Closed-form Solution Research Articles

Localization in Modified Polar Representation: Hybrid Measurements and Closed-Form Solution

Localization in Modified Polar Representation: Hybrid Measurements and Closed-Form Solution

On the resonance frequency of the irregular shaped visco-acoustic Helmholtz cavity problem

In this paper we develop a closed-form mathematical solution for the resonance frequency of the axisymmetric irregular shaped visco-acoustic Helmholtz cavity problem. The solution is based on a conformal mapping that transforms the governing visco-acoustic equations of motion for the irregular cavity geometry into a new domain described by a set of orthogonal curvilinear coordinates. These orthogonal curvilinear coordinates have an important characteristic relative to the transformation of the physical boundary: the mapped domain is constructed such that the locus of all points for the length-wise profile of the physical boundary coincides with a constant value for a (quasi radial) coordinate in the new mapped domain. In these orthogonal curvilinear coordinates we obtain an analytical solution for the visco-acoustic equations of motion and exactly satisfy the no-slip fluid boundary conditions over the irregular shaped cavity walls by matching the coefficients of an orthogonal Legendre polynomial series representation for the non-uniform velocity on the cavity walls. The mathematical solution is an approximation to the extent that the scale factors chosen for the mapping consist of strictly the mutual-coupling terms of the true scale factors. Thus we analyze some results to understand the correlation of the analytical solution frequency response spectra with numerical finite element analyses of the idealized boundary conditions with various irregular shaped cavity geometries. The results for fundamental resonance frequency were found to exhibit good correlation with the results of finite element analyses over a significant range of fluid properties with different irregular cavity geometries. As the primary motivation for the derivation stems from the lack of a rapid yet accurate alternative to computationally intensive MultiPhysics FEA in the formulation of acoustic trendlines for sensor development, the solution’s acoustic impedance was applied as dispersive loading in a simple lumped parameter model of a fluid identification sensor for comparison. The resulting acoustic trendlines of the reduced (5 DOF) model were found to exhibit good correlation with the results of the MultiPhysics FEA (>105 DOF). Thus the solution was found to enable a closed-form analytical method for rapid construction of parametric studies in the practical design of fluid analysis sensors based on complex resonant cavity geometries.

Improved weighted nuclear norm with total variation for removing multiplicative noise

This paper introduces an improved weighted nuclear norm with a total variation model tailored for removing multiplicative noise. The model incorporates a weight matrix to regularize the residual matrix, effectively leveraging image redundancy to differentiate various statistical properties of the noise. Since there is no guarantee of a unique solution, the model is reformulated as a linear equality constraint problem and decomposed into two subproblems. These are addressed by using the alternating direction method of multipliers and the split Bregman method, respectively. In addition, each alternative update step has a closed-form and convergent solution. After obtaining the denoised image in the log-domain, the recovered image is given by using the exponential function and bias correction. Experimental evaluations demonstrate the efficacy of our algorithms in enhancing image restoration quality.

Fourier Domain Robust Denoising Decomposition and Adaptive Patch MRI Reconstruction.

The sparsity of the Fourier transform domain has been applied to magnetic resonance imaging (MRI) reconstruction in k -space. Although unsupervised adaptive patch optimization methods have shown promise compared to data-driven-based supervised methods, the following challenges exist in MRI reconstruction: 1) in previous k -space MRI reconstruction tasks, MRI with noise interference in the acquisition process is rarely considered. 2) Differences in transform domains should be resolved to achieve the high-quality reconstruction of low undersampled MRI data. 3) Robust patch dictionary learning problems are usually nonconvex and NP-hard, and alternate minimization methods are often computationally expensive. In this article, we propose a method for Fourier domain robust denoising decomposition and adaptive patch MRI reconstruction (DDAPR). DDAPR is a two-step optimization method for MRI reconstruction in the presence of noise and low undersampled data. It includes the low-rank and sparse denoising reconstruction model (LSDRM) and the robust dictionary learning reconstruction model (RDLRM). In the first step, we propose LSDRM for different domains. For the optimization solution, the proximal gradient method is used to optimize LSDRM by singular value decomposition and soft threshold algorithms. In the second step, we propose RDLRM, which is an effective adaptive patch method by introducing a low-rank and sparse penalty adaptive patch dictionary and using a sparse rank-one matrix to approximate the undersampled data. Then, the block coordinate descent (BCD) method is used to optimize the variables. The BCD optimization process involves valid closed-form solutions. Extensive numerical experiments show that the proposed method has a better performance than previous methods in image reconstruction based on compressed sensing or deep learning.

