Exact Analytical Solution Research Articles

Features of the surface wave propagation along the interface between the hyperbolic graded-index layer and nonlinear medium with a step change in the dielectric constant

A new exact analytical solution describing the transverse profile of the surface wave in the case of contact with nonlinear medium with the step-wise nonlinearity and the hyperbolic index profile and describing new features of the light localization near such an interface are found. The formation of the near interface cladding layer with the dielectric constant different from its value in the rest nonlinear medium is described. It is found that light localization can be effectively controlled by the parameters of the hyperbolic profile such as the interface dielectric constant and the effective refractive index. The depth of field penetration decreases inside the hyperbolic graded-index layer and it increases in the nonlinear medium with an increase in the effective refractive index. The depth of field penetration in the hyperbolic graded-index layer increases and it decreases in the nonlinear medium with an increase in the value of the dielectric constant at the medium interface. Thickness of the formed cladding layer decreases with increase in the interface dielectric constant and it increases with increase in the effective refractive index.

Quantum state engineering in a five-state chainwise system by generalized coincident pulse technique.

In this paper, an exact analytical solution is presented for achieving coherent population transfer and creating arbitrary coherent superposition states in a five-state chainwise system by a train of coincident pulses. We show that the solution of a five-state chainwise system can be reduced to an equivalent three-state Λ-type one with the simplest resonant coupling under the assumption of adiabatic elimination together with a requirement of the relation among the four coincident pulses. In this method, the four coincident pulses at each step all have the same time dependence, but with specific magnitudes. The results show that, by using a train of appropriately coincident pulses, this technique not only enables complete population transfer, but also creates any desired coherent superposition between the initial and final states, while the population in all intermediate states is effectively suppressed. Furthermore, this technique can also exhibit a one-way population transfer behavior. The results are of potential interest in applications where high-fidelity multi-state quantum control is essential, e.g., quantum information, atom optics, formation of ultracold molecules, cavity QED, nuclear coherent population transfer, and light transfer in waveguide arrays.

The role of conjugacy in the dynamics of time of arrival operators

The construction of time of arrival (TOA) operators canonically conjugate to the system Hamiltonian entails finding the solution of a specific second-order partial differential equation called the time kernel equation (TKE). In this paper, we provide an exact analytic solution of the TKE for a special class of potentials satisfying a specific separability condition. The solution enables us to investigate the time evolution of the eigenfunctions of the conjugacy-preserving TOA operators (CPTOA) and show that they exhibit unitary arrival at the intended arrival point at a time equal to their corresponding eigenvalues. We also compare the dynamics between the TOA operators constructed by quantization and those independent of quantization for specific interaction potentials. We find that the CPTOA operator possesses smoother and sharper unitary dynamics over the Weyl-quantized one within numerical accuracy.

A High-Order Numerical Method Based on a Spatial Compact Exponential Scheme for Solving the Time-Fractional Black–Scholes Model

This paper investigates a high-order numerical method based on a spatial compact exponential scheme for solving the time-fractional Black–Scholes model. Firstly, the original time-fractional Black–Scholes model is converted into an equivalent time-fractional advection–diffusion reaction model by means of a variable transformation technique. Secondly, a novel high-order numerical method is constructed with (2−α) accuracy in time and fourth-order accuracy in space based on a spatial compact exponential scheme, where α is the fractional derivative. The uniqueness of solvability of the derived numerical method is rigorously discussed. Thirdly, the unconditional stability and convergence of the derived numerical method are rigorously analyzed using the Fourier analysis technique. Finally, numerical examples are presented to test the effectiveness of the derived numerical method. The proposed numerical method is also applied to solve the time-fractional Black–Scholes model, whose exact analytical solution is unknown; numerical results are illustrated graphically.

Group foliations of the β-plane barotropic vorticity equation

Despite the large number of publications on symmetry analysis of the barotropic vorticity equation on the β-plane, its group foliations have not been considered previously. The present publication aims to address this shortcoming. Group foliations are constructed for the equation, and based on them, invariant solutions are derived, some of which generalize previously known exact analytical solutions. There is also a discussion of the pros and cons of the group foliation approach including consideration of some numerical issues.

