Analytical Solution Method Research Articles

Analytical Solutions and Computer Modeling of a Boundary Value Problem for a Nonstationary System of Nernst–Planck–Poisson Equations in a Diffusion Layer

This article proposes various new approximate analytical solutions of the boundary value problem for the non-stationary system of Nernst–Planck–Poisson (NPP) equations in the diffusion layer of an ideally selective ion-exchange membrane at overlimiting current densities. As is known, the diffusion layer in the general case consists of a space charge region and a region of local electroneutrality. The proposed analytical solutions of the boundary value problems for the non-stationary system of Nernst–Planck–Poisson equations are based on the derivation of a new singularly perturbed nonlinear partial differential equation for the potential in the space charge region (SCR). This equation can be reduced to a singularly perturbed inhomogeneous Burgers equation, which, by the Hopf–Cole transformation, is reduced to an inhomogeneous singularly perturbed linear equation of parabolic type. Inside the extended SCR, there is a sufficiently accurate analytical approximation to the solution of the original boundary value problem. The electroneutrality region has a curvilinear boundary with the SCR, and with an unknown boundary condition on it. The article proposes a solution to this problem. The new analytical solution methods developed in the article can be used to study non-stationary boundary value problems of salt ion transfer in membrane systems. The new analytical solution methods developed in the article can be used to study non-stationary boundary value problems of salt ion transport in membrane systems.

Predicting the collapse direction of large-scale pulsating bubbles based on Kelvin impulse theory

Predicting the collapse direction of large-scale pulsating bubbles is crucial for evaluating the safety performance of marine vessels in ocean applications involving underwater explosions and high-pressure bubble detection. This study develops a rapid forecasting method for large-scale pulsating bubbles under the influence of hybrid boundaries (free surface, bottom, and sidewall) based on the Kelvin impulse theory. The boundary element method was used to simulate bubble jets and clarify the applicability of the analytical solution in predicting the direction of large-scale bubbles. The analytical solution of the Kelvin impulse underestimates the buoyancy effects of bubbles near the bottom. However, near the free surface, the strong interaction between the bubble and free surface strengthens the downward movement of the bubble, resulting in an analytical solution with improved accuracy. A buoyancy correction factor was introduced to rectify inaccuracies in the analytical solution near the bottom. The correction factor was obtained under the condition of a vertically neutral collapse for the bubbles. Comparison of the simulation results with theoretical values across various buoyancy parameters indicate that the modified analytical solution can effectively predict the direction of bubble collapse across most parameter domains. The modification method for analytical solution proposed in this study may serve as a reference for practical operations aimed at protecting marine vessels near underwater explosions or marine seismic sources.

Nonlinear effects in quantum field theory: Applications of the Pochhammer–Chree equation

This study aims to solve the nonlinear Pochhammer–Chree ([Formula: see text]) equation to understand its physical implications and establish connections with other nonlinear evolution equations, particularly in plasma dynamics. By using the Khater II ([Formula: see text]hat. II) method for analytical solutions and validating these solutions numerically with the variational iteration method, this study offers a detailed understanding of the equation’s behavior. The results demonstrate the effectiveness of these methods in accurately modeling the system, highlighting the importance of combining analytical and numerical approaches for reliable solutions. This research significantly advances the field of nonlinear dynamics, especially in plasma physics, by employing multidisciplinary methods to tackle complex physical processes. Moreover, the [Formula: see text] equation is relevant in various physical contexts beyond plasma dynamics such as optical and quantum fields. In optics, it models the propagation of nonlinear waves in fiber optics, where similar nonlinear evolution equations describe wave interactions. In quantum field theory, the [Formula: see text] equation helps in understanding the behavior of quantum particles and fields under nonlinear effects, making it a versatile tool for studying complex phenomena across different domains of physics.

Application of Digital Technology for Oil and Gas Fields

The paper investigates the problem of isothermal filtration using an approximate method for solving the theory of partial differential equations. The mathematical model under consideration is nonlinear and does not lend itself to analytical methods of solution. The results obtained indicate the need for wide application in the development of oil and gas fields in the Republic of Kazakhstan. In particular, the results of the study make it possible to solve the problems of adapting mathematical models and evaluating changes in technological indicators, which are necessary attributes in the digital technology ""Information System for the Analysis of oil and Gas Field Development"" (ISAR). Many problems and mathematical problems of filtration theory arose while working on specific oil and gas fields in the western region of the Republic of Kazakhstan. The above approximate solution methods have found applications not only in filtration theory, but also in other problems (geophysics, ecology, etc.)"

