natural frequency to varied degrees. Unifying the final distribution of parameters might lead to optimizing the impact of parameters on the natural frequencies of the system.

Method of exact solutions code verification of a superelastic constitutive model in a commercial finite element solver

The superelastic constitutive model implemented in the commercial finite element code ABAQUS is verified using the method of exact solutions (MES). An analytical solution for uniaxial strain is first developed under monotonic loading. Numerical simulations are then performed, and simulation predictions are compared to the exact analytical solutions. Results reveal the superelasticity model agrees with the analytical solution to within one ten-thousandth of a percent (0.0001%) or less for stress and strain quantities of interest when using displacement-driven boundary conditions. Full derivation of the analytical solution is provided in an Appendix, and simulation input files and post-processing scripts are provided as supplemental material.

Controlling the charge-transfer dynamics of two-level systems around avoided crossings.

Two-level quantum systems are fundamental physical models that continue to attract growing interest due to their crucial role as a building block of quantum technologies. The exact analytical solution of the dynamics of these systems is central to control theory and its applications, such as that to quantum computing. In this study, we reconsider the two-state charge transfer problem by extending and using a methodology developed to study (pseudo)spin systems in quantum electrodynamics contexts. This approach allows us to build a time evolution operator for the charge transfer system and to show new opportunities for the coherent control of the system dynamics, with a particular emphasis on the critical dynamic region around the transition state coordinate, where the avoided crossing of the energy levels occurs. We identify and propose possible experimental implementations of a class of rotations of the charge donor (or acceptor) that endow the electronic coupling matrix element with a time-dependent phase that can be employed to realize controllable coherent dynamics of the system across the avoided level crossing. The analogy of these rotations to reference frame rotations in generalized semiclassical Rabi models is discussed. We also show that the physical rotations in the charge-transfer systems can be performed so as to implement quantum gates relevant to quantum computing. From an exquisitely physical-mathematical viewpoint, our approach brings to light situations in which the time-dependent state of the system can be obtained without resorting to the special functions appearing in the Landau-Zener approach.

Stagnant points and uniform film analysis of Eyring fluid film flow on a vertically upward moving slippery flat plate

In this paper, we analyze how Eyring fluid film behavior on a vertically upward moving slippery flat plate can help predictive models in engineering, notably in coating and lubrication operations. This paper deals with the stagnant points and uniform film analysis of Eyring fluid film flow on a vertically upward moving slippery flat plate. The formulated ordinary differential equation is solved for exact analytic solution. Exact analytic expressions for velocity, flow rate, average velocity, shear stress components, and stagnant points are derived. A highly nonlinear algebraic equation is derived for film thickness. Equation is solved by Newton’s method using Maple code. The thickness of film widens with increasing relaxation time, constant plate velocity, and fluid density, while it decreases with increasing fluid viscosity and constant slip parameter. The analysis delineates that the positions of stagnant points tend to relocate closer the slippery plate as the Stokes number, Deborah number, and constant slip parameter increase. A high Deborah number enhances drainage, causing stagnant points to relocate closer to the slippery plate and the development of a stable elastic layer adjacent to the plate. For stagnant points and fluid film thickness, a comparison between the Eyring fluid and existing studies (EPTT, LPTT, UCM, and Newtonian fluid) is also provided. The validity of this paper with the existing studies is also presented by reducing Eyring fluid model to Newtonian model. The results of this research are significant for a wide range of biofluid applications, including agrochemical uses, paint and surface coating flow behavior, thin films on the cornea and lungs, and chemical and nuclear reactor design.

Quantum dynamics of spin-0 particles in a cosmological space-time

In this paper, our focus is on investigating the impact of cosmological constant on relativistic quantum systems comprising spin-0 scalar particles. Our analysis centers around the Klein-Gordon equation, and we obtain both approximate and exact analytical solutions for spin-0 particles of the quantum system. Afterwards, we explore quantum oscillator fields by considering the Klein-Gordon oscillator within the same space-time characterized by a cosmological constant. We obtain an approximate expression for the energy eigenvalue of the oscillator fields. In fact, the energy spectra in both scenarios are examined and show the influences of the cosmological constant and geometry's topology. Our investigation is situated within the context of a magnetic universe-a four-dimensional cosmological space-time recognized as the Bonnor-Melvin universe.

