Special Analytical Solutions Research Articles

A Discontinuous Galerkin–Finite Element Method for the Nonlinear Unsteady Burning Rate Responses of Solid Propellants

The unsteady combustion of solid propellants under oscillating environments is the key to understanding the combustion instability inside solid rocket motors. The discontinuous Galerkin–finite element method (DG-FEM) is introduced to provide an efficient yet flexible numerical platform to investigate the combustion dynamics of solid propellants. The algorithm is developed for the classical unsteady model, the Zel’dovich–Novozhilov model. It is then validated based on a special analytical solution. The DG-FEM algorithm is then compared with the classical spectral method based on Laguerre polynomials. It is shown that the DG-FEM works more efficiently than the traditional spectral method, providing a more accurate solution with a lower computational cost.

Remarks on blowup of solutions for one‐dimensional compressible Navier–Stokes equations with Maxwell's law

AbstractIn this note, we present some blowup results of solutions to the one‐dimensional compressible Navier–Stokes equations with Maxwell's law. First, we improve the blowup result of Hu and Wang [Math. Nachr. 92 (2019), 826–840] with initial density away from vacuum by removing two restrictions. Next, we give a blowup result for the solutions with decay at far fields. Finally, we construct some special analytical solutions to exhibit the blowup or non‐blowup phenomena for the relaxed system.

Thermal optimization and magnetization of nanofluid under shape effects of nanoparticles

The measurements of thermal transport are complex to optimize from high temperatures; this is because increased thermal-management performance have typically been accompanied for the sake of applicability in thermal barriers and heat dissipation. In this manuscript, an investigation for the flow of Brinkman nanofluid is carried out in which shapes of nanoparticles are suspended in ethylene glycol and their thermophysical properties have been emphasized. In order to have an optimized heat and mass transfer, the mathematical modeling is traced out for temperature, concentration and velocity profiles through non-singularized fractional technique. The analytical solutions for temperature, concentration and velocity profiles have been derived by invoking integral transforms. Special analytical solutions have been traced out for Stoke's first and second problems by varying boundary conditions of the problem. The rheological parameters have shown the pertinent effects on the basis of numerical variations and similarities. The results suggested that velocity of ethylene glycol-alumina is slower than ethylene glycol-copper, ethylene glycol-gold and ethylene glycol-silver during non-magnetization process.

Complex plane representations and stationary states in cubic and quintic resonant systems

Weakly nonlinear energy transfer between normal modes of strongly resonant PDEs is captured by the corresponding effective resonant systems. In a previous article, we have constructed a large class of such resonant systems (with specific representatives related to the physics of Bose–Einstein condensates and Anti-de Sitter (AdS) spacetime) that admit special analytic solutions and an extra conserved quantity. Here, we develop and explore a complex plane representation for these systems modelled on the related cubic Szegő and LLL equations. To demonstrate the power of this representation, we use it to give simple closed form expressions for families of stationary states bifurcating from all individual modes. The conservation laws, the complex plane representation and the stationary states admit furthermore a natural generalization from cubic to quintic nonlinearity. We demonstrate how two concrete quintic PDEs of mathematical physics fit into this framework, and thus directly benefit from the analytic structures we present: the quintic nonlinear Schrödinger equation in a one-dimensional harmonic trap, studied in relation to Bose–Einstein condensates, and the quintic conformally invariant wave equation on a two-sphere, which is of interest for AdS/CFT-correspondence.

Optoelectronic Device Simulations Based on Macroscopic Maxwell–Bloch Equations

AbstractDue to their intuitiveness, flexibility, and relative numerical efficiency, the macroscopic Maxwell–Bloch (MB) equations are a widely used semiclassical and semi‐phenomenological model to describe optical propagation and coherent light–matter interaction in media consisting of discrete‐level quantum systems. This review focuses on the application of this model to advanced optoelectronic devices, such as quantum cascade and quantum dot lasers. The Bloch equations are here treated as a density matrix model for driven quantum systems with two or multiple discrete energy levels, where dissipation is included by Lindblad terms. Furthermore, the 1D MB equations for semiconductor waveguide structures and optical fibers are rigorously derived. Special analytical solutions and suitable numerical methods are presented. Due to the importance of the MB equations in computational electrodynamics, an emphasis is placed on the comparison of different numerical schemes, both with and without the rotating wave approximation. The implementation of additional effects which can become relevant in semiconductor structures, such as spatial hole burning, inhomogeneous broadening, and local‐field corrections, is discussed. Finally, links to microscopic models and suitable extensions of the Lindblad formalism are briefly addressed.

