Explicit Analytical Solutions Research Articles

Wavelet Based Solutions to the Poisson and the Helmholtz Equations on the n-Dimensional Unit Sphere

We present a method of solving partial differential equations on the n-dimensional unit sphere using methods based on the continuous wavelet transform derived from approximate identities. We give an explicit analytical solution to the Poisson equation and to the Helmholtz equations. For the first one and for some special values of the parameter in the latter one, we derive a closed formula for the generalized Green function.

Model‐free predictive control for a class of asymmetric nonlinear systems

AbstractA model‐free predictive control (MFPC) method is proposed for a kind of discrete nonlinear asymmetric systems (NASs) whose models are undefined. According to the dynamic characteristics of the asymmetric system, the compact format dynamic linearization (CFDL) technology and the improved projection algorithm are adopted to set up models for the controlled NAS. Then, based on the estimation models, the explicit analytical solution of the controller is obtained by solving the quadratic function of finite‐time domain rolling optimization. Finally, a switching law is designed to make the system switch reasonably between the positive and negative models, thus ensuring the stability of the whole system. The effectiveness of the proposed method is testified by a simulation example.

Dipole–Monopole Crossover and Chargeless Half-Mode in an Integrable Exciton–Phonon Nonlinear Dynamical System on a Regular One-Dimensional Lattice

A new form of the integrable nonlinear exciton–phonon dynamical system characterized by two physically independent parameters is suggested. The system is settled along an infinite one-dimensional regular lattice, and it admits the semi-discrete Lax representation in terms of 3 × 3 auxiliary spectral and evolution matrices. The explicit analytic four-component solution to the system’s dynamical equations found by means of the Darboux–Backlund dressing technique turns out to be of broken PT-symmetry. Each component of the solution consists of two nonlinearly superposed traveling waves that inspires the dipole–monopole crossover for the equal values of two physically distinct spatial scaling parameters of the nonlinear wave packet. The phenomenon of the dipole–monopole alternative for the spatial distribution of pseudoexcitons is shown to initiate the partial splitting between the pseudoexcitonic and vibrational subsystems at the threshold point manifested by the complete elimination of one pseudoexcitonic component and the conversion of another pseudoexcitonic component into the pseudoexcitonic chargeless half-mode.

Thermoelectric Field Analysis of Trapezoidal Thermoelectric Generator Based on the Explicit Analytical Solution of Annular Thermoelectric Generator

The geometrical configuration is one of the main factors that affect the thermoelectric performance of a device. Research on the trapezoidal thermoelectric generator (TTEG) with varied cross section is mainly based on finite element simulation and experiment. In this paper, an explicit analytical solution of the maximum output power of annular thermoelectric generators (ATEG) is proposed, which has been proved to have high accuracy. Then, the maximum output power between ATEG and TTEG is compared. Results show that, for the appropriate geometric parameter δ, the relative error of maximum output power between explicit analytical ATEG and the simulated solution of TTEG can reach the order of 10−3. When the hot end is at the a side, the high temperature and θ is 510 K and 10°, respectively. For Bi2Te3 material and PbTe material, the relative error of maximum output power between the explicit analytical and simulated solution is 0.0261% and 0.074%, respectively. Under suitable working conditions, the explicit analytical results of ATEG can provide some reference for the performance optimization of TTEG.

Dipole–monopole alternative as the precursor of pseudo-excitonic chargeless half-mode in an integrable nonlinear exciton–phonon system on a regular one-dimensional lattice