A comparative analysis of three distinct fractional derivatives for a second grade fluid with heat generation and chemical reaction

This article provides a comparison among the generalized Second Grade fluid flow described by three recently proposed fractional derivatives i.e. Atangana Baleanu fractional derivative in Caputo sense (ABC), Caputo Fabrizio (CF) and Constant Proportional-Caputo hybrid (CPC) fractional derivative. The heat mass transfer is observed during the flow past a vertical porous plate that is accelerated exponentially under the effects of the Magneto hydro dynamics. The effects of the heat generation and exponential heating in the temperature boundary layer and chemical reaction at the concentration boundary layer are also analyzed in this article. The flow model is described by three partial differential equations and the set of non-dimensional PDE’s is transformed into ODE’s by utilization of the integral transform technique (Laplace transform). For the better understanding of the rheological properties of the Second Grade fluid we used the CF, ABC and CPC operators to describe the memory effects. The analytical exact solution of the problem is obtained in the form of G-functions and Mittag Leffler functions. For the physical significance of flow parameters, different parameters are graphed. From this analysis it is concluded that the CPC is the most suitable operator to describe the memory effects.

Extended landslide velocity and analytical drag

The landslide velocity plays a dominant role in estimating the impact force and devastated area. Here, based on Pudasaini and Krautblatter (Earth Surf Dyn 10:165–189, 2022. https://doi.org/10.5194/esurf-10-165-2022), I develop a novel extended landslide velocity model that includes the force induced by the hydraulic pressure gradient, which was neglected by all the existing analytical landslide velocity models. By a rigorous conversion between this force and inertia, which facilitates constructing exact analytical solutions for velocity, I develop two peer systems expecting to produce the same result. However, this contradicts with our conventional wisdom. This raises a legitimate question of whether we should develop some new balance equations such that these phenomena can be better explained naturally. I compare the two velocity models that neglect and include the force induced by the hydraulic pressure gradient. Analytical solutions produced by the two systems are fundamentally different. The new model is comprehensive, elegant, and yet an extraordinary development as it reveals serendipitous circumstance resulting in a pressure–inertia paradox. Surprisingly, the mass first moves upstream for quite a while; then, it winds back and continues accelerating down slope. The difference between the extended and simple solution is significant, and widens strongly as the force associated with the hydraulic pressure gradient increases, demonstrating the importance of this force in the landslide velocity. The viscous drag is an essential dissipative force mechanism and plays an important role in controlling the landslide dynamics. However, no explicit mechanical and analytical model exists to date for this. The careful sagacity of the graceful form of new velocity equation results in a plain, yet mechanically extensive, analytical model for viscous drag, the first of this kind. It contains several physical and geometrical parameters, and evolves dynamically as it varies inversely with the flow depth. A dimensionless drag number is constructed characterizing the drag dynamics. Importance of the drag model is explained. In contrast with the prevailing practices, I have proved that drags are essentially different for the expanding and contracting motions. This is an entirely novel revelation. Drag coefficients are close to the empirical or numerical values often used in practice. But, now, I offer an innovative, physically founded analytical model for the drag that can be instantly applied in mass flow simulations.

