Analytical Solution Research Articles

Analytical Solution for Transient Two-Dimensional Filtration nearby a Fracture of Finite Conductivity

Analytical Solution for Transient Two-Dimensional Filtration nearby a Fracture of Finite Conductivity

Closed-form analytical solution for the transfer matrix based on Pendry-MacKinnon discrete Maxwell's equations

Closed-form analytical solution for the transfer matrix based on Pendry-MacKinnon discrete Maxwell's equations

Acoustic analysis of a three-dimensional cylindrical shell model under electromagnetic vibration

This paper presents the acoustic analysis of a three-dimensional cylindrical shell model under electromagnetic vibration, a critical factor affecting the performance of electric motors in various applications such as automotive, aerospace, and industrial systems. The study provides a multidisciplinary approach that integrates electromagnetics, structural vibration, and acoustics, solved using the fast multipole boundary element method (FMBEM). The results summarize the validation of the analytical models and numerical simulations, offering insights into effective vibration reduction methods. The conclusions indicate that the 3-D numerical analysis using FMBEM aligns well with the analytical solution for the sound pressure in the exterior acoustic domain of the cylindrical shell model. The paper contributes valuable insights for the design of low-noise motors and the control of electromagnetic vibration and noise in electric motors.

Analytical solution of surrounding rock stress and plastic zone of rectangular roadway under non-uniform stress field

The shape and extent of the plastic zone can effectively reflect the degree of deformation and failure of the surrounding rock. By applying complex variable functions and the theory of elasticity, a mechanical model for the surrounding rock of a rectangular roadway is established. The analytical solution for the stress in the surrounding rock of a rectangular roadway under a non-uniform stress field is derived, and the spatial stress distribution pattern in the surrounding rock of the rectangular roadway is obtained. Furthermore, using the Mohr-Coulomb criterion, the boundary equation of the plastic zone in the surrounding rock of a rectangular roadway is derived, and the morphological characteristics of the plastic zone are obtained. The results indicate that the plastic zones on both sides of the rectangular roadway are arc-shaped. The asymmetry of the stress environment has a significant impact on the plastic zones in the roof and floor of the rectangular roadway, causing the shape of the plastic zones to resemble a “petal” pattern. The lateral pressure coefficient and the height-to-width ratio have a significant impact on the stress distribution and the morphological characteristics of the plastic zone in the surrounding rock of the rectangular roadway. By validating the computational accuracy of the analytical solution through numerical simulations and conducting an engineering applicability analysis based on a coal mine roadway, it has been demonstrated that the analytical calculation method is both rational and possesses good engineering applicability. This provides reliable theoretical guidance for the design calculations and construction planning of similar roadway projects.

A Novel Approach for Time-Local Fractional Solutions of Certain Nonlinear Partial Differential Equations in Fractal Dimension

Time-local fractional approaches for nonlinear partial differential equations in fractal dimensions are essential for capturing the complex, irregular behaviors found in fractal systems. In this paper, a new modification of the local fractional Laplace variational iteration method (MLFLVIM) for obtaining analytical approximate solutions to the fractional gas dynamics equation, fractional Stefan equation, and fractional Newell-Whitehead-Segel equation within the context of fractal time space is presented. The proposed method (MLFLVIM) elegantly combines the local fractional Laplace transform (LFLT) with modified variational iteration method. Specifically, we first apply the (LFLT) to the given local fractional PDEs, yielding a transformed system of equations. We then apply modified variational iteration to this system. Finally, we use the inverse of (LFLT) to obtain the desired solution. To demonstrate the effectiveness of this approach, we implement it on three numerical physical problems. The results show that the (MLFLVIM) can successfully handle these nonlinear LFPDEs and provide accurate analytical approximation solutions.

Propagation of traveling wave solution of the strain wave equation in microcrystalline materials

Abstract This study focuses on the propagation behavior of traveling wave solution in microcrystalline materials using the polynomial complete discriminant system method. By establishing a complete discriminant system, we systematically analyze the formation and evolution process of traveling wave solution in microcrystalline materials. Specifically, we apply the cubic polynomial extension to the strain wave equation to obtain more accurate analytical solutions. Additionally, two-dimensional, three-dimensional, and contour plots are generated to visually illustrate the characteristics of the obtained solutions, facilitating a more intuitive understanding of their physical significance. These findings not only help reveal the propagation mechanism of traveling wave solution but also provide a theoretical foundation for the application of microcrystalline materials.

