Explicit Analytical Solutions Research Articles

Global Existence and Nonexistence of Solutions to a Cross Diffusion System with Nonlocal Boundary Conditions

Mathematical models of nonlinear cross diffusion are described by a system of nonlinear partial parabolic equations associated with nonlinear boundary conditions. Explicit analytical solutions of such nonlinearly coupled systems of partial differential equations are rarely existed and thus, several numerical methods have been applied to obtain approximate solutions. In this paper, based on a self-similar analysis and the method of standard equations, the qualitative properties of a nonlinear cross-diffusion system with nonlocal boundary conditions are studied. We are constructed various self-similar solutions to the cross diffusion problem for the case of slow diffusion. It is proved that for certain values of the numerical parameters of the nonlinear cross-diffusion system of parabolic equations coupled via nonlinear boundary conditions, they may not have global solutions in time. Based on a self-similar analysis and the comparison principle, the critical exponent of the Fujita type and the critical exponent of global solvability are established. Using the comparison theorem, upper bounds for global solutions and lower bounds for blow-up solutions are obtained.

Emergence of ghosts in Horndeski theory

We show that starting from initial conditions with stable perturbations, evolution of a galileon scalar field results in the appearance of a ghost later on. To demonstrate this, we consider a theory with k-essence and cubic galileon Lagrangians on a fixed Minkowski background. Explicit analytical solutions of simple waves are constructed, on top of which a healthy scalar degree of freedom is shown to be converted onto a ghost. Such a transformation is smooth and moreover perturbations remain hyperbolic all the time (until a caustic forms). We discuss a relation between the ghost and the appearance of closed causal curves for such solutions.

Simplified Analytical Solution of the Contact Problem on Indentation of a Coated Half-Space by a Conical Punch

The contact problem on indentation of an elastic coated half-space by a conical punch is considered. To obtain an explicit analytical solution suitable for applications, the bilateral asymptotic method is used in a simplified form. For that purpose, kernel transform of the integral equation is approximated by a ratio of two quadratic functions containing only one parameter. Such an approach allows us to obtain explicit analytical expressions for the distribution of contact stresses and relations between the indentation force, depth, stiffness and contact radius. The obtained solution is suitable both for homogeneous and functionally graded coatings. The dependence of the characteristics of contact interaction on a relative Young’s modulus of the coating and relative coating thickness is analyzed and illustrated by the numerical examples. Ranges of values of elastic and geometrical parameters are obtained, for which the presence of a coating sufficiently changes the contact characteristics. The accuracy of the obtained simplified expressions is studied in detail. Results of the paper sufficiently simplify engineering calculations and are suitable for inverse analysis, e.g., analysis of indentation experiments of coated materials using either a conical or a pyramidal (Berkovich) indenter.

Exploration on traveling wave solutions to the 3rd-order klein–fock-gordon equation (KFGE) in mathematical physics

In this paper, the -expansion method has been applied to find the new exact traveling wave solutions of the nonlinear evaluation equations (NLEEs) by utilizing 3rd-order Klein–Gordon Equation (KFGE). With the collaboration of symbolic commercial software maple, the competence of this method for inventing these exact solutions has been more exhibited. As an upshot, some new exact solutions are obtained and signified by hyperbolic function solutions, different combinations of trigonometric function solutions, and exponential function solutions. Moreover, the -expansion method is a more efficient method for exploring essential nonlinear waves that enrich a variety of dynamic models that arises in nonlinear fields. All sketching is given out to show the properties of the innovative explicit analytic solutions. Our proposed method is directed, succinct, and reasonably good for the various nonlinear evaluation equations (NLEEs) related treatment and mathematical physics also.Â