Solvable Cubic Resonant Systems

Weakly nonlinear analysis of resonant PDEs in recent literature has generated a number of resonant systems for slow evolution of the normal mode amplitudes that possess remarkable properties. Despite being infinite-dimensional Hamiltonian systems with cubic nonlinearities in the equations of motion, these resonant systems admit special analytic solutions, which furthermore display periodic perfect energy returns to the initial configurations. Here, we construct a very large class of resonant systems that shares these properties that have so far been seen in specific examples emerging from a few standard equations of mathematical physics (the Gross-Pitaevskii equation, nonlinear wave equations in Anti-de Sitter spacetime). Our analysis provides an additional conserved quantity for all of these systems, which has been previously known for the resonant system of the two-dimensional Gross-Pitaevskii equation, but not for any other cases.

Minimal time splines on the sphere

An interesting problem in geometric control theory arises from robotics and space science: find a smooth curve, controlled by a bounded acceleration, connecting in minimum time two prescribed tangent vectors of a Riemannian manifold Q. The state equation is $$ \nabla _{\dot{\gamma }} \dot{\gamma } = u \in TQ, |u| \le A$$ . Applying Pontryagin’s principle one gets a Hamiltonian system in $$T^*(TQ)$$ . We consider this problem in $$S^2(r)$$ . Seemingly, it has not been addressed before. Via the SO(3) symmetry, we reduce the four degrees of freedom system to the five variables $$(a,v, M_1,M_2,M_3),$$ where v is the scalar velocity, conjugated to a costate variable a and $$(M_1,M_2,M_3)$$ are costate variables that satisfy $$\{ M_i, M_j\} = \epsilon _{ijk} M_k$$ . We derive the reduced equations and find special analytical solutions, that are organizing centers for the dynamics. Reconstruction of the curve $$\gamma (t)$$ is achieved by a time dependent linear system of ODEs for the orthogonal matrix R whose first column is the unit tangent vector of the curve and whose last column is the unit normal vector to the sphere.

Anisotropic mesh adaptation for finite element solution of Anisotropic Porous Medium Equation

Anisotropic Porous Medium Equation (APME) is developed as an extension of the Porous Medium Equation (PME) for anisotropic porous media. A special analytical solution is derived for APME for time-independent diffusion. Anisotropic mesh adaptation for linear finite element solution of APME is discussed and numerical results for two dimensional examples are presented. The solution errors using anisotropic adaptive meshes show second order convergence.

Magnetohydrodynamic Models of Molecular Tornadoes

Abstract Recent observations near the Galactic Center (GC) have found several molecular filaments displaying striking helically wound morphology that are collectively known as molecular tornadoes. We investigate the equilibrium structure of these molecular tornadoes by formulating a magnetohydrodynamic model of a rotating, helically magnetized filament. A special analytical solution is derived where centrifugal forces balance exactly with toroidal magnetic stress. From the physics of torsional Alfvén waves we derive a constraint that links the toroidal flux-to-mass ratio and the pitch angle of the helical field to the rotation laws, which we find to be an important component in describing the molecular tornado structure. The models are compared to the Ostriker solution for isothermal, nonmagnetic, nonrotating filaments. We find that neither the analytic model nor the Alfvén wave model suffer from the unphysical density inversions noted by other authors. A Monte Carlo exploration of our parameter space is constrained by observational measurements of the Pigtail Molecular Cloud, the Double Helix Nebula, and the GC Molecular Tornado. Observable properties such as the velocity dispersion, filament radius, linear mass, and surface pressure can be used to derive three dimensionless constraints for our dimensionless models of these three objects. A virial analysis of these constrained models is studied for these three molecular tornadoes. We find that self-gravity is relatively unimportant, whereas magnetic fields and external pressure play a dominant role in the confinement and equilibrium radial structure of these objects.

Examination of Stochastic and Deterministic Perturbations in Reactor Physics

ABSTRACTThe general solution of the perturbed diffusion equation in case of localized random perturbation is discussed in details. The auto-covariance function of the flux perturbation and the reaction rate have been calculated by Green-function technique, and the algorithm is tested with some special analytic solutions. The reactivity and the variance of reactivity in cases of different perturbations has been determined numerically.

