General Closed-form Solution Research Articles

Closed-form Expressions for Multiscatter Dark Matter Capture Rates

Any astrophysical object can, in principle, serve as a probe of the interaction between dark matter (DM) and regular baryonic matter. This method is based on the potential observable consequences annihilations of captured DM have on the surface temperature of the object itself. In a series of previous papers we developed and validated simple analytic approximations for the total capture rates of DM valid in four distinct regions of the DM–nucleon scattering cross section (σ) versus DM particle mass (m X ) parameter space. In this work, we summarize those previous results and extend them significantly by deriving a completely general closed-form solution for the total capture rate of DM in the multiscatter regime. Moreover, we demonstrate the existence of a region in the σ versus m X parameter space where the constraining power of any astrophysical object heated by annihilations of captured DM is lost. This corresponds to a maximal temperature (T crit) any astrophysical object can have, such that it can still serve as a DM probe. Any object with observed temperature T obs > T crit loses its DM constraining power. We provide analytic formulae that can be used to estimate T crit for any object.

A closed–form general solution for the distance of point-to-hyperbola in two dimensions

We consider the simple problem, in a two dimensional plane, the shortest Euclidean distance from a point to a hyperbola. Get rid of existing analytical or approximating methods from the literature, we give a closed-form general equation to this problem. Adhered in the end of the paper, the C++ and Python codes for the resolved general solution can facilitate practitioners in all fields to conduct the solution to their applications immediately.

Uncertainty propagation in orbital dynamics via Galerkin projection of the Fokker-Planck Equation

The Fokker–Planck equation is a partial differential equation that describes how the probability density function of an object state varies, when subject to deterministic and random forces. The solution to this equation is crucial in many space applications, such as space debris trajectory tracking and prediction, guidance navigation and control under uncertainties, space situational awareness, and mission analysis and planning. However, no general closed-form solutions are known and several methods exist to tackle its solution. In this work, we use a known technique to transform this equation into a set of linear ordinary differential equations in the context of orbital dynamics. In particular, we show the advantages of the applied methodology, which allows to decouple the time and state-dependent components and to retain the entire shape of the probability density function through time, in the presence of both deterministic and stochastic dynamics. With this approach, the probability density function values at future times and for different initial conditions can be computed without added costs, provided that some time-independent integrals are solved offline. We showcase the efficacy and use of this method on some orbital dynamics example, by also leveraging the use of automatic differentiation for efficiently computing the involved derivatives.

Unified viscoelastic injectivity model: Analytical solutions predicting polymer excess pressure and fracture initiation

The most pressing question to embark on polymer flood projects is whether a polymer can be injected at economic rates while below reservoir parting pressure. There are several published examples of field polymer floods with unexpected injectivity and/or fracture initiation/propagation. Current commercial reservoir simulators lack the capability to model viscoelastic polymer rheology and fracture initiation and often lead to misleading expectations. The laboratory tests to select a polymer are performed without considering fracture or shear-thickening behavior near wellbores. This paper presents a physical-mathematical model to estimate excess pressure due to viscoelastic polymers and assist field operators in better polymer selection and injection design. One of the primary purposes of this paper is to develop predictive analytical tools based on the well-known Unified Viscoelastic Model (UVM), capable of estimating excess pressure and fracture initiation, independent of numerical solutions and reservoir simulators. Asymptotic techniques and special functions theory are utilized to construct a set of general closed-form analytical solutions. It is found that viscoelastic pressure drop can be determined by merely knowing the polymer rheology and slug radius from the wellbore. The derived viscoelastic pressure drop solutions can be easily implemented in MS Excel, which is convenient for engineers participating in the design and execution of polymer flooding projects. Analytical solutions are compared with numerical solutions, laboratory experiments, and field data, and excellent agreement is found.

Dynamic response of floors induced by a single walking pedestrian including walking path variability

This paper proposes a general non-dimensional closed-form solution to calculate the dynamic response of simply supported rectangular floors crossed by a single pedestrian schematized as a deterministic moving multi-harmonic load. The critical walking paths, which can provide the maximum vibration response, are identified and then a simplified approximated closed-form solution for the dynamic response due to a moving resonant harmonic load along the critical walking paths is proposed. Finally, the accuracy of simplified methods suggested by current guidelines is investigated by comparing the maximum non-dimensional acceleration response obtained from design guidelines with the corresponding value calculated by the analytical solution.