Hard-magnetic elastica

Recently, ferromagnetic soft continuum robots – a type of slender, thread-like robots that can be steered magnetically – have demonstrated the capability to navigate through the brain's narrow and winding vasculature, offering a range of captivating applications such as robotic endovascular neurosurgery. Composed of soft polymers with embedded hard-magnetic particles as distributed actuation sources, ferromagnetic soft continuum robots produce large-scale elastic deflections through magnetic torques and/or forces generated from the intrinsic magnetic dipoles under the influence of external magnetic fields. This unique actuation mechanism based on distributed intrinsic dipoles yields better steering and navigational capabilities at much smaller scales, which differentiate them from previously developed continuum robots. To account for the presence of intrinsic magnetic polarities, this emerging class of magnetic continuum robots provides a new type of active structure – hard-magnetic elastica – which means a thin, elastic strip or rod with hard-magnetic properties. In this work, we present a nonlinear theory for hard-magnetic elastica, which allows accurate prediction of large deflections induced by the magnetic body torque and force in the presence of an external magnetic field. From our model, explicit analytical solutions can be readily obtained when the applied magnetic field is spatially uniform. Our model is validated by comparing the obtained solutions with both experimental results and finite element simulations. The validated model is then used to calculate required magnetic fields for the robot's end tip to reach a target point in space, which essentially is an inverse problem challenging to solve with a linear theory or finite-element simulation. Providing facile routes to analyze nonlinear behavior of hard-magnetic elastica, the presented theory can be used to guide the design and control of the emerging class of magnetically steerable soft continuum robots.

Stability of perforated nanobeams incorporating surface energy effects

This paper aims to present an analytical methodology to investigate influences of nanoscale and surface energy on buckling stability behavior of perforated nanobeam structural element, for the first time. The surface energy effect is exploited to consider the free energy on the surface of nanobeam by using Gurtin-Murdoch surface elasticity theory. Thin and thick beams are considered by using both classical beam of Euler and first order shear deformation of Timoshenko theories, respectively. Equivalent geometrical constant of regularly squared perforated beam are presented in simplified form. Problem formulation of nanostructure beam including surface energies is derived in detail. Explicit analytical solution for nanoscale beams are developed for both beam theories to evaluate the surface stress effects and size-dependent nanoscale on the critical buckling loads. The closed form solution is confirmed and proven by comparing the obtained results with previous works. Parametric studies are achieved to demonstrate impacts of beam filling ratio, the number of hole rows, surface material characteristics, beam slenderness ratio, boundary conditions as well as loading conditions on the non-classical buckling of perforated nanobeams in incidence of surface effects. It is found that, the surface residual stress has more significant effect on the critical buckling loads with the corresponding effect of the surface elasticity. The proposed model can be used as benchmarks in designing, analysis and manufacturing of perforated nanobeams.

Coherence properties of high-frequency wave field propagating through inhomogeneous ionosphere with anisotropic random irregularities of electron density: 1. Theoretical background

Rigorous analytic solution is constructed for the first of 2 s-order space position and frequency coherence function (which is alternatively also termed as symmetric) of the high-frequency wave field propagating through the stochastic transionospheric channel with the inhomogeneous layer of the background ionosphere and anisotropic fluctuations of the electron density in the approximation of the appropriate Markov equation. It is obtained employing the anisotropic parabolic model of the effective structure function of the fractional electron density fluctuations with two different scales along and across the Earth's magnetic field. The explicit analytic solution is presented, which is capable of describing the coherence properties of the field for any given inhomogeneous distribution of the electron density in the background ionosphere along a path of propagation. The procedure of calibrating the spatial scales of the anisotropic parabolic model of the electron density fluctuations along and across the Earth's magnetic field lines in order to provide the adequate solution to correspond to the realistic anisotropic inverse power law spectrum of the electron density fluctuations is presented in the companion paper.

The direct method for multisolitons and two-hump solitons in the Hirota-Satsuma system

The coupled Korteweg-de Vries (cKdV) system, originally proposed by Hirota and Satsuma [1], modeling the interaction of two long waves evolving with different dispersion relations, is considered. Multisoliton solutions are explicitly found in the Hirota-Satsuma system by means of an extension of the direct method due to Hirota [2]. Such multisolitons are found to be a complete family of interacting classical solitons consisting of the trivial ones corresponding to the decoupled N-soliton solution of the KdV equation and the ones coupled to the M-cKdV solitons, here simply referred to as the classical (N, M)-soliton solutions which are N+M interacting solitons indeed. Explicit analytical solutions, expressed in terms of hyperbolic functions, are found for the (0, 1), (1, 1) and (0, 2) cases while the process to generate the general (N, M)-soliton solutions is systematically described in detail. In the particular scenario of the (1, 1)-soliton, a novel two-hump one-soliton solution is obtained when a negative prestressing is prescribed for one of the waves in the merging of the 1-KdV and 1-cKdV solitons. Special attention is emphasized to the phase shift for the (1, 1) and (0, 2) interaction cases to asymptotically get a particular remarkable long range interaction phenomenon. Finally, some numerical evolutions are also considered to illustrate the interaction dynamics of the multisolitons and the new two-hump soliton in the different situations considered.