Joint Anchor Graph Embedding and Discrete Feature Scoring for Unsupervised Feature Selection.

The success of existing unsupervised feature selection (UFS) methods heavily relies on the assumption that the intrinsic relationships among original high-dimensional (HD) data samples exist in the discriminative low-dimension (LD) subspace. However, previous UFS methods commonly construct pairwise graphs and employ l2,1 -norm regularization to severally preserve the local structure and calculate the score of features, which is computationally complex and easy to get stuck into local optimum, so that those approaches cannot be applied in dealing with large-scale datasets in practice. To overcome this challenge, we propose a novel UFS method, in which a novel anchor graph embedding paradigm is designed to extract the local neighborhood relationships among data samples by reducing the computational complexity of graph construction to be linear in the number of data. Moreover, to improve the optimality of selected features as well as the performance of downstream tasks, we propose a discrete feature scoring mechanism, which imposes orthogonal l2,0 -norm constraints on learned projections, in order to enhance the distinction of feature scores as well as reduce the probability of falling into local optimum. In addition, solving the proposed nonconvex and nonsmooth NP-hard problem is challenging, and we present an efficient optimization algorithm to address it and acquire a closed-form solution of the transformation matrix. Extensive experiments demonstrate the effectiveness and efficiency of the proposed UFS by comparison with several state-of-the-art approaches to clustering and image segmentation tasks.

Closed-form asymptotic solution for the transport of chlorine concentration in composite pipes

Recently, some researchers have revisited the analysis of chlorine transportation in cylindrical pipes by deploying a coupling between the Laplace transform method and the complex analysis’ residue approach for inverting complex integrals. This method yielded interesting results after the incorporation of root-finding numerical schemes. Thus, away from incorporating numerical tools, the present study makes consideration of the same formulation of chlorine transport in a single-layered pipe and further extends it to the case of a bi-layered pipe using the hybrid of the Laplace transform method and the asymptotic approximations method. The need for asymptotic approximations for the modified Bessel functions, which arise in the reduced ordinary differential equations, necessitates the quest for closed-form analytical solutions, which are largely considered benchmark solutions for numerical investigations. Moreover, the obtained closed-form asymptotic solutions have been examined graphically; where it was observed that both the radial diffusion coefficient η and the spatial radial variable are contributory in the transport of chorine concentration in the media.

Nonlinear dynamic wave characteristics of optical soliton solutions in ion-acoustic wave

Traveling wave solutions are utilized to depict reaction-diffusion and analyze electrical signal transmission and propagation in space-time nonlinear fractional order partial differential equations like the space-time fractional Telegraph and Kolmogorov-Petrovsky Piskunov equations, and that are used in physical science to model combustion, biological research to model nerve impulse propagation, chemical dynamics to model concentration in order wave propagation, and plasma to model the progression of a set of duffing oscillators. In this study, the new generalized (G′/G)−expansion technique was employed to construct some novel and more universal closed-form traveling wave solutions in the sense of conformable derivatives which explain the above-stated phenomena properly. By utilizing complex fractional transformation, the ordinary differential equations are generated from fractional order differential equations. The recommended technique allowed us to produce some dynamical wave patterns of kink, single soliton, compacton, periodic shape, multiple periodic waves, anti-kink, and other structures are developed, which are shown using 3D plots and contour plots to illustrate the physical layout clearly. The traveling waveform responses can be defined in terms of functions based on trigonometry, hyperbolic operations, and rational functions and that are quick, flexible, and simple to reproduce. Furthermore, the obtained closed-form solutions for nonlinear fractional evolution equations make stability analysis and accuracy comparison amongst numerical solvers easier, which asserts that the new generalized (G′/G)−expansion technique is one of the most proficient and effective approaches.

Estimating Position-Dependent and Anisotropic Diffusivity Tensors from Molecular Dynamics Trajectories: Existing Methods and Future Outlook.