Analytical Framework for Tension Characterization in Submerged Anchor Cables via Nonlinear In-Plane Free Vibrations

Submerged tensioned anchor cables (STACs) are pivotal components utilized extensively for anchoring and supporting offshore floating structures. Unlike tensioned cables in air, STACs exhibit distinctive nonlinear damping characteristics. Although existing studies on the free vibration response and tension identification of STACs often employ conventional Galerkin and average methods, the effect of the quadratic damping coefficient (QDC) on the vibration frequency remains unquantified. This paper re-examines the effect of bending stiffness on the static equilibrium configuration of STACs, and establishes the in-plane transverse free motion equations considering bending stiffness, sag, and hydrodynamic force. By introducing the bending stiffness influence coefficient and the Irvine parameter, the exact analytical solutions of symmetric and antisymmetric frequencies and modal shapes of STACs are derived. An improved Galerkin method is proposed to discretize the nonlinear free motion equations ensuring the accuracy and applicability of the analytical results. Additionally, this paper presents an analytical solution for the nonlinear free vibration response of the STACs using the improved averaging method, along with improved frequency formulas and tension identification methods considering the QDC. Through a case study, it is demonstrated that the improved methods introduced in this paper offer higher accuracy and wider applicability compared to the conventional approaches. These findings provide theoretical guidance and reference for the precise dynamic analysis, monitoring, and evaluation of marine anchor cable structures.

Exact solution of the nonlinear boson diffusion equation for gluon scattering

An exact analytical solution of the nonlinear boson diffusion equation is presented. It accounts for the time evolution toward the Bose–Einstein equilibrium distribution through inelastic and elastic collisions in the case of constant transport coefficients. As a currently interesting application, gluon scattering in relativistic heavy-ion collisions is investigated. An estimate of the time-dependent gluon-condensate formation in overoccupied systems through number-conserving elastic scatterings in Pb–Pb collisions at relativistic energies is given.

Exact analytical solution of the Flory-Huggins model and extensions to multicomponent systems.

The Flory-Huggins theory describes the phase separation of solutions containing polymers. Although it finds widespread application from polymer physics to materials science to biology, the concentrations that coexist in separate phases at equilibrium have not been determined analytically, and numerical techniques are required that restrict the theory's ease of application. In this work, we derive an implicit analytical solution to the Flory-Huggins theory of one polymer in a solvent by applying a procedure that we call the implicit substitution method. While the solutions are implicit and in the form of composite variables, they can be mapped explicitly to a phase diagram in composition space. We apply the same formalism to multicomponent polymeric systems, where we find analytical solutions for polydisperse mixtures of polymers of one type. Finally, while complete analytical solutions are not possible for arbitrary mixtures, we propose computationally efficient strategies to map out coexistence curves for systems with many components of different polymer types.

Rheological effects in peristaltic flow of Prandtl fluid through elliptical duct: A comprehensive analysis

AbstractThis research venture comprehends a theoretical examination of non‐Newtonian fluid flowing peristaltic via an elliptical channel. Furthermore, the Prandtl fluid method for this elliptic duct problem is thoroughly considered. This mathematical inquiry adopts a non‐Newtonian Prandtl fluid model. A polynomial methodology is used to analyze partial differential equations that appear in nondimensional form and deliver an exact analytical solution for the temperature and velocity profile. This study is the first to utilize a novel order polynomial of degree eight having eleven constants to precisely solve the temperature equation for Prandtl fluid flow via an elliptic domain. A comprehensive graphical analysis is also provided to understand the mathematical conclusions fully. The graphs of the velocity profiles clearly show that the non‐Newtonian effects are more potent along the minor axis of the elliptical duct. The streamlined graphs accentuating the trapping phenomenon show specific closed contours close to the boundary wall of the peristaltic duct.

Torsional buckling of a tape-spring: Review and renew

Tape-springs – straight strips with transversely curved cross-sections – undergo a flattening of the cross-section upon bending, and accompanying torsional buckling in equal-sense bending which then “recedes” at greater deformation. The exact analytical solution for tape-spring bending was solved by Mansfield; however, the resulting expression for the twisting curvature at the onset and recession of torsional buckling cannot be solved in closed-form. Approximate closed-form expressions are therefore sought: firstly, by utilising Taylor series approximations of Mansfield’s extensive functions; and secondly, by assuming a uniform transverse curvature, thereby rendering the problem algebraic in nature, and deriving new governing expressions. Further approximations assume the cross-section does not deform upon bending, and that the tape-spring remains a developable surface in the inextensible limit of zero thickness. All approximate solutions are shown to agree closely with Mansfield’s solution at small curvatures, with the uniform curvature approximation also capturing the recession of torsional buckling to a reasonable degree of accuracy. The accuracy of the uniform curvature approximation is related to a pertinent dimensionless parameter, which governs the size of a boundary layer in Mansfield’s exact solution: as this parameter tends to infinity, the uniform curvature approximation and Mansfield’s exact solution are shown to converge; as it tends to zero, all solutions are shown to converge on the inextensible limit.