A Comparison of General Solutions to the Non-Axisymmetric Frictionless Contact Problem with a Circular Area of Contact: When the Symmetry Does Not Matter

The non-axisymmetric problem of frictionless contact between an isotropic elastic half-space and a cylindrical punch with an arbitrarily shaped base is considered. The contact problem is formulated as a two-dimensional Fredholm integral equation of the first type in a fixed circular domain with the right-hand side being representable in the form of a Fourier series. A number of general solutions of the contact problem, which were published in the literature, are discussed. Based on the Galin–Mossakovskii general solution, new formulas are derived for the particular value of the contact pressure at the contact center and the contact stress-intensity factor at the contour of the contact area. Since the named general solution does not employ the operation of differentiation of a double integral with respect to the coordinates that enter it as parameters, the form of the general solution derived by Mossakovskii as a generalization of Galin’s solution for the special case, when the contact pressure beneath the indenter is bounded, is recommended for use as the most simple closed-form general solution of the non-axisymmetric Boussinesq contact problem.

A closed-form general solution for the distance of point-to-parabola in two dimensions

A closed-form general solution for the distance of point-to-parabola in two dimensions

A closed-form general solution for the distance of point-to-parabola in two dimensions

A closed-form general solution for the distance of point-to-parabola in two dimensions

A Poisson equation method for prescribing fully developed non-Newtonian inlet conditions for computational fluid dynamics simulations in models of arbitrary cross-section

Prescribing inlet boundary conditions for computational fluid dynamics (CFD) simulations of internal flow in complex geometries such as anatomical vascular models is challenging. In the absence of patient-specific inlet velocity data, a common approach for long blood vessels is to assume that the inlet flow is fully developed. In vessels of irregular cross section, however, prescribing fully developed conditions is complicated due to the lack of a general closed-form analytical solution. In this study, we develop a simple Poisson equation method for prescribing fully developed inlet conditions for the flow of either Newtonian or non-Newtonian fluids in CFD models of arbitrary cross-section. We first derive the generalized Poisson equation for fully developed flow of a non-Newtonian fluid and we then develop and verify a methodology for numerically computing the solution on any planar boundary domain. In addition, we develop a simple extension of the method for prescribing a non-orthogonal inlet velocity that represents fully developed flow from an upstream tube that is connected to the CFD inlet at a non-orthogonal angle. This may be used to investigate a common source of uncertainty in CFD simulations of internal flow that is due to a lack of information concerning the exact streamwise flow direction at the inlets. Comparison to several Newtonian and non-Newtonian benchmark verification solutions shows the method to be extremely accurate. As a practical demonstration case, we use the method to prescribe fully developed conditions on multiple non-circular inlets for the non-Newtonian flow of blood in a patient-specific model of the inferior vena cava (IVC). Finally, we further demonstrate the utility of the method by performing a sensitivity study using the patient-specific IVC model, wherein we investigate the influence of inlet velocity flow direction on the non-Newtonian IVC hemodynamics. Given its simplicity and computational efficiency, the method is shown to be far superior to alternative approaches for prescribing fully developed inlet conditions in such complicated geometries. To facilitate the adoption of our Poisson equation method, we have distributed our OpenFOAM source code and the associated test cases from this study as open-source software.

On the solutions and conservation laws of the 2D breaking soliton equation of fluid mechanics

In this article, we study two-dimensional generalized breaking soliton equation, which describes two-dimensional interchange of Riemann wave disseminating alongside y -axis with a long wave disseminating alongside x -axis. We derive Lie symmetry generators of this nonlinear partial differential equation and then utilize them to perform symmetry reductions. Travelling wave variables are used to obtain most general closed-form solutions of this equation by using two procedures. In addition, we compute the conserved vectors of this equation by engaging the classical Noether’s theorem.

Design of Thin and Wideband Microwave Absorbers Using General Closed-Form Solutions

A new design method for RLC reactive absorbers is presented. This method is based on closed-form solutions to help realize the widest absorption bandwidth for an arbitrarily specified thickness. The solutions for the RLC values of the reactive screen are derived using an equivalent circuit in which the thickness of the substrate used, the permittivity and tangential loss of the substrate, and the frequency are all considered. A perfect match and maximum bandwidth at a design frequency can be achieved with the proposed method. Various aspects of the absorber characteristics, depending on the thickness and loss of the substrate, are analyzed using the presented solutions and electromagnetic (EM) simulations. To validate the proposed design method, an X-band microwave absorber with a crossed-dipole structure patterned on a silver nanowire resistive film is designed, fabricated, and measured. The substrate electrical thickness of the absorber is 70° at 10 GHz, with a permittivity of 2.2. The 90% absorption bandwidth is 8 GHz in the frequency range of 8.2–16.2 GHz. The measured absorption agrees well with the results obtained using circuit and EM simulations.