The new form of an integrable exciton–phonon nonlinear dynamical system distinguished by the two physical parameters is suggested. The system is settled along an infinite one-dimensional regular lattice and it admits the semi-discrete Lax representation in terms of 3 × 3 auxiliary spectral and evolution matrices. The explicit analytical four-component solution to the system’s dynamical equations found by means of Darboux–Bäcklund dressing technique turns out to be of broken PT-symmetry giving rise to the crossover between the monopole and dipole regimes of system’s dynamics. The crossover effect is proved to be inseparable from the mutual influence between the interacting subsystems in the form of specific nonlinear superposition of two qualitatively distinct types of traveling waves characterized by the two spatial scales and by the two velocities of two physically distinct origins. The threshold value of localization parameter separating the monopole and dipole dynamical regimes is strictly established in terms of the basic physical parameters. The phenomenon of dipole–monopole alternative for the spatial distribution of pseudo-excitons is shown to initiate the partial splitting between the pseudo-excitonic and vibrational subsystems in the so-called threshold dynamical regime specified by the threshold value of localization parameter. This partial splitting is manifested by the complete elimination of one pseudo-excitonic component accompanied by the actual conversion of another pseudo-excitonic component into the pseudo-excitonic chargeless half-mode. Relying upon the inherent PT-symmetry of system’s dynamical equations the four-component symmetry broken PT-conjugated counterpart solution is also analytically recovered. The integrable nonlinear exciton–phonon system under study is expected to be useful in modeling the nonlinear dynamical properties of PT-symmetric metamaterials. The critical overviews of several relevant and several irrelevant model systems are given to rebuff certain misleading but dogmatically accepted notions and statements circulating in scientific literature.

Intrinsic decoherence dynamics in the three-coupled harmonic oscillators interaction

A system of three-coupled quantized fields is studied under the intrinsic decoherence scheme given by the Milburn equation. Taking the closed analytical form of this relation, without second-order approximation, we arrive at an explicit analytical solution for the wavefunction dynamics. We show the decaying behavior in the number of modes of each quantized field for suitable initial conditions. One of the modes goes coherently driven, and the others are vacuum states. We discuss the redistribution of the excitations.

Static and free vibration analysis of functionally graded annular plates using stress-driven nonlocal theory

This study presents analytical and numerical solutions of the bending and the free vibration analysis of a functionally graded annular nanoplate based on the stress-driven nonlocal theory. The nonlocal equation is obtained using the classical plate theory; the power law distribution is assumed to model changes in material properties throughout the thickness. The governing differential equation is analytically solved for different types of edge supports. The explicit analytical solution is obtained in terms of special functions, namely, hypergeometric and Meijer functions. Furthermore, based on the Galerkin technique, a finite element method is introduced to figure out the nanoplate’s flexural deformation and natural frequencies. The analytical and numerical solutions are compared with those available in the literature. The size effects in the nonlocal theory and FGM’s gradient properties have been discussed, considering clamped and simply supported boundary conditions. According to the stress-driven nonlocal model, size-dependent flexural responses of the plate demonstrate stiffening behavior with increasing nonlocal parameters. Which, in general, leads to a decrease in the value of transversal deformation and an increase in natural frequencies when compared to the plate’s local solutions.

Tidally Forced Planetary Waves in the Tachocline of Solar-like Stars

Can atmospheric waves in planet-hosting solar-like stars substantially resonate to tidal forcing, perhaps at a level of impacting the space weather or even being dynamo-relevant? In particular, low-frequency Rossby waves, which have been detected in the solar near-surface layers, are predestined to respond to sunspot cycle-scale perturbations. In this paper, we seek to address these questions as we formulate a forced wave model for the tachocline layer, which is widely considered as the birthplace of several magnetohydrodynamic planetary waves, i.e., Rossby, inertia-gravity (Poincaré), Kelvin, Alfvén, and gravity waves. The tachocline is modeled as a shallow plasma atmosphere with an effective free surface on top that we describe within the Cartesian β-plane approximation. As a novelty to former studies, we equip the governing equations with a conservative tidal potential and a linear friction law to account for viscous dissipation. We combine the linearized governing equations into one decoupled wave equation, which facilitates an easily approachable analysis. Analytical results are presented and discussed within several interesting free, damped, and forced wave limits for both midlatitude and equatorially trapped waves. For the idealized case of a single tide-generating body following a circular orbit, we derive an explicit analytic solution that we apply to our Sun for estimating leading-order responses to Jupiter. Our analysis reveals that Rossby waves resonating to low-frequency perturbations can potentially reach considerable velocity amplitudes on the order of 101–102 cm s−1, which, however, strongly rely on the yet unknown frictional damping parameter.