An explicit analytical solution for temperature rise in a half-space induced by ellipsoidal inclusions and its application for problems related to friction heating

Green functions for temperature rise in a semi-infinite space containing an ellipsoidal inclusion are obtained in the present study. Explicit expression for disturbed temperature rise generated by eigen-temperature gradients uniformly distributed within a domain is derived. Further, the proposed analytical solution method is utilized to deal with temperature rise in heterogeneous half-space subjected to friction heating via applying the equivalent inclusion method (EIM), whose results are proven to be in good agreements with those of the benchmarks. Influences of heat load velocity, spatial orientation and aspect ratio of ellipsoidal inhomogeneity on temperature rise in a semi-infinite space are discussed. Finally, a model of semi-infinite medium with embedded dispersed ellipsoidal inhomogeneities of arbitrary spatial orientation is adopted to explore the application scope of the proposed solution method.

Compressible liquid/gas inclusion with high initial pressure in plane deformation: Modified boundary conditions and related analytical solutions

In the analysis of the elastic behavior of a composite system composed of a solid matrix and a number of compressible liquid/gas inclusions, it is customary to incorporate the change in the magnitude of an inclusion's internal pressure (induced by the change in its volume) during deformation. However, when an inclusion has a relatively high initial pressure, the change in the direction of the pressure during deformation may also be significant, particularly in establishing an accurate boundary condition on the inclusion-matrix interface. This can be attributed to the fact that, during deformation, the change in the direction of the normal to the inclusion's boundary is of the same order as the change in its volume. Accordingly, in this paper, we propose two modified boundary conditions for a general compressible liquid/gas inclusion surrounded by a solid matrix subjected to plane deformation. Specifically, we incorporate changes in both the magnitude and direction of the inclusion's internal pressure during deformation. The corresponding boundary value problems are formulated using complex variable methods with explicit analytical solutions obtained in the particular case when the initial shape of the inclusion is circular and the surrounding matrix is infinite. Concise explicit formulae are also given for evaluating the effective moduli of the corresponding composite system based on the dilute model and the Mori-Tanaka method. Numerical results are presented to illustrate our solutions and to draw comparison with the classical solution in the case of an air bubble inclusion in a soft gel matrix. We find that when the inclusion has a relatively high initial pressure, the classical solution may indeed overestimate the external loading-induced stress concentration around the inclusion and underestimate the effective moduli of the corresponding composite system. In particular, it is shown that a porous material could be significantly strengthened in resisting shear deformation when filled with liquid/gas inclusions of a relatively high pressure even if surface tension effects are neglected: this phenomenon is not captured in classical models.

Vertical vibration of a circular foundation in a transversely isotropic poroelastic soil

Abstract Analytical methods based on linear elasticity have been used to model the dynamic response of foundations. These solutions commonly assume that soils are isotropic and elastic. Incorporation of anisotropy and the two-phased nature of soils (solid skeleton with pores filled with water) is important in the study of dynamic response of foundations. This paper presents the explicit analytical solutions for a transversely isotropic poroelastic soil half-space under a buried time-harmonic vertical load and a time-harmonic pore pressure discontinuity. These versatile fundamental solutions are derived by using Hankel integral transforms. They can be used to analyze a variety of dynamic problems in geomechanics. The fundamental solutions are then applied to solve the time-harmonic vertical vibration of a flexible circular foundation by using variational methods. Selected numerical results are presented to demonstrate the influence of soil anisotropy, poroelasticity, foundation flexibility, depth of embedment and frequency of excitation on the vertical dynamic response of foundation and the force transmitted to soil.