Modeling of Heat Transfer and Airflow Inside Evacuated Tube Collector With Heat Storage Media: Experimental Validation Powered by Artificial Neural Network

ABSTRACTSolar air heater (SAH) is a widely employed technology for harnessing solar thermal energy in numerous applications. Contemporary research optimizes designs within identical spatial constraints to maximize energy output. However, there is a recognized need to conduct analytical validation to stimulate the experimental setup and formulate an artificial neural network (ANN) model to govern and predict the operation system. This investigation involved developing and assessing an evacuated tube solar air heater (ETSAH) integrated with annulus‐filled heat storage media. Furthermore, this study introduced an ANN model and analytical solution to predict performance parameters, representing a noteworthy contribution. The proposed ANN model achieved its optimal validation performance with a mean square error of 5.4018 × 10−6 after 11 epochs within 31 of training. Also, correlation findings show that the optimal architecture of a feed‐forward backpropagation is achieved when the 5‐40‐40‐40‐2 model architecture is used. In this case, there are five neural nodes in the input layer that represent timing, temperature, radiation, and airflow rate. The power output of the ETSAH device shows a strong correlation with the flow rate, reaching its peak at 0.05 kg/s with a value of 2261 W and dropping to 368 W at 0.006 kg/s. Correspondingly, the greatest energy efficiency was measured at airflow rates of 0.05, 0.01, and 0.006 kg/s, accounting for 48.38%, 27.32%, and 19.65%, respectively.

Nonequilibrium Molecular Dynamics Method to Generate Poiseuille-Like Flow between Lipid Bilayers.

There are various flows inside and outside cells in vivo. Nonequilibrium molecular dynamics (NEMD) simulation is a useful tool for understanding the effects of these flows on the dynamics of biomolecules. We propose an NEMD method to generate a Poiseuille-like flow between lipid bilayers. We extended the conventional equilibrium MD method to produce a flow by adding constant external force terms to the water molecules. Using the Lagrange multiplier method, the center of mass of the lipid bilayer is constrained so that the flow does not sweep away the lipid bilayer, but the individual lipid molecules fluctuate. The temperature of the system is controlled properly in the solution and membrane by using the Nosé-Hoover thermostat. We found that the flow velocity increases linearly as the applied external force term increases. It is possible to estimate the appropriate value of acceleration to generate a flow with an arbitrary velocity using this proportional relation once a single short MD simulation is performed. We also found that the flow between two lipid bilayers is slower than the analytical solution of the Navier-Stokes equations between rigid parallel plates due to the interactions between water molecules and the membrane. This method can be applied not only to a flow on lipid membranes but also to a flow on soft surfaces generally.

Effects of rotation and chemical reactions on an electroconvection in Maxwell dielectric nanofluids within anisotropic porous media

ABSTRACT This study explores the complex interactions between rotation, chemical reactions, and electroconvection in Maxwell dielectric nanofluids within anisotropic porous media. The research aims to elucidate how these factors influence the heat transfer efficiency and fluid dynamics under the influence of external alternating current (AC) electric fields. The normal mode method is employed to analyse the stability of the system, allowing for the examination of oscillatory modes and their growth rates. By introducing perturbations in the physical parameters, the study derives a set of ordinary linear differential equations that characterize the system's response to external influences. The coupled differential equations were solved using the Galerkin approach for first approximation while meeting the necessary boundary conditions. Analytical solution of the representative equation for stress-free boundary states yields Rayleigh number formulas for nonoscillatory and oscillatory mode occur. Oscillatory modes are seen for both bottom and top-heavy nanoparticle distributions. The electric Rayleigh number, thermal Prandtl number, and stress relaxation parameter increases, whereas the Brinkman-Darcy number delays the onset of stationary and oscillatory convection. The study finds that positive concentration Rayleigh numbers stabilize the system, while negative ones lead to instability. Additionally, it demonstrates that external alternating current electric fields influence convection behavior and provides formulas for critical Rayleigh numbers, enhancing our understanding of fluid interactions in various engineering and technological applications.