RETRACTED ARTICLE: Magnetic System and Equilibrium Reconstruction for ITER using the GSE Solution and TEQ Code

The superconducting magnet system of ITER consists of four main sub-systems: toroidal field coils, central solenoid coils; poloidal field coils; and correction coils. Like many other ITER systems, the magnet components are supplied in-kind by six domestic agencies. The technical specifications, manufacturing processes and procedures required to fabricate these components are particularly challenging. The management structure and organization to realize this procurement within the tight ITER construction schedule is very complex. On the other hand, all plasma processes including linear and early nonlinear stages of MHD instabilities, transport and plasma flows, waves, micro-instabilities and turbulence, represent different kinds of deviations from MHD equilibrium and thus require accurate calculations of equilibrium configurations. The simplest useful mathematical model to describe equilibrium in fusion plasmas is achieved by combining MHD equations with Maxwell’s equations. The final result is the Grad–Shafranov (GS) equation. Analytical solutions to the GS equation are an aid to theoretical investigations into plasma equilibrium, stability and transport in axisymmetric plasmas. In this paper a special analytical solution for GS equation and also simulation of equilibrium by TEQ code for ITER were presented.

Half of internal inductance plus poloidal beta and plasma position in a circular cross-section tokamak

A special analytical solution of the Grad–Shafranov equation (GSE) is presented with source functions, where the plasma pressure is linear in ψ and the squared poloidal current has both a quadratic and a linear ψ term. Half of internal inductance plus poloidal beta and plasma position have been calculated for a typical discharge of IR-T1 tokamak with circular cross-section. In the presented solution, six parameters are measured by the magnetic measurements. The calculated parameters approach the values measured by discrete magnetic coils in the middle of IR-T1 discharge.

Reactive silica transport in fractured porous media: Analytical solutions for a system of parallel fractures

A general analytical solution is derived by using the Laplace transformation to describe transient reactive silica transport in a conceptualized 2‐D system involving a set of parallel fractures embedded in an impermeable host rock matrix, taking into account of hydrodynamic dispersion and advection of silica transport along the fractures, molecular diffusion from each fracture to the intervening rock matrix, and dissolution of quartz. A special analytical solution is also developed by ignoring the longitudinal hydrodynamic dispersion term but remaining other conditions the same. The general and special solutions are in the form of a double infinite integral and a single infinite integral, respectively, and can be evaluated using Gauss‐Legendre quadrature technique. A simple criterion is developed to determine under what conditions the general analytical solution can be approximated by the special analytical solution. It is proved analytically that the general solution always lags behind the special solution, unless a dimensionless parameter is less than a critical value. Several illustrative calculations are undertaken to demonstrate the effect of fracture spacing, fracture aperture and fluid flow rate on silica transport. The analytical solutions developed here can serve as a benchmark to validate numerical models that simulate reactive mass transport in fractured porous media.

To the analytical theory of the supercompensation phenomenon

The effect of supercompensation and the conditions of its origination have been analyzed using a special analytical solution of the equation of the overdamped Duffing oscillator. The threshold values of the functional shift after the exposure of a living organism to intensive external load have been found. If these values are exceeded, the restoration process goes through the stages of enhanced and lowered working capacity. Expressions for the times of transition to these phases after the onset of recovery and for the time of occurrence of the maxima of these phases have been obtained and analyzed.

Typhoon eye trajectory based on a mathematical model: Comparing with observational data

Abstract We propose a model based on the primitive system of the Navier–Stokes equations in a bidimensional framework, as the l -plane approximation, which allows us to explain the variety of tracks of tropical cyclones (typhoons). Our idea is to construct special analytical solutions with a linear velocity profile for the Navier–Stokes systems. The evidence of the structure of linear velocity near the center of vortex is provided by the observational data. We study solutions with the linear-velocity property for both barotropic and baroclinic cases and show that they follow the same equations in describing the trajectories of the typhoon eye at the equilibrium state (that relates to the conservative phase of the typhoon dynamics). Moreover, at the equilibrium state, the trajectories can be viewed as a superposition of two circular motions: one has period 2 π / l , the other has period 2 π / b 0 , where l is the Coriolis parameter and b 0 is the height-averaged vorticity at the center of the cyclone. Also, we compare our theoretical trajectories based on initial conditions from the flow with tracks obtained from the observational database. It is worth mentioning that, under certain conditions, our results are still compatible with the observational data, although we did not truly consider the influence of the the ambient pressure field (the “steering” effect). Finally, we propose a parameter-adopting method so that one can correct trajectory prediction in real-time. Examples of our analysis and the use of a parameter-adopting method for the historic trajectories are provided.

Monopole Blocking Governed by a Modified KdV Type Equation

A type of coupled variable coefficient modified Korteweg‐de Vries system is derived from a two‐layered fluid system. It is known that the formation, maintenance, and collapse of an atmospheric blocking are always related with large‐scale weather or short term climate anomalies. One special analytical solution of the obtained system successfully features the evolution cycle of an atmospheric monopole type blocking event. In particular, our theoretical results capture a real monopole type blocking case that happened during 19 February 2008 to 26 February 2008 can be well described by our analytical solution.