Confinement can substantially alter the physicochemical properties of materials by breaking translational isotropy and rendering all physical properties position-dependent. Molecular dynamics (MD) simulations have proven instrumental in characterizing such spatial heterogeneities and probing the impact of confinement on materials' properties. For static properties, this is a straightforward task and can be achieved via simple spatial binning. Such an approach, however, cannot be readily applied to transport coefficients due to lack of natural extensions of autocorrelations used for their calculation in the bulk. The prime example of this challenge is diffusivity, which, in the bulk, can be readily estimated from the particles' mobility statistics, which satisfy the Fokker-Planck equation. Under confinement, however, such statistics will follow the Smoluchowski equation, which lacks a closed-form analytical solution. This brief review explores the rich history of estimating profiles of the diffusivity tensor from MD simulations and discusses various approximate methods and algorithms developed for this purpose. Besides discussing heuristic extensions of bulk methods, we overview more rigorous algorithms, including kernel-based methods, Bayesian approaches, and operator discretization techniques. Additionally, we outline methods based on applying biasing potentials or imposing constraints on tracer particles. Finally, we discuss approaches that estimate diffusivity from mean first passage time or committor probability profiles, a conceptual framework originally developed in the context of collective variable spaces describing rare events in computational chemistry and biology. In summary, this paper offers a concise survey of diverse approaches for estimating diffusivity from MD trajectories, highlighting challenges and opportunities in this area.

Closed-Form Solution for Scale-Dependent Frequencies of Shear Deformable Cellular Microplates Coated by Piezoelectric Polymeric Skins Consisting FG Distributed CNTs

Closed-Form Solution for Scale-Dependent Frequencies of Shear Deformable Cellular Microplates Coated by Piezoelectric Polymeric Skins Consisting FG Distributed CNTs

Analytical poro‐elastic solution of deep lined tunnels in anisotropic rock with consideration of tunnel face advance

AbstractThis paper aims at deriving a closed‐form solution for deep lined tunnels within a saturated anisotropic poro‐elastic medium. The derivation of the solution is carried out with the assumption of plane strain conditions along the tunnel axis, an isotropic elastic liner, a steady‐state flow, and perfect contact between the liner and rock. For this purpose, the complex potential function approach pioneered by Lekhnitskii for the anisotropic elastic body was used to develop the mechanical response of the rock mass. A complex hydraulic potential function is also introduced to solve the steady state fluid flow. Then the well‐known Biot theory is chosen to describe the hydro‐mechanical coupling of the anisotropic saturated rock. The stress and displacement solutions of both the liner and the surrounding rock, considering the tunnel face advance with respect to the considered section, are derived based on the principle of the convergence‐confining method. Comparisons with previous closed‐form solutions for isotropic case and finite element solution for anisotropic case were made to show the accuracy of the current closed form solution. Sensitivity analysis is performed based on this analytical solution to highlight the effect of some rock parameters on the stress and displacement of the liner.

A neural network inverse problem solution for a MIMO magnetic levitation actuator

A multiple-input and multiple-output planar actuator is proposed, which utilizes mechanically driven stator magnet arrays to levitate a permanent magnet mover. A state of levitation and actuation is obtained by mechanically altering the orientation of the stator magnets to control the forces and torques on the mover. A challenge for the design and control of the actuator is inverting the relationship between the force and stator magnet rotation angles, as there is no closed-form analytical solution. In this study, a feed-forward neural network is applied to model the forward relation between stator magnet angle input and a force and torque output to reduce the forward computation time for the design process and for error estimation in real-time applications. Additionally, the neural network is considered for inverting the solution for a motion profile sampled at 1000 Hz. The developed forward model is able to calculate the forces and torques on the mover a factor 10 faster than the equivalent charge or Fourier model with an absolute error of 3 mN and 0.1 mNm for the forces and torques, respectively, and a feed-forward neural network is able to accurately learn an inverse solution for small motion profiles.

Optimal investment for asset–liability management with delay and partial information under Ornstein–Uhlenbeck process

In this paper, we investigate the optimal investment strategy of asset liability management (ALM) with bounded memory and partial information. Suppose that investors invest their assets in a financial market consisting of a risk-free bond and a risk-free stock, while also taking on liabilities, in which the value of liabilities and the price of risky assets satisfy the Ornstein–Uhlenbeck (O–U) processes whose drift terms are unobserved. By constructing a dynamic portfolio of risk-free bonds, risky stocks and liabilities, a stochastic delay differential equation is obtained to depict the surplus process of investor. The ALM problem is formulated as finding the best strategy to maximize the terminal utility of the sum of terminal surplus and some historical wealth under partial information, and the corresponding full information case is also studied as a supplement. For both cases of partial information and full information, we apply the dynamic programming method to derive HJB equations, verification theorems, and closed-form solutions of optimal strategies and value functions. Moreover the relationship between optimal strategy and value function under full information and partial information is also given. Finally, numerical examples are carried out to illustrate the influence of some important parameters on the obtained results.