Analytical study on the seepage field of different drainage and pressure relief options for tunnels in high water-rich areas

The arrangement of tunnel drainage measures is closely related to the tunnel seepage field and water pressure outside the lining, especially for tunnels in water-rich areas with high water tables. A reasonable drainage hole that actively controls drainage pressure relief is an important measure that ensures the safety of the surrounding rock and lining structure. Since no analytical studies of the seepage field of a tunnel with an arbitrary number of drainage holes have been conducted, this paper derives an explicit analytical solution for the two-dimensional steady-state seepage field of a tunnel with an arbitrary number of drainage holes by means of conformal mapping and the separation of variables method and solves for the unknowns in the analytical solution using orthogonality. The correctness and superiority of the analytical solution proposed in this paper are verified by comparing it with numerical software and existing pipe leakage solutions from various aspects. The influence of the drainage hole arrangement scheme, the number of drainage holes and the diameter of the drainage holes on the seepage field was explored through extensive parametric analysis, and the results show that a uniform arrangement of drainage holes in the tunnel lining centred on the bottom of the arch is the best solution for reducing the water pressure in the lining. The number of drainage holes and the diameter of the holes have a significant effect on the water pressure and seepage volume of the tunnel lining, and based on the results of the analysis, design recommendations for the drainage hole arrangement are provided.

New Explicit and Approximate Solutions of the Newton-Schrödinger System

In this paper, we consider the Newton-Schrödinger system ∇2Ψ=γΦ+a(x)Ψ,∇2Φ=∣Ψ∣2,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\left\\{ \\begin{array}{rcl} \ abla ^{2} \\Psi & =& \\left( \\gamma \\Phi +a(x)\\right) \\Psi ,\\\\ \ abla ^{2} \\Phi & =& \\mid \\Psi \\mid ^{2}, \\end{array} \\right. \\end{aligned}$$\\end{document}which arises in certain quantum transport and chemistry problems. Explicit analytic solutions, which contain an auxiliary parameter, are obtained. An existence and uniqueness theorem to this nonlinear system subject to the boundary conditions is proved. Also, we introduce approximate solutions to the modified Newton-Schrödinger system in the case of spherically-symmetric stationary and time-independence by the Adomian decomposition method.

Incommensurate multi‐term fractional differential equations with variable coefficients with respect to functions

Linear fractional differential equations with variable coefficients and various types of fractional derivatives can be solved explicitly, using either fixed‐point theorems and successive approximations or Green's functions. In this paper, we consider a multi‐term fractional differential equation with continuous variable coefficients and incommensurate fractional orders, and we construct an explicit solution by a direct method of successive approximations, which is made rigorous by checking the solution function explicitly via substitution. The results are obtained for the most general initial value problem of an inhomogeneous equation with inhomogeneous initial conditions, the first time that an explicit analytical solution has been found for this general problem. Special cases are provided to illustrate the main results, and connections are discussed with previous results in the literature.

Numerical approach to an age-structured Lotka-Volterra model.

We study the impact of an age-dependent interaction in a structured predator-prey model. We present two approaches, the PDE (partial differential equation) and the renewal equation, highlighting the advantages of each one. We develop efficient numerical methods to compute the (un)stability of steady-states and the time-evolution of the interacting populations, in the form of oscillating orbits in the plane of prey birth-rate and predator population size. The asymptotic behavior when species interaction does not depend on age is completely determined through the age-profile and a predator-prey limit system of ODEs (ordinary differential equations). The appearance of a Hopf bifurcation is shown for a biologically meaningful age-dependent interaction, where the system transitions from a stable coexistence equilibrium to a collection of periodic orbits around it, and eventually to a stable limit cycle (isolated periodic orbit). Several explicit analytical solutions are used to test the accuracy of the proposed computational methods.