Exploring Wave Interactions and Conserved Quantities of KdV–Caudrey–Dodd–Gibbon Equation Using Lie Theory

This study introduces the KdV–Caudrey–Dodd–Gibbon (KdV-CDGE) equation to describe long water waves, acoustic waves, plasma waves, and nonlinear optics. Employing a generalized new auxiliary equation scheme, we derive exact analytical wave solutions, revealing rational, exponential, trigonometric, and hyperbolic trigonometric structures. The model also produces periodic, dark, bright, singular, and other soliton wave profiles. We compute classical and translational symmetries to develop abelian algebra, and visualize our results using selected parameters.

On a similarity solution for lock-release gravity currents affected by slope, drag and entrainment

We consider the long-time propagation of a Boussinesq inertia–buoyancy (large-Reynolds- number) gravity current released from a lock over a downslope of angle $\gamma$ , affected by entrainment and drag. We show that the shallow-water (depth-averaged) equations with a Benjamin-type front-jump condition admit a similarity solution $x_N(t) = K t^{2/3}$ while $h, \phi, u$ change like $t$ to the power of $2/3, -4/3, -1/3$ , respectively; here $x_N, h, \phi, u$ and $t$ are the position of the nose (distance from backwall), thickness, concentration of dense fluid, velocity and time, respectively, and K is a constant. Assuming that $\gamma$ and the coefficients of entrainment and drag are constant, we derive an analytical exact solution for the similarity profiles and show that $K \propto (\tan \gamma )^{1/3}$ ; the driving of the slope is balanced by entrainment and/or drag. The predicted $t^{2/3}$ propagation is in agreement with previously published experimental data but a conclusive quantitative assessment of the present theory cannot be performed due to various uncertainties (discussed in the paper) that must be resolved by future work.

Adaptive multi-patch isogeometric analysis with truncated hierarchical B-splines in isotropic/orthotropic media

We present an adaptive multi-patch isogeometric analysis on the basis of truncated hierarchical B-splines (THB-splines) for solving two-dimensional complex isotropic/orthotropic elasticity. The THB-splines have local refinement property and their basis functions exhibit linear independence, so these features are highly applicable for adaptive isogeometric analysis. Guided by a posterior error estimator based on stress recovery, the adaptive algorithm is utilized in isogeometric analysis. In order to further extend the proposed method to solve complex geometry problems, the multi-patch technique is adopted to achieve exact modeling with Nitsche’s method as a multi-patch coupling approach. An isotropic numerical example with exact analytical solutions and three orthotropic numerical examples are presented to verify the effectiveness and accuracy of the developed method. Numerical solutions show that the developed adaptive isogeometric analysis method has high computational efficiency.

Element-based finite volume method applied to autogenous bead-on-plate GTAW process using the enthalpy energy equation

The welding process is the most used technique for metal joining. Understanding the temperature variation along the welded part during the process can prevent the appearance of failures. The experimental investigation process is quite time-consuming and costly. Therefore, numerical simulation processes based on the finite difference method, finite element method, finite volume method, or meshless Element-Free Galerkin (EFG) methods are important tools to optimize the welding process. The main goal of the present study is to show the feasibility of the Element-based Finite-Volume Method (EbFVM) approach for actual engineering applications. To solve the unsteady 2D and 3D thermal energy equation using enthalpy as an independent variable, an in-house Fortran code has been developed based on the EbFVM approach in conjunction with unstructured and structured meshes. The numerical simulations, with four types of different heat sources, were performed for applications of real welding processes with variations in density and enthalpy as a function of temperature. The results are presented in terms of thermal cycles and temperature fields. Furthermore, the developed code was confronted against experimental works from the literature, simulated and lab-controlled experiments with AISI 409 ferritic workpieces, and exact analytical solutions with thermocouples fixed in different positions. In general, the numerical results from the current investigation are in close agreement with the results from the literature and the experimental results performed by the authors. The numerical results also highlighted the differences between the 2D and 3D models for thermal cycles near the bead weld.

Analytical solutions and physical interpretation of a predator–prey system with Allee effect using fractional derivative operators

This study investigates the dynamics of a predator–prey system by incorporating the Allee effect. The exact analytical solutions of the system are explored using the generalized exponential function method and the Bäcklund transformation approach. The system is discussed using Conformable, M-truncated, and Beta time fractional derivative operators to provide a physical interpretation. The effects of these derivative operators on the solutions are demonstrated through various 2D and 3D graphical simulations. Furthermore, the impacts of physical parameters on the model are analysed, providing valuable insights into the dynamics of Predator–prey systems with the Allee effect. Our obtained solutions differ from those in the literature, owing to the methodologies employed and the choice of differential operators utilized. Nevertheless, we believe that the comparisons presented for the differential operators are effective and beneficial.