Design of optimum spacecraft reorientation control with a combined criteria of quality based on the quaternions

Specifical original problem of attitude controlling for spacecraft was proposed in this paper. Problem of optimal rotation from a known initial state in a prescribed spatial orientation was studied in detail (turnaround time is not fixed). Design of optimal program of reorientation is based on new indicator of quality that combines energy costs including the contribution of controlling torques and integral of rotary energy (in a known proportion) and reorientation time; presence of duration factor bounds time of rotation finish. To construct an optimal control of angular momentum changing, quaternionic method and the maximum principle were applied. Differential equation that relates spacecraft angular momentum and quaternion of spacecraft orientation is a base to obtain analytic solution to a problem. We reveal the properties of optimum control program analytically, and study key features of optimum motion in details. Also, we write the formalized equations, mathematical formulas to design optimal law for change of spacecraft’s angular momentum. Analytic relations and equations are given for finding the optimal solution. Control law (in as explicit dependence between phase variables and con-trolling variables has been formulated. Main relations determining optimum values of parameters for rotation control algorithm were given. The closed-form law for rotation was obtained for dynamically symmetric solids. Numerical example as well as results of mathematical modeling of spacecraft motion that formed using optimum control are presented. This data as addition to the made theoretic descriptions shows reorientation process (in virtual form) and demonstrates practical feasibility of the developed control method. A designed algorithm for optimal control of rotation improves an efficiency of attitude system, and originates more economical performing of space vehicle during its flight along orbit.

Numerical solution based on the Haar wavelet collocation method for partial integro-differential equations of Volterra type

In this paper, a numerical investigation of a class of parabolic Volterra integro-differential equations has been carried out. Basically, the finite difference method associated with the Haar wavelet collocation technique is pursued. The main idea relies on semi-discretizing the parabolic-VIDEs by considering the finite differences in time and approximating the integral term within the trapezoidal rule. Afterwards, using a uniform mesh, the spatial derivative is approximated by the Haar wavelet method. Stability and convergence of the proposed approach are established in the frame of Sobolev space. Numerical examples are used to illustrate the effectiveness of this approach under different test scenarios. The error analysis, using both L 2 and L ∞ norms, shows that the method has high efficiency computationally with a great deal of accuracy. It is observed that these numerical results are in excellent agreement with the analytical solution. In addition, the methods and results obtained in this study are compared with those reported in recent literature.

Nonlinear Dynamics of an Electromagnetically Actuated Cantilever Beam Under Harmonic External Excitation

The present work is devoted to the study of nonlinear vibrations of an electromagnetically actuated cantilever beam subject to harmonic external excitation. The soft actuator that controls the vibratory motion of such components of a robotic structure led to a strongly nonlinear governing differential equation, which was solved in this work by using a highly accurate technique, namely the Optimal Auxiliary Functions Method. Comparisons between the results obtained using our original approach with those of numerical integration show the efficiency and reliability of our procedure, which can be applied to give an explicit analytical approximate solution in two cases: the nonresonant case and the nearly primary resonance. Our technique is effective, simple, easy to use, and very accurate by means of only the first iteration. On the other hand, we present an analysis of the local stability of the model using Routh–Hurwitz criteria and the eigenvalues of the Jacobian matrix. Global stability is analyzed by means of Lyapunov’s direct method and LaSalle’s invariance principle. For the first time, the Lyapunov function depends on the approximate solution obtained using OAFM. Also, Pontryagin’s principle with respect to the control variable is applied in the construction of the Lyapunov function.