Using Rotations to Control Observable Relativistic Effects

This paper examines the possibility of controlling the outcome of measured (flat space-time) relativistic effects, such as time dilation or length contractions, using pure rotations and their nontrivial interactions with Lorentz boosts in the isometry group SO+(3,1). In particular, boost contributions may annihilate leaving only a geometric phase (Wigner rotation), which we see in the complex solutions of the generalized Euler decomposition problem in R3. We consider numerical examples involving specific matrix factorizations, along with possible applications in special relativity, electrodynamics and quantum scattering. For clearer interpretation and simplified calculations we use a convenient projective biquaternion parametrization which emphasizes the geometric phases and for a large class of problems allows for closed-form solutions in terms of only rational functions.

Multiple scattering of local nonlinear resonators on a thin plate

Developing nonlinear resonant metamaterial plates requires an effective wave simulation method to analyze the wave interactions between multiple nonlinear scatterers. The multiple scattering method has the advantages of high computational efficiency and closed-form displacement solutions in modeling the finitely periodic scatterer array. However, the multiple scattering method of nonlinear scatterers is absent in the available studies of plate structures. In this paper, a nonlinear multiple scattering approach is formulated and analyzed for thin plates with nonlinear resonant scatterers. The resonant scatterer consists of an inner plate with a local resonator modeled by the nonlinear mass-spring system. The motion equation of resonant scatterers is solved using the perturbation technology and wave expansion method. The T-matrix of the resonant scatterer of each order is then derived at the arbitrary harmonic frequency from continuous interfacial conditions of resonant scatterers. Finally, the multiple scattering is formulated for a finitely periodic array of resonant scatterers on a thin plate. Numerical simulations capture the scattering characteristics and stop-band formations of fundamental and second-harmonic waves. Furthermore, the finitely periodic array of nonlinear and linear resonant scatterers achieves the nonreciprocal transmission of flexural waves. These numerical results demonstrate the potential applications of the proposed method in designing metamaterial-based plates with complex transmission properties.

Analysis of shock wave propagation in two-layered blood flow model via Lie symmetry

In this paper, we study the two-layered blood flow model within the framework of Lie symmetry analysis to investigate a range of closed-form solutions. Our investigation involves optimal classification, revealing the existence of six optimal subalgebras within the model. Further, we streamline the model by reducing it into a system of ordinary differential equations (ODEs), thereby deriving several exact solutions. Our analysis establishes the presence of the nonlinear self-adjointness property in the governing equation, leading to the development of various conservation laws. Furthermore, leveraging one of the exact solutions we obtained, we delve into the behaviour of characteristic shocks, C1 waves, and their interactions, providing a comprehensive understanding of their dynamics.

Linear and nonlinear vibrations of nonlinearly elastically constrained functionally graded porous microbeams with rough surface

Linear and nonlinear free vibrations of nonlinearly elastically constrained functionally graded porous (FGP) microbeams with rough surface are analytically studied in this paper via the modified couple stress theory. A surface roughness model of the FGP microbeams is developed. The material properties including porous property are supposed to continuously vary along the thickness direction. The higher-order nonlinear governing equation and elastic boundary conditions are derived through the Hamilton principle. A closed-form solution for linear frequency and mode shape is presented. Subsequently, an analytical solution for nonlinear frequency is achieved by employing the Galerkin procedure. Finally, the effects of surface roughness, size parameter, power-law index, porosity volume fraction, and nonlinearly elastic constraints on the linear and nonlinear vibrations of functionally graded porous microbeams are examined.