The step-wise construction of solitary solutions to Riccati equations with diffusive coupling

<abstract><p>A novel scheme based on the generalized differential operator and computer algebra was used to construct solitary solutions to a system of Riccati differential equations with diffusive coupling. The presented approach yields necessary and sufficient existence conditions of solitary solutions with respect to the system parameters. The proposed stepwise approach enabled the derivation of the explicit analytic solution, which could not be derived using direct balancing techniques due to the complexity of algebraic relationships. Computational experiments were used to demonstrate the efficacy of proposed scheme.</p></abstract>

Analytical studies on complete, lag and anticipation synchronization in cascaded circuits with numerical and experimental confirmation

We report in this paper, the explicit analytical solutions developed for studying the different types of synchronization behaviors such as complete, lag and anticipation synchronization observed in cascaded circuits without any delay factors in the system. The Pecora–Carroll method of synchronization of unidirectionally coupled systems, which has not been earlier studied through explicit analytical solutions is reported in this paper. The emergence of different synchronization phenomena and the nature of evolution of the synchronization behavior has been discussed. Further, it has also been confirmed that the evolution of anticipation from lag through complete synchronization along with the attenuation, amplification of the subsystem signal occurs as a generalized phenomenon irrespective of the nature of the chaotic attractor state of the drive and the system parameters. The Hamilton energy of the cascaded system for different types of synchronization is reported. The numerical simulation and electronic circuit experimental results are presented to confirm the analytical results.

Explicit Analytic Solutions for the Subsurface Stress Field in Single Plane Contacts of Elastically Similar Truncated Cylinders or Wedges

As has been pointed out recently, a possible solution strategy to the wear–fatigue dilemma in fretting, operating on the level of contact mechanics and profile geometries, can be the introduction of “soft” sharp edges to the contact profiles, for example, by truncating an originally smooth profile. In that regard, analysis of possible mechanical failure of a structure, due to the contact interaction, requires the knowledge of the full subsurface stress state resulting from the contact loading. In the present manuscript, a closed-form exact solution for the subsurface stress state is given for the frictional contact of elastically similar truncated cylinders or wedges, within the framework of the half-plane approximation and a local-global Amontons–Coulomb friction law. Moreover, a fast and robust semi-analytical method, based on the appropriate superposition of solutions for parabolic contact, is proposed for the determination of the subsurface stress fields in frictional plane contacts with more complex profile geometries, and compared with the exact solution. Based on the analytical solution, periodic tangential loading of a truncated cylinder is considered in detail, and important scalar characteristics of the stress state, like the von-Mises equivalent stress, maximum shear stress, and the largest principal stress, are determined. Positive (i.e., tensile) principal stresses only exist in the vicinity of the contact edge, away from the pressure singularity at the edge of the profile, and away from the maxima of the von-Mises equivalent stress, or the maximum shear stress. Therefore, the fretting contact should not be prone to fatigue crack initiation.

A Variety of New Explicit Analytical Soliton Solutions of q-Deformed Sinh-Gordon in (2+1) Dimensions

In this paper, the (2+1)-dimensional q-deformed Sinh-Gordon model has been investigated via (G′G,1G)-expansion and Sine-Gordon-expansion methods. These techniques successfully retrieve trigonometric as well as hyperbolic solutions, along necessary restricted conditions applied on parameters. In addition to these solutions, dark solitons and complexiton solutions have also been obtained. The proposed equation expands the possibilities for modeling physical systems in which symmetry is broken. The obtained solutions are graphically illustrated. A Painlevé analysis for the proposed model has also been discussed in this paper.

An Analysis of the Fractional-Order Option Pricing Problem for Two Assets by the Generalized Laplace Variational Iteration Approach

An option is the right to buy or sell a good at a predetermined price in the future. For customers or financial companies, knowing an option’s pricing is crucial. It is well recognized that the Black–Scholes model is an effective tool for estimating the cost of an option. The Black–Scholes equation has an explicit analytical solution known as the Black–Scholes formula. In some cases, such as the fractional-order Black–Scholes equation, there is no closed form expression for the modified Black–Scholes equation. This article shows how to find the approximate analytic solutions for the two-dimensional fractional-order Black–Scholes equation based on the generalized Riemann–Liouville fractional derivative. The generalized Laplace variational iteration method, which incorporates the generalized Laplace transform with the variational iteration method, is the methodology used to discover the approximate analytic solutions to such an equation. The expression of the two-parameter Mittag–Leffler function represents the problem’s approximate analytical solution. Numerical investigations demonstrate that the proposed scheme is accurate and extremely effective for the two-dimensional fractional-order Black–Scholes Equation in the perspective of the generalized Riemann–Liouville fractional derivative. This guarantees that the generalized Laplace variational iteration method is one of the effective approaches for discovering approximate analytic solutions to fractional-order differential equations.