On the Andromeda-Milky Way Future Encounter: Thrice Faster Over Time

This inquiry accurately provides an analytical solution of the orbital free-falling time for the system of galaxies Andromeda-Milky Way within a non-expanding tired-light-dominated framework. The mathematical and physical background involves a two-body problem based on the orbital dynamics in which, in the first scenario, the gravitational interaction between local galaxies dominates over the expanding space. The latter is the standard astrophysical approach undertaken by public fund-based universities all over the world. However, in the second alternative scenario, Andromeda’s blueshift has to be corrected for the apparent recession velocity provided by the photon energy loss as a result of multiple interactions between photons and crystallized electrons through the intergalactic medium provided by a Tired Light, specifically, a New Tired Light process. This leads towards a substantial temporal deviation between the classic research approach and this unconventional and independent mindset. Hence, the outcome gives out a disagreement consisting in an orbital free-falling time equal to 1.7 rather than 4.5 billion years. Accordingly, the encounter of the two main galaxies of the Local Group will occur much earlier than estimated thus far. Accordingly, this implies an urgent shift in the scientific mindset as well as a course change in the application of the boundary conditions into forthcoming computational methods.

Analytical solution of time-fractional Emden-Fowler equations arising in astronomy and mathematical physics

The time-fractional Emden-Fowler model is an interesting mathematical model because it is widely used and important in mathematics and applied sciences. The current investigation compiles the Elzaki transform homotopy perturbation method (ET-HPM) and the Laplace transform homotopy perturbation method (LT-HPM) for the analytical solution of the considered model. In their developments, we show that fractional orders are simple to compute, and the resulting series lead to accurate results after a few iterations. We consider the fractional orders in the context of the Caputo form. The primary aim of these approaches is to compare the analytical results of the models under consideration. These schemes do not impose any constraints on the parameters or make any assumptions that could complicate the process of obtaining the analytical results. We show that the derived outcomes are in convergent sequence with a rapid iterative formula. The iterative series and the graphical illustrations in various fractional orders confirm the efficiency of the proposed approaches. The derived results are also compared to the precise solutions, which show that our suggested approaches are simple, effective, and accurate.

An analytical solution of MHD non-Newtonian fluid flow over a stretching surface with mixed convection and porous media

An analytical solution of MHD non-Newtonian fluid flow over a stretching surface with mixed convection and porous media

Optimal sampling strategy for probability estimation: An application to the Agricultural Quarantine Inspection Monitoring program.

Imported agricultural pests can cause substantial damage to agriculture, food security, and ecosystems. In the United States, the Agricultural Quarantine Inspection Monitoring (AQIM) program conducts random sampling to estimate the probabilities that cargo and passengers arriving at ports of entry carry pests. Assessing these risks accurately is critical to enable effective policies and operational procedures. This study introduces a pathway-level analysis with an objective function aligned with AQIM's goal, offering a new perspective compared to the current container-by-container approach, which relies on heuristics to set inspection rates. We formulate an optimization model that minimizes the mean squared error of the probability estimates that AQIM obtains. The central decision-making tradeoff that the model explores is whether it is preferable to sample more arriving containers (and fewer boxes per container) or more boxes per container (and fewer containers), given limited resources. We first derive an analytical solution for the optimal sampling strategy by leveraging several approximations. Then, we apply our model to a numerical case study of maritime cargo sampling at the Port of Long Beach. Across a wide range of parameter settings, the optimal strategy samples more containers (but fewer boxes per container) than the current AQIM protocol. The difference between the two strategies and the accuracy improvement with the optimal approach are larger if the pest statuses of boxes in the same container are more strongly correlated. We recommend that AQIM record box-level (beyond only container-level) inspection data, which could be used to estimate this correlation and other modelparameters.

Data management solutions for real-time analytics in retail cloud environments

In today's rapidly evolving retail landscape, the importance of real-time data analytics has surged, driven by the need for businesses to make informed decisions and enhance customer experiences. This paper explores the critical role of effective data management solutions within cloud environments, focusing on retailers' unique challenges, such as data integration, latency, security, and storage costs. It presents practical solutions, including data partitioning, caching, AI-driven analytics, and the strategic use of data warehousing and data lakes. By adopting hybrid data management models, retailers can optimize their data processes and leverage real-time insights to drive operational efficiency and competitive advantage. This paper offers valuable recommendations for retail businesses and data architects to enhance their cloud-based data management strategies, ensuring they can navigate the complexities of real-time data effectively and sustainably. Keywords: Real-time analytics, Data management, Retail, Cloud environments, Data integration, AI-driven solutions.