Fermion-antifermion pairs in a magnetized space-time with non-zero cosmological constant

We analyze a fermion-antifermion pair within the (2+1)-dimensional Bonnor-Melvin magnetic (BMM) spacetime featuring a nonzero cosmological constant (Λ>0). To do this, we rigorously investigate analytical solutions to the corresponding form of a fully-covariant two-body Dirac equation. We derive a non-perturbative wave equation and obtain an exact closed-form analytical solution for the system under consideration. In this magnetic background spacetime, described by a surface with positive Gaussian curvature, we noticed that fermion-antifermion pairs display a collective behavior resembling that of a single quantum oscillator, carrying a total rest mass of the particles and oscillating with a frequency (ωo∝Λ), even in the absence of mutual interaction between the particles. This indicates a compelling correlation between the cosmological constant and the energy spectrum of fermion-antifermion systems, implying connections between quantum realms and cosmology.

A novel analytical technique for analyzing the (3+1)-dimensional fractional calogero- bogoyavlenskii-schiff equation: investigating solitary/shock waves and many others physical phenomena

This work presents a thorough analysis of soliton wave phenomena in the (3+1)-dimensional Fractional Calogero-Bogoyavlenskii-Schiff equation (FCBSE) with Caputo’s derivatives through the use of a novel analytical technique known as the modified Extended Direct Algebraic Method (mEDAM). By converting nonlinear Fractional Partial Differential equations (FPDE) into integer-order Nonlinear Ordinary Differential equations (NODE), and then using closed-form series solutions to translate the NODE into an algebraic system of equations, this method allows us to derive families of soliton solutions, which include kink waves, lump waves, breather waves, and periodic waves, exposing new insights into the behavior and distinctive features of soliton waves in the FCBSE. By including contour and 3D graphics, the behaviors of a few selected soliton solutions are well depicted, showcasing their amplitude, shape, and propagation characteristics. The results enhance our understanding of the FCBSE and show that the mEDAM is a valuable tool for studying soliton wave phenomena. This work creates new opportunities for studying wave phenomena in more intricately constructed nonlinear FPDEs (NFPDEs).

Anisotropic functionally graded nano-beam models and closed-form solutions in plane gradient elasticity

This study delves into the investigation of exact analytical solutions for the plane stress and displacement fields within linear homogeneous anisotropic nano-beam models of gradient elasticity. It focuses on solving the Helmholtz equation, which encompasses a second-order non-homogeneous linear partial differential equation in plane gradient elasticity theory, utilizing polynomial series-type solutions. The analysis centers on the utilization of gradient Airy stress functions to derive stress fields in the gradient theory. The research yields closed-form analytical solutions for Airy stress functions, stress, and strain fields in both classical and gradient theories. The study considers five distinct types of two-dimensional functionally graded cantilever beams with various boundary conditions: a cantilever anisotropic nano-beam subjected to a concentrated force at the free end, a cantilever anisotropic nano-beam under a uniform load, a simple anisotropic nano-beam under a uniform load, a propped cantilever anisotropic nano-beam under a uniform load, and a fixed end cantilever anisotropic nano-beam under a uniform load. General analytical solutions for the gradient stress and displacement fields of two-dimensional and one-dimensional anisotropic nano-beams under different boundary conditions are provided. The study showcases significant strain gradient size effects at the nano-scale through the derived analytical solutions for anisotropic beams. Additionally, it demonstrates that the strain gradient theory results for the limit case of the gradient coefficient c precisely align with results for isotropic and anisotropic materials in elasticity theory and classical theory. Furthermore, real-world applications are discussed, considering stress and displacement fields in real anisotropic materials such as TiSi2 single crystals and orthotropic materials, which are special cases of anisotropic materials like wood and epoxy as documented in the literature.

Anisotropic Hyperelastic Strain Energy Function for Carbon Fiber Woven Fabrics.

The present paper introduces an innovative strain energy function (SEF) for incompressible anisotropic fiber-reinforced materials. This SEF is specifically designed to understand the mechanical behavior of carbon fiber-woven fabric. The considered model combines polyconvex invariants forming an integrity basisin polynomial form, which is inspired by the application of Noether's theorem. A single solution can be obtained during the identification because of the relationship between the SEF we have constructed and the material parameters, which are linearly dependent. The six material parameters were precisely determined through a comparison between the closed-form solutions from our model and the corresponding tensile experimental data with different stretching ratios, with determination coefficients consistently reaching a remarkable value of 0.99. When considering only uniaxial tensile tests, our model can be simplified from a quadratic polynomial to a linear polynomial, thereby reducing the number of material parameters required from six to four, while the fidelity of the model's predictive accuracy remains unaltered. The comparison between the results of numerical calculations and experiments proves the efficiency and accuracy of the method.