A mathematical modelling of multiphysics-based propagation characteristics of surface wave in piezoelectric - hydrogel layer on an elastic substrate

This paper presents a mathematical modelling of a novel multiphysics (electro-mechanic coupling) problem of the shear wave propagation in laminated structures (piezoelectric - hydrogel - elastic substrate) using the wave mode method while finding the general solution for each medium. A dynamic mechanical equilibrium equation (for transverse deflection) and Maxwell equation (for electric potential) are solved in a coupled manner over each domain resulting a general closed-form solution. A specific analytical solution is then obtained enforcing the boundary conditions at the top of piezoelectric layer and the interface continuity requirements between each layer and the elastic half-space. A novel approach, to uncouple the coupled equations, is presented resulting in a final systematic solution. The effect of x coordinate (thickness) on the field variables is carefully examined for the electrically open and short cases. The jump in the stress and electric displacement components (causing delamination due to shearing mode of fracture) is present across all the interfaces due to the different bulk material constants. The key contribution of the current work is demonstrating the influence of fixed-charge concentration inside the hydrogel layer on the shear wave propagation. The present study thus provides a new concept for adjusting and controlling the elastic wave propagation in the composite structures, and provides a proper theoretical understanding of wave propagation in the damage tolerance-based design of piezotronics devices.

Symbolic Variable Elimination for Discrete and Continuous Graphical Models

Probabilistic reasoning in the real-world often requires inference incontinuous variable graphical models, yet there are few methods for exact, closed-form inference when joint distributions are non-Gaussian. To address this inferential deficit, we introduce SVE -- a symbolic extension of the well-known variable elimination algorithm to perform exact inference in an expressive class of mixed discrete and continuous variable graphical models whose conditional probability functions can be well-approximated as piecewise combinations of polynomials with bounded support. Using this representation, we show that we can compute all of the SVE operations exactly and in closed-form, which crucially includes definite integration w.r.t. multivariate piecewise polynomial functions. To aid in the efficient computation and compact representation of this solution, we use an extended algebraic decision diagram (XADD) data structure that supports all SVE operations. We provide illustrative results for SVE on probabilistic inference queries inspired by robotics localization and tracking applications that mix various continuous distributions; this represents the first time a general closed-form exact solution has been proposed for this expressive class of discrete/continuous graphical models.

Traveling wave solutions of Benny Luke equation via the enhanced (G'/G)-expansion method

In this article, we execute the enhanced (G'/G)-expansion method to search for new and further general closed-form wave solutions to the nonlinear partial differential equation, namely the Benny Luke equation. This method is one of the powerful techniques that come into view in recent time for establishing more exact wave solutions to nonlinear partial differential equations. We have achieved some new exact solutions including soliton and periodic wave solutions with arbitrary parameters and the solutions are expressed in terms of hyperbolic and trigonometric functions. The efficiency of this method for finding exact solutions has been established. It is shown that the enhanced (G'/G)-expansion method is direct, effective and can be used for many other nonlinear partial differential equations in mathematical physics and engineering.

Adequate wide-ranging closed-form wave solutions to a nonlinear biological model

The auxiliary equation method and the extended tanh function method are put in use to establish wide-ranging, further general and standard closed-form traveling wave solutions to the time fractional nonlinear biological model. The fractional derivative is considered in the sense of beta-derivative. The ascertained solutions are in the form of exponential, rational, hyperbolic and trigonometric functions with significant physical meaning. The achieved solutions demonstrate that the methods are effective, compatible and efficient mathematical tools. We analyze the attained solutions through sketching graphs for the defined values of the associated parameters to perceive the internal structure by making use the Mathematica software. The developed traveling wave solutions might be helpful to illustrate the internal infrastructure of tangible phenomena associated with the consider model.

Closed-form solutions for the coupled deflection of anisotropic Euler–Bernoulli composite beams with arbitrary boundary conditions