Thermally developing streaming potential-mediated pressure-driven flow of Phan–Thien–Tanner fluid in a microchannel

In this study, we have conducted a semi-analytical investigation into the streaming potential-mediated pressure-driven flow of hydrodynamically fully developed and thermally developing viscoelastic fluid in a parallel plate microchannel. We have utilized a simplified Phan–Thien–Tanner model to describe the rheology of the viscoelastic fluid. Our approach delved into the full Poisson–Boltzmann equation, deriving exact analytical solutions for the electrostatic potential distribution and velocity profile. Furthermore, we concurrently derived semi-analytical solutions for the temperature distribution and Nusselt number, accounting for the effects of heat generation from viscous dissipation and Joule heating in thermally developing flows. We have demonstrated that an increase in the degree of surface charge triggers the streaming potential field, while the volumetric flow rate escalates with the viscoelastic parameter εWik¯2. Moreover, we have observed that the magnitude of the dimensionless temperature decreases with increasing values of the effective Joule heating parameter Speff. Our analysis reveals that the streaming potential effect hampers fluid flow, resulting in an increase in the bulk fluid temperature and consequently reducing the heat transfer rate. We observe that the magnitude of the Nusselt number decreases with increasing Speff. The entropy generation analysis reveals that increasing the Peclet number amplifies flow and temperature gradients, leading to higher fluid irreversibility in microchannels. The Bejan number experiences a significant decrease across the channel, reaching its minimum at a specific axial location before stabilizing further downstream. We find that heat transfer irreversibility predominantly influences system irreversibility, except for the Brinkmann number Br = 0.01, where convective heat transfer dominates at the entrance region, transitioning to friction losses beyond the thermal entry zone.

Non-Hermitian transport in Glauber-Fock optical lattices

The effect of a non-unitary transformation on an initial Hermitian operator is studied. The initial (Hermitian) optical system is a Glauber-Fock optical lattice. The resulting non-Hermitian Hamiltonian models an anisotropic (Glauber-Fock) waveguide array of the Hatano-Nelson-type. Several cases are analyzed and exact analytical solutions for both the Hermitian and non-Hermitian Schrödinger problems are given, as they are simply connected. Indeed, such transformation can be regarded as a non-unitary Supersymmetric transformation and the resulting non-Hermitian Hamiltonian can be considered as representing an open system that interchanges energy with the environment.

Efficient Analysis of Unlined Elliptic Tunnels: An Approximate Method for Dynamic Response to Incident Plane SH Waves

ABSTRACT Dynamic analysis of underground structures, such as tunnels, is crucial to ensure their safety and stability during seismic loading. This paper proposes a more general method of large elliptic arc assumption-based wave function expansion, which is attributed to the important role of the large circular arc assumption-based wave function expansion method in the elastic wave scattering problem. The study is conducted within elliptic coordinate systems, employing the Mathieu function and Mathieu function addition theorem. The applicability and limitations of this method are analysed by comparing it with the existing exact analytical solutions obtained through the mirror-based wave function expansion method.

Exact solution to band structures of one-dimensional viscoelastic functionally graded phononic crystals

In this article, longitudinal wave (P-wave) band structures of functionally graded viscoelastic phononic crystals (PnCs) with exponential distribution properties are analytically investigated via an exact method. Differing from the laminated model, displacement and stress vectors of normal incident P-wave are obtained exactly by utilizing a stress-transform technique and the state space method. According to the continuous condition and Bloch theory, the eigenvalue equation of band structure is derived by applying transfer matrix method. The storage modulus is considered to be frequency-dependent while the loss modulus is neglected for the consideration of components viscoelasticity. Dispersion curves, obtained from the calculation of eigenvalue equation, are validated by the results of exist laminated model and transmission spectra which are also present to reveal the propagation characteristic of normal incident P-wave. Numerical examples indicate that band width and center frequencies of band gaps can be effectively altered by changing magnitude of exponential function, ratio between initial modulus and final modulus. Other new band gaps are opened as well as wider band width are achieved with the increase of gradient index к . The exact analytical solution presented in this paper can be the benchmark for those approximation solutions.

Analytical solution to some boundary value problems of elasticity in bipolar coordinates

Analytical solutions of two-dimensional statics problems of elasticity are presented in bipolar coordinates for homogeneous isotropic bodies bounded by the coordinate lines of a bipolar coordinate system. In particular, various boundary problems for an eccentric circular ring, a half-plane with circular holes, and others are considered. The equilibrium equations and Hooke’s law are expressed using bipolar coordinates. This paper does not address the static equilibrium requirement of the external load at each circular boundary of the study area. This requirement, which significantly limits the range of problems that can be solved, typically appears in papers dealing with the aforementioned problems. In addition, the proposed method for obtaining an exact (analytical) solution is much simpler compared to the traditional approach. The exact solutions are derived using the method of separation of variables. By utilizing MATLAB software, the numerical results of some boundary value problems for an eccentric semi-ring are obtained, and the corresponding diagrams are presented.