Utilization of the Modified Adomian Decomposition Method on the Bagley-Torvik Equation Amidst Dirichlet Boundary Conditions

The Bagley-Torvik equation is an imperative differential equation that considerably arises in various branches of mathematical physics and mechanics. However, very few methods exist for the treatment of the model analytically; in fact, researchers frequently shop for semi-analytical and numerical methods in their studies. Therefore, the main goal of this research is to find theexact analytical solution for the fractional Bagley-Torvik equation fitted with Dirichlet boundary data, as well as a system of fractional Bagley-Torvik equations. Thus, this research aims to show that the modified Adomian decomposition method (MADM) via the proposed two algorithms is a very effective method for treating a class of Bagley-Torvik equations endowed with Dirichletboundary data. Certainly, MADM is a very powerful approach for solving dissimilar functional equations without the need for either linearization, discretization, perturbation, or even unnecessary restraining postulations. Additionally, the method reveals exact analytical solutions whenever obtainable or closed-form series solutions whenever exact solutions are not feasible. Lastly, some illustrative test problems of the governing model are examined to demonstrate the superiority of the proposed algorithms.

An exact analytical solution to unsteady population balance equation with particles coagulation

We consider the final step of a phase transition process in saturated solutions when particles coalescence (Ostwald ripening) and coagulation are possible to happen concurrently. A parametric solution to unsteady Smoluchowski integro-differential equation with constant coagulation kernel is found taking the diffusion mechanism of particles in the space of their volumes into account. The particle-size distribution is given by an exponentially decreasing bell-shaped function describing the experimental data at long times. The theory is extended to describe the cases of different coagulation mechanisms, hydrodynamic flows and external forces.

Analytic Electrically Charged Black Holes in F(R)-ModMax Theory

Abstract Motivated by a new model of nonlinear electrodynamics known as Modified Maxwell (ModMax) theory, an exact analytical solution for black holes is obtained by coupling ModMax nonlinear electrodynamics and F(R) gravity. Then, the effects of the system’s parameters (F(R)-ModMax gravity parameters) on the event horizons are analyzed. The obtained black hole thermodynamic properties in the F(R)-ModMax theory are investigated by extracting their thermodynamic quantities such as Hawking temperature, electric charge, electric potential, entropy, and also total mass. The first law of thermodynamics for the system under study is evaluated. Next, by considering these black holes, the impacts of various parameters on both the local stability and global stability are investigated by examining the heat capacity and the Helmholtz free energy, respectively. Finally, the thermodynamic geometry of the black hole in F(R)-ModMax gravity is investigated by applying the Hendi–Panahiyan–Eslam Panah–Momennia thermodynamic metric (HPEM’s metric).

Estimation of Energy Separation in Tunelling States in Symmetric-Hyperbolicus Double-Well Potential Problem using Filter Method

Double well potential (DWP) plays an important role in quantum physics but its analytical exact solutions are incomplete, except for some limited quasi-exact solvable (QES) and therefore numerical solutions are still required. This paper is aimed to obtain numerical solution of the recently proposed symnetric-hyperbolicus DWP and by using our newly developed filter method. The filter method is able to produce eigen-energies ​​and eigenfunctions those are in accordance with the exact analytical results. Next, we obtain the dependence of the ground state energy and the energy separation between the 2 lowest states on the DWP parameter k. For large k, the 2 lowest energy levels are below the potential barrier so the eigenfunctions penetrate the barrier, so the 2 lowest energy levels can be considered as the result of a single energy split. In this case, we observe that the energy separation is an exponential function k, as opposed to a linear function for smaller k in the non tunelling region. The exponential trend of the numerical results is in good agreement with the theoretical approach and allows one to estimate the 2 lowest energies for a given k. HIGHLIGHTS For small , the two lowest energy levels lie above the barrier potential, where both ground state and the energy separation is a linier function of For large , the two lowest energy levels lie below the barrier potential so that the eigenfunction penetrates the barrier. As a result, the two lowest energy levels can be thought of as splitting energy states. In this case, we observe that the energy separation is an exponential function . Since also depends on , it is possible to estimate the two lowest energy levels The exponential trend of the numerical results for large agrees well with the theoretical approach, shown by the linear relationship between and the classical actions in the classical forbidden regions, GRAPHICAL ABSTRACT