Interface stress effect tuning and enhancing the energy dissipation of staggered nanocomposites

ABSTRACTNatural biological composites and artificial biomimetic staggered composites with nanoscale internal structures can exhibit extraordinary energy dissipation, compared with conventional composites. It is believed that the interface stress effects of the interfaces between hard platelets and a viscous matrix play an important role in the extraordinary damping properties of such nanocomposites. In this study, a viscoelastic model is established to investigate the mechanism of the influence that the interface stress effect has on the damping properties, based on the Gurtin-Murdoch interface model and the tension-shear chain model. An explicit analytical solution of the effective dynamic moduli characterising the damping properties is obtained by using the correspondence principle, which is also validated by comparison with a finite element analysis. From the obtained analytical solution, an interface factor is abstracted to characterise the synergistic effect of the feature size and material parameters on the damping properties. Based on the model established, the optimal size of the platelets and the optimal loading frequency can be designed to achieve superior energy dissipation, when the staggered nanocomposites bear the dynamic load. Therefore, the findings of the present study not only reveal the damping mechanism of biological structures at nanoscale but also provide useful guidelines for the design of biomimetic nanocomposites from the nanoscale to the macroscopic scale.

Solitary wave, M-lump and localized interaction solutions to the (4+1)-dimensional Fokas equation

At present, many researchers pay close attention to the solutions of nonlinear partial differential equations. Because these solutions can explain physical phenomena well. In this paper, soliton solutions, fissionable wave solutions, M-lump solutions and interaction solutions of the (4+1)-dimensional Fokas equation are explicitly generated. Taking advantage of the Hirota bilinear method, N-soliton solutions are constructed. Then, on bases of the obtained solutions, fissionable wave solutions are ascertained by appropriate selections of parameters. Applying the long wave limit method and restricting the conjugation conditions on the related solitons, M-lump solutions are derived. Afterwards, the interaction solutions which describe interactions of a lump and a soliton, a lump and two solitons, a lump and fissionable solitary waves are gained. Moreover, some graphs are given out to demonstrate the dynamic properties of the explicit analytical localized wave solutions.

Non-uniform global-buckling and local-folding in thin film of stretchable electronics

Thin film can either buckle globally or fold locally to assemble stretchable electronics, the variation of thickness in thin film leads to tunable structures and promises the diverse design of stretchable electronics. In this paper, theoretical models are developed to investigate the global-buckling and local-folding in the non-uniform thin film of stretchable electronics. The models are capable of giving concise and explicit analytical solutions of the deformed configuration and maximum strain in thin film. The effectiveness and accuracy of analytical solutions are verified by the finite element analyses (FEA) and molecular dynamics (MD) simulations. Our study provides useful guidance for the diverse design and precise control of stretchable electronics.

Non-equilibrium electronic transport through a quantum dot with strong Coulomb repulsion in the presence of a magnetic field

The non-equilibrium electronic transport through a nanoscale device composed of a single quantum dot between two metallic contacts is studied theoretically within the framework of the Keldysh formalism. The quantum dot consists of a single energy level subject to an applied magnetic field. Correlations due to the Coulomb repulsion between electrons on the dot are treated with a Green’s function decoupling scheme which, although similar to the Hubbard-I approximation, captures some of the dynamics beyond. The scheme is exact in the so-called atomic limit, defined by vanishing tunneling between contacts and dot, and in the non-interacting limit, where the on-dot Coulomb repulsion is zero. Explicit analytic solutions, valid for arbitrary magnetic fields, are obtained for two important setups: (i) the stationary regime, with constant voltage bias between the leads, and (ii) the time-dependent regime for metallic leads with constant density of states of infinite width. In these regimes, the current through the dot is evaluated numerically for various parameter sets and its main features interpreted in terms of the underlying physical processes. The results are compared to the non-crossing approximation (NCA) and diagrammatic non-equilibrium quantum Monte-Carlo (QMC) where available.

Analytical treatment for cyclic three-state dynamics on static networks.