Analysis of Gravity‐Driven Infiltration With the Development of a Saturated Region

AbstractUnderstanding the controls on the infiltration of precipitation into soil is an important problem in hydrology and its representation in heterogeneous media is still challenging. Here we investigate the reduction of gravity‐driven infiltration by the development of a saturated region due to the downward decrease in porosity and/or hydraulic conductivity. The formation of a saturated region chokes the flow and leads to a rising perched water table that causes ponding even when rainfall intensity is lower than the surface infiltration capacity. Mathematically this problem is interesting, because its governing partial differential equation switches from hyperbolic in the unsaturated region to elliptic in the saturated region. To analyze this coupled unsaturated‐saturated flow we develop an extended kinematic wave analysis in the limit of no capillary forces. This theory provides a general framework to solve gravity‐dominated infiltration problems for arbitrary downward decrease in porosity and/or conductivity. We apply the framework to three soil profiles (two‐layer, exponential and power‐law decay with depth) and develop (semi‐) analytic solutions for evolution of the water saturation. For the case of a two‐layer soil the saturated flux, and therefore the front speeds, are constant which allows explicit analytic solutions that agree well with Hydrus‐1D. All solutions show excellent agreement with our numerical solutions of the governing equations in the limit of no capillary forces. Similarly, our solutions compare well with experimental data for infiltration into a multi‐layer soil with declining hydraulic conductivity.

Sparse Monte Carlo Method for Nonlocal Diffusion Problems

A class of evolution equations with nonlocal diffusion is considered in this work. These are integro-differential equations arising as models of propagation phenomena in continuum media with nonlocal interactions including neural tissue, porous media, peridynamics, and models with fractional diffusion, as well as continuum limits of interacting dynamical systems. The principal challenge of numerical integration of nonlocal systems stems from the lack of spatial regularity of the data and solutions intrinsic to nonlocal models. To overcome this problem we propose a semidiscrete numerical scheme based on the combination of sparse Monte Carlo and discontinuous Galerkin methods. Our method requires minimal assumptions on the regularity of the data. In particular, the kernel of the nonlocal diffusivity is assumed to be a square integrable function and may be singular or discontinuous. An important feature of our method is the use of sparsity. Sparse sampling of points in the Monte Carlo approximation of the nonlocal term allows us to use fewer discretization points without compromising the accuracy. For kernels with singularities, more points are selected automatically in the regions near the singularities. We prove convergence of the numerical method and estimate the rate of convergence. There are two principal ingredients in the error of the numerical method related to the use of Monte Calro and Galerkin approximations, respectively. We analyze both errors. Two representative examples of discontinuous kernels are presented. The first example features a kernel with a singularity, while the kernel in the second example experiences jump discontinuity. We show how the information about the singularity in the former case and the geometry of the discontinuity set in the latter translate into the rate of convergence of the numerical procedure. In addition, we illustrate the rate of convergence estimate with a numerical example of an initial value problem, for which an explicit analytic solution is available. Numerical results are consistent with analytical estimates.

Thermodynamics of Einstein-Aether black holes

We analyze several spherically symmetric exterior vacuum solutions allowed by the Einstein-Aether (EA) theory with a non-static aether and study the thermodynamics of their Killing and universal horizons. We show that there are five classes of solutions corresponding to different values of a combination of the free parameters, c_{2}, c_{13}=c_1+c_3 and c_{14}=c_1+c_4, which are: (A) c_2 ne 0 and c_{13} ne 0 and c_{14}=0, (B) c_2 ne 0 and c_{13} = 0 and c_{14} = 0, (C) c_2 = 0 and c_{13} ne 0 and c_{14} = 0, (D) c_2 = 0, c_{13} = 0 and c_{14} ne 0, and (E) c_2 = - c_{13} ne 0 and c_{14} ne 0. We present explicit analytical solutions for these five cases. All these cases have singularities at r=0 and are asymptotically flat spacetimes and possess both Killing and universal horizons with the universal horizons always being inside the Killing horizons. Finally, we compute the surface gravity, the temperature, the entropy and the first law of thermodynamics for the universal horizons.