The fully anisotropic response of composite beams is an important consideration in diverse applications including aeroelastic responses of helicopter rotor and wind turbine blades. Our goal is to present exact analytical solutions for the first time for coupled deflection of Euler–Bernoulli composite beams. Towards this goal, two approaches are proposed: (1) obtaining the exact analytical solutions directly from the governing equations of Euler–Bernoulli composite beams and (2) extraction of the solutions from Timoshenko composite beam solutions. For the direct solution approach, based on Euler–Bernoulli theory, new variationally-consistent field equations are obtained, in which four degrees of freedom, i.e. in-plane bending, out-of-plane bending, twist and axial elongation are fully coupled. By expressing the coupled system of differential equations in a compact matrix form, a novel expression for the eccentricity of neutral axes from the midplane, as well as the shift in shear centre from the centre of beam, is obtained. This eccentricity matrix serves to decouple the bending in the two principal directions from in-plane and twist deformations. Then, the general closed-form analytical solutions for the decoupled system are derived simply using direct integration. Additionally, the analogous closed-form analytical solutions are retrieved from the previously obtained Timoshenko composite beam solution and it is proven that they are identical to those obtained from the current direct approach for conditions where Euler–Bernoulli beam theory apply. To study the effects of anisotropy, numerical results are obtained for a number of examples with different composite stacking sequences showing various coupled behaviours. The results are compared against the Chebyshev collocation method as well as against less comprehensive analytical solutions available in the literature, noting that excellent agreement is observed, where expected. The present exact solutions can serve as benchmark problems for assessing the accuracy and convergence of various analytical and numerical methods.

Plenty of soliton solutions to the DNA Peyrard-Bishop equation via two distinctive strategies

Here, the Deoxyribo-Nucleic Acid (DNA) dynamic equation that arises from the oscillator chain named the Peyrard-Bishop model for plenty of solitary wave solutions is presented. The efficacy of newly designed algorithms are investigated, namely, the extended Auxiliary equation method and Kudryashov expansion method for constructing the new solitary wave solutions of the DNAdynamic Peyrard-Bishop model with beta-derivative. Here, the proposed methods contribute to a range of accurate solutions for soliton, including light, dark, and other solutions are obtained. In addition, some results are also clarified by computer simulations demonstrating the uniqueness of our work relative to the existing literature on the classic Peyrard-Bishop model. These solutions lead to the issue of the possibility to expand the method to deal with other non-linear equations of fractional space-time derivatives in non-linear science. It is noted that the newly proposed approach is accurate and is used to create new general closed-form solutions for all other fractional NPDEs.

Development of 2-D Generalized Tri-Focal Rotman Lens Beamforming Network to Excite Conformal Phased Arrays of Antennas for General Near/Far-Field Multi-Beam Radiations

In this paper, we present a general closed-form solution of trifocal Rotman lens beamforming network (TFRL-BFN) design to excite a conformal phased array of antennas for multibeam radiations. This TFRL-BFN is a two-dimensional (2-D) configuration and is realizable in a dielectric substrate sandwiched by a pair of metal parallel-plates. In the development, instead of employing the geometric optic (GO) ray paths external to the Rotman lens to form an equal-phase wavefront by equal path lengths for directional beam radiation, we consider the phase need of antenna array excitation as the design target of TFRL-BFN. As a result, the TFRL-BFN concept can be extended to treat conformal arrays of antennas with well-planned excitation schemes, ranging from conventional near- and far-field focus multibeam radiations, as well as multibeam with shaped patterns. Basic theoretical concepts and foundations are presented with simulation and measurement examples to demonstrate the phase production characteristics of this TFRL-BFN. When used to feed a conformal array of antennas, the resulting radiation characteristics validate this TFRL-BFN generalization concept's feasibility.

Uniformly accelerated traveler in an FLRW universe

This paper provides an analytical treatment of accelerated and geodesic motion within the framework of the Friedmann -Lemaitre-Robertson-Walker (FLRW) spacetime. By employing conformal time transformations we manage to convert second order differential equations of motion in FLRW spacetime to first order equations in the conformally transformed spacetime. This allows us to derive a general analytical solution in closed-form for accelerated motion in spatially curved FLRW spacetime. We provide few examples of this general solution for the spatially flat cases. The last part of our work focuses on the return journey for a traveler exploring a FLRW universe. We derive certain condition for de Sitter universe that must be satisfied in order to have an actual return journey.

Quantum Electromagnetic Finite-Difference Time-Domain Solver

We employ another approach to quantize electromagnetic fields in the coordinate space, instead of the mode (or Fourier) space, such that local features of photons can be efficiently, physically, and more intuitively described. To do this, coordinate-ladder operators are defined from mode-ladder operators via the unitary transformation of systems involved in arbitrary inhomogeneous dielectric media. Then, one can expand electromagnetic field operators through the coordinate-ladder operators weighted by non-orthogonal and spatially-localized bases, which are propagators of initial quantum electromagnetic (complex-valued) field operators. Here, we call them QEM-CV-propagators. However, there are no general closed form solutions available for them. This inspires us to develop a quantum finite-difference time-domain (Q-FDTD) scheme to numerically time evolve QEM-CV-propagators. In order to check the validity of the proposed Q-FDTD scheme, we perform computer simulations to observe the Hong-Ou-Mandel effect resulting from the destructive interference of two photons in a 50/50 quantum beam splitter.