Eccentric mergers in AGN Discs: Influence of the supermassive black-hole on three-body interactions

Abstract There are indications that stellar-origin black holes (BHs) are efficiently paired up in binary black holes (BBHs) in Active Galactic Nuclei (AGN) disc environments, which can undergo interactions with single BHs in the disc. Such binary-single interactions can potentially lead to an exceptionally high fraction of gravitational-wave mergers with measurable eccentricity in LIGO/Virgo/KAGRA. We here take the next important step in this line of studies by performing post-Newtonian N-body simulations between migrating BBHs and single BHs set in an AGN disc-like configuration, with a consistent inclusion of the central supermassive black hole (SMBH) in the equations of motion. With this setup, we study how the fraction of eccentric mergers varies in terms of the initial size of the BBH semi-major axis relative to the Hill sphere, as well as how it depends on the angle between the BBH and the incoming single BH. We find that the fraction of eccentric mergers is still relatively large, even when the interactions are notably influenced by the gravitational field of the nearby SMBH. However, the fraction as a function of the BBH semi-major axis does not follow a smooth functional shape, but instead shows strongly varying features that originate from the underlying phase-space structure. The phase-space further reveals that many of the eccentric mergers are formed through prompt scatterings. Finally, we present the first analytical solution to how the presence of an SMBH in terms of its Hill sphere affects the probability for forming eccentric BBH mergers through chaotic three-body interactions.

Mathematical model of velocity and distribution law in gas lift reverse circulation well washing flow field for drilling shaft sinking

To reveal the velocity distribution law of a gas lift reverse circulation well washing flow field in drilling shaft sinking, a velocity mathematical model of well washing flow field is established. The theoretical analytical solutions and velocity distribution laws in the drill pipe and bottom hole were provided and validated through numerical simulations and similar model tests. Furthermore, this study explores the impacts of mud circulation, wind pressure, rock debris, and mud properties on the flow field distribution, presenting the sensitivity order of the factors. This study indicated the following findings. (1) The velocity theoretical model of well washing flow field was validated through numerical simulation and similar model tests, and the relative error ranged from 4.2 to 18.5%, demonstrating the high consistency. (2) The rock debris and mud experienced slight and consistent deceleration within the drill pipe. Upon reaching the gas input end, the axial upward return of the multiphase flow resulted in a “jumping” sharp increase. (3) The mud circulation, wind pressure, and void fraction positively correlated with the multiphase fluid velocity in the drill pipe, but negatively correlated with the mud density and slag content. The axial upward velocity of the rock debris decreased with increasing particle size and density. (4) The sensitivity of each factor to the discharge speed of rock debris was summarized as follows: air content > mud circulation > wind pressure > mud density > rock debris density > rock debris particle size > slag content. These findings could offer a theoretical basis for selecting well washing parameters and enhancing drilling and washing efficiency.

Lie-symmetry analysis of a three dimensional flow due to unsteady stretching of a flat surface with non-uniform temperature distribution

Various mathematical frameworks have been developed to study dynamics of the fluid flow in stretching films. In this work, a three–dimensional flow induced by an unsteady stretching of an infinite flat sheet is analyzed through the Lie symmetry approach. Twelve Lie point symmetries for the nonlinear partial differential equations describing the considered flow and heat transfer phenomena are derived. Invariants for these Lie symmetries are obtained to construct a new class of similarity transformations. With the deduced similarity transformations, the flow equations are converted to ordinary differential equations which are solved using the Homotopy analysis method. The deduced analytic solution enables an exploration of the effects of various parameters on the flow and heat transfer, which have not been revealed previously using the Lie method as per the authors’ knowledge. The influences of the stretching rate, parameters that maintain the non-uniformity and unsteadiness of sheet temperature, and the Prandtl number, are demonstrated with the help of graphs and tables. These results show that for certain values of the system parameters, heat transfer reverses its direction and occurs from the fluid to the sheet. At these values, maximum heat transfer does not occur at the sheet but rather slightly above it.