Whenever a dynamical process unfolds on static networks, the dynamical state of any focal individual will be exclusively influenced by directly connected neighbors, rather than by those unconnected ones, hence the arising of the dynamical correlation problem, where mean-field-based methods fail to capture the scenario. The dynamic correlation coupling problem has always been an important and difficult problem in the theoretical field of physics. The explicit analytical expressions and the decoupling methods often play a key role in the development of corresponding field. In this paper, we study the cyclic three-state dynamics on static networks, which include a wide class of dynamical processes, for example, the cyclic Lotka-Volterra model, the directed migration model, the susceptible-infected-recovered-susceptible epidemic model, and the predator-prey with empty sites model. We derive the explicit analytical solutions of the propagating size and the threshold curve surface for the four different dynamics. We compare the results on static networks with those on annealed networks and made an interesting discovery: for the symmetrical dynamical model (the cyclic Lotka-Volterra model and the directed migration model, where the three states are of rotational symmetry), the macroscopic behaviors of the dynamical processes on static networks are the same as those on annealed networks; while the outcomes of the dynamical processes on static networks are different with, and more complicated than, those on annealed networks for asymmetric dynamical model (the susceptible-infected-recovered-susceptible epidemic model and the predator-prey with empty sites model). We also compare the results forecasted by our theoretical method with those by Monte Carlo simulations and find good agreement between the results obtained by the two methods.

Optimization of Band Selection in Multispectral and Narrow-Band Imaging: An Analytical Approach.

The contrast ratio has been proposed for quantification of image quality of subsurface inhomogeneities in the skin. Based on the two-flux Kubelka-Munk model, we developed an analytical approach which links the contrast ratio with optical tissue parameters. We obtained an explicit analytical solution for the dependence of maximal contrast ratio on optical tissue parameters. Then, we linked the minimally observable contrast ratio (cmin) with the bit depth of the camera, d: cmin=1/(2d-1). Based on this analysis we were able to derive an explicit expression, which links camera properties with the minimally detectable changes in optical tissue parameters (both scattering and absorption). The proposed analytical model can be used for rapid assessment and optimization of multispectral and narrow band imaging techniques and for estimation of the accuracy of imaging techniques. The developed model confirms the utility of the contrast ratio for tissue imaging.

New explicit analytical solutions of equations for heat and mass transfer in a cooling tower energy system

The aim of this article is to develop an accurate and fast analytical method for heat and mass transfer model in a cooling tower energy system. Some algebraic explicit analytical solutions of the one-dimensional differential equation sets describing the coupled heat and mass transfer process in a cooling tower are derived. The explicit solutions have not yet been published before. The explicit equations of heat and mass transfer are expressed in elementary functions. By solving these differential equations in a cooling tower, the temperature distribution of liquid and gas, the moisture content in the air can be obtained in each section over the vertical height of the tower. A comparison of analytical and experimental results was given in this article, and good agreements were shown for the typical cases studied. The analytical solutions can serve as a benchmark to check the results of numerical calculation.

Time fractional Kupershmidt equation: symmetry analysis and explicit series solution with convergence analysis

Abstract In this work, the fractional Lie symmetry method is applied for symmetry analysis of time fractional Kupershmidt equation. Using the Lie symmetry method, the symmetry generators for time fractional Kupershmidt equation are obtained with Riemann-Liouville fractional derivative. With the help of symmetry generators, the fractional partial differential equation is reduced into the fractional ordinary differential equation using Erdélyi-Kober fractional differential operator. The conservation laws are determined for the time fractional Kupershmidt equation with the help of new conservation theorem and fractional Noether operators. The explicit analytic solutions of fractional Kupershmidt equation are obtained using the power series method. Also, the convergence of the power series solutions is discussed by using the implicit function theorem.

Two-dimensional distribution of streamwise velocity in open channel flow using maximum entropy principle: Incorporation of additional constraints based on conservation laws

This study derived the vertical and transverse distribution of streamwise velocity in open channels using the information-theoretic concept involving the maximum entropy (MaxEnt) principle, moment constraints expressed by the conservation of mass, momentum, and energy and a hypothesis invoking the connection between probability and space domains through a generalized coordinate system. Explicit analytical solutions were obtained for velocity profiles using a non-perturbation approach along with Padé approximation. The convergence of the series solution was shown both theoretically and numerically. The Lagrange multipliers, arising through the application of MaxEnt principle, were obtained by minimizing a convex potential, and the quasi-Newton method along with BFGS scheme was applied for solving the unconstrained optimization problem. For applying the conservation laws, the momentum and energy coefficients were found to influence the velocity profile in a channel cross-section. With the expressions of these coefficients, the derived velocity equations were validated for a wide range of experimental and field data.