A new stable finite volume method for predicting thermal performance of a whole building

Discretised governing equations involving only temperatures and heat fluxes at both surfaces of a solid wall layer were obtained by combining a new stable finite volume scheme for the two inner nodes of the wall layer with the surface diffusion equations (discretised by third order equations). The finite volume scheme for the inner nodes of the layer is proved to be stable with its truncation error being O ( Δ x 4 , Δ x 2 Δ t 2 ) . A special analytical solution for a solid wall was used to evaluate different schemes for the inner nodes, showing that the new proposed scheme performs better than all other schemes for time steps of 3600 and 600 s. Finally, this scheme was used to simulate a whole house and the predicted zonal air temperature, and surface temperatures agreed well with measured values.

Optical-parametric-oscillator solitons driven by the third harmonic

We introduce a model of a lossy second-harmonic-generating (chi(2)) cavity externally pumped at the third harmonic, which gives rise to driving terms of a new type, corresponding to a cross-parametric gain. The equation for the fundamental-frequency (FF) wave may also contain a quadratic self-driving term, which is generated by the cubic nonlinearity of the medium. Unlike previously studied phase-matched models of chi(2) cavities driven at the second harmonic or at FF, the present one admits an exact analytical solution for the soliton, at a special value of the gain parameter. Two families of solitons are found in a numerical form, and their stability area is identified through numerical computation of the perturbation eigenvalues (stability of the zero solution, which is a necessary condition for the soliton's stability, is investigated in an analytical form). One family is a continuation of the special analytical solution. At given values of the parameters, one soliton is stable and the other one is not; they swap their stability at a critical value of the mismatch parameter. The stability of the solitons is also verified in direct simulations, which demonstrate that an unstable pulse rearranges itself into a stable one, or into a delocalized state, or decays to zero. A soliton which was given an initial boost C starts to move but quickly comes to a halt, if the boost is smaller than a critical value C(cr) . If C > C(cr) , the boost destroys the soliton (sometimes, through splitting into two secondary pulses). Interactions between initially separated solitons are investigated, too. It is concluded that stable solitons always merge into a single one. In the system with weak loss, it appears in a vibrating form, slowly relaxing to the static shape. With stronger loss, the final soliton emerges in the stationary form.

NavierStokes solutions for parallel flow in rivulets on an inclined plane

We investigate the solutions of the Navier–Stokes equations that describe the steady flow of rivulets down an inclined surface. We find that the shape of the free surface is given by an analytic formula obtained by solving the equation that expresses the condition of static equilibrium under the action of gravity and surface tension, independently of the velocity field and of any assumption concerning the rheology of the liquid. The velocity field is then obtained by solving (in general numerically) a Poisson equation in the domain defined by the cross-section of the rivulet. The isovelocity contours are perpendicular to the free surface. Various properties of the solutions are given as functions of the parameters of the problem. Two special analytic solutions are presented. The exact solutions suggest that the lubrication approximation, frequently employed to investigate problems similar to the present one, predicts reasonably well the global properties of the rivulet provided the static contact angle is not too large.

The limiting ideal theory for shear-index cohesionless granular materials

AbstractTo model cohesionless granular flow using continuum theory, the usual approach is to assume the cohesionless Coulomb-Mohr yield condition. However, this yield condition assumes that the angle of internal friction is constant, when according to experimental evidence for most powders the angle of internal friction is not constant along the yield locus, but decreases for decreasing normal stress component σ from a maximum value of π/2. For this reason, we consider here the more general yield function which applies for shear-index granular materials, where the angle of internal friction varies with σ. In this case, failure due to frictional slip between particles occurs when the shear and normal components of stress τ and σ satisfy the so-called Warren Spring equation (|τ|/c)n = 1 − (σ/t), where c, t and n are positive constants which are referred to as the cohesion, tensile strength and shear-index respectively, and experimental evidence indicates for many materials that the value of the shear-index n lies between 1 and 2. For many materials, the cohesion is close to zero and therefore the notion of a cohesionless shear-index granular material arises. For such materials, a continuum theory applying for shear-index cohesionless granular materials is physically plausible as a limiting ideal theory, and any analytical solutions might provide important benchmarks for numerical schemes. Here, we examine the cohesionless shearindex theory for the problem of gravity flow of granular materials through two-dimensional wedge-shaped hoppers, and we attempt to determine analytical solutions. Although some analytical solutions are found, these do not correspond to the actual hopper problem, but may serve as benchmarks for purely numerical schemes. The special analytical solutions obtained are illustrated graphically, assuming only a symmetrical stress distribution.