Nonlinear effects of partitioning and diffusion-limiting phenomena on the response and sensitivity of three-layer amperometric biosensors

The influence of the partitioning and diffusion limitations on the response of amperometric biosensors is investigated analytically and computationally using a three-compartment model: an enzyme layer, a diffusion limiting membrane and an outer diffusion layer. The biosensors are modelled by a system of reaction–diffusion equations involving a nonlinear term related to Michaelis–Menten kinetics. Exact steady state analytical solutions for the substrate and reaction product concentrations and the output current are presented for specific cases of first and zero-order reaction rates. The performance of the biosensor is numerically investigated at the transition conditions. Application of different specific types of the interface conditions, perfect contact and partition conditions, at which the steady state biosensor response is the same for both types of the interface conditions is analysed. The effective diffusion coefficients merging two diffusion layers are defined for reducing the three-compartment to the corresponding two-compartment model. The nonlinear and non-monotonic behaviour of the biosensor response is discussed.

Exact analytical solution for deep tunnels in viscoelastic–plastic rock considering the actual loading path

The rock rheology and time process of tunnel construction make the mechanical responses of rock a function of time and are closely related to the loading path, however, these factors are not properly or correctly considered in analytical studies. The analytical study for deep tunnels in rheological rock is performed in this study, strictly taking into account the viscoelastic‒plastic characteristic of surrounding rocks and the actual loading path during excavation and supporting stages.The plane-strain problem of a circular tunnel in infinite plane under hydraulic far-field stress is simplified, with the time-dependent supporting pressure exerted at a specific time to consider the effect of delay installation of yielding support. The solving procedure as well as the time-dependent analytical solution of displacement, stress, and plastic radius in the excavation and supporting stages are presented in detail when the rocks satisfy the Unified strength theory and perfect plasticity. The analytical solutions agree very well with the numerical results, where the time procedure of tunnel excavation and support is simulated in the numerical model. Furthermore, the proposed solution can predict well the long-term deformation of the Shangxinzhai tunnel, which validates its applicability in real engineering. It is found that the tunnel deformation is underestimated if the unloading stage is not correctly considered. Based on the analytical solution, the influence of the supporting time, start time of the yielding stage and viscosity coefficient on the time dependence of tunnel convergence and stresses is investigated.

Invariant approach to the driven Jaynes-Cummings model

We investigate the dynamics of the driven Jaynes-Cummings model, where a two-level atom interacts with a quantized field and both, atom and field, are driven by an external classical field. Via an invariant approach, we are able to transform the corresponding Hamiltonian into the one of the standard Jaynes-Cummings model. Subsequently, the exact analytical solution of the Schrödinger equation for the driven system is obtained and employed to analyze some of its dynamical variables.

Shear Waves in an Elastic Plate with a Hole Resting on a Rough Base

The article is devoted to the analytical and numerical study of the pattern of propagation and attenuation, due to Coulomb friction, of shear waves in an infinite elastic thin plate with a circular orifice of radius r0 lying on a rough base. Considering the friction forces and their influence on the sample of wave propagation in extended rods or thin plates is important for calculating the stress–strain state in them and the size of the area of motion. An exact analytical solution of a nonlinear boundary value problem for tangential stresses and velocities is obtained in quadratures by the Laplace transform, with respect to time. It turned out that the complete exhaustion of the wave front of a strong rupture occurs at a finite distance r* from the center of the orifice, and an elementary formula is given for this distance (the case of tangential shock stresses suddenly applied to the orifice boundary is considered). For various ratios of the magnitude of the limiting friction force to the amplitude of the applied load, the stopping (trailing) wave fronts are calculated. After passing them, a state of static equilibrium between the elastic and friction forces with a nonlinear distribution of residual stresses is established in the region r0≤r≤r*. For the first time, a precise analytical solution was obtained for the boundary value problem of the propagation of elastic shear waves in an infinite isotropic space with a cylindrical cavity, when a tangential shock load is set on its surface.