A study on analytical solutions of one of the important shallow water wave equations and its stability analysis

Abstract The present research uses the modified w g -expansion technique to give analytical solutions for the nonlinear time fractional integro-differential Kadomtsev-Petviashvili (KP) hierarchy equation with beta fractional derivative. The modified w g -expansion technique is an important method for finding analytical solutions to nonlinear equations. The suggested approach produces three sorts of solutions: trigonometric, hyperbolic, and rational. These answers have been discovered with the aid of a software tool. Additionally, stability analysis of observed governing model solutions is used to validate scientific calculations. Furthermore, the 3-dimensional, contour, and 2-dimensional images are provided for a better understanding of the behavior of the solutions.

Analysis of tunnel lining internal forces under the influence of S and P-waves: An analytical solution and quasi-static numerical method

This paper employs analytical and pseudo-static approaches to analyze the tunnel response under the compression (P) and shear (S) waves. In the first step, Einstein and Schwartz’s method is revised for calculating Tunnel Lining Internal Forces (TLIFs) under P-wave. Next, a comprehensive comparison is performed between TLIFs under S and P-waves in two extreme contact interfaces of no-slip (NS) and full-slip (FS) conditions. Lastly, the effect of the intermediate layer was investigated by quasi-static finite element numerical modeling. The results showed that the maximum value of the axial force under the P-wave exceeds that of the S-wave in both the NS and FS conditions. Also, the amount of bending moment and shear force in both the NS and FS conditions under the S-wave is almost twice the P-wave. In general, the weak interlayer causes a decrease in the maximum axial force and the axial force values in the range of placement of the weak interlayer with the tunnel. Besides, it increases the maximum bending moment and shear force compared to the homogeneous medium. It was also observed that the weak interlayer with low thickness causes unpredictable behavior under S and P-waves. Overall, the presence of a layer with different stiffness led to a significant effect on the TLIFs under S and P-waves and increased the complexity of the dynamic analysis of tunnel lining. Therefore, it should be simulated separately under NS and FS conditions.

Investigation of solution behavior of Differential Equations by Sumudu methods of random complex Partial Differential Equations

In this study, solutions of random complex partial differential equations were found using the two-dimensional Sumudu transformation method(STM). The initial conditions of a deterministic equation or the non-homogeneous part of the equation are transformed into random variables to obtain a random complex partial differential equation. With the help of the properties of two-dimensional Sumudu and inverse Sumudu transformation, an approximate analytical solution of a complex partial differential equation with random constant coefficients was obtained by selecting a random variable with an initial condition of Normal and Gamma distribution. The probability characteristics of the resulting solutions, such as expected value and variance, were obtained and graphically shown with the help of the Maple package program.

Analytical solutions of the kinetic equation for rounded spirals in effectively isotropic systems

A method for finding analytical solutions to the Cabrera and Levine differential equation is proposed for the case of the spiral formation on a screw dislocation when there is no difference between the growth and evaporation of a crystal. The method is based on finding asymptotic solutions for small spiral angles, exact solutions in the interval of dimensionless spiral radii equal and exceeding unity, and asymptotic solutions for large radii. Then, the method of matching the obtained solutions and their derivatives by dimensionless radius is used for two radius values determined from the problem conditions. In the particular case, for angular velocity ω1=0.33, the exact solution is the result of matching the asymptotic solution of the original equation for small spiral angles, the exact solution in the interval of dimensionless spiral radii equal and exceeding unity, and the asymptotic solution for large radii. For large spiral radii, the asymptotic solution is the general one of the second kind Abel equation. Analysis of the obtained solution showed that the spiral pitch is not constant and equal to 19, but monotonically decreases with increasing spiral radius.

Nonlinear evolution equations in dispersive media: Analyzing the semilinear dispersive-fisher model

The nonlinear Semilinear Dispersive-Fisher (SDF) model is an important equation describing wave dynamics in dispersive media influenced by nonlinear and diffusive effects. This study enhances both analytical and numerical methods to solve the SDF model and validate the solutions. We utilize the Modified Khater (MKhat) and Unified (UF) methods for analytical solutions, and He’s Variational Iteration (HVI) scheme for numerical approximations. Our investigation connects the SDF model to other nonlinear equations like the Korteweg–de Vries (KdV) and Fisher–Kolmogorov (FK) equations, demonstrating the consistency between analytical and numerical solutions. This research contributes to the understanding and modeling of wave phenomena in dispersive media with nonlinear and diffusive dynamics, offering refined analytical and numerical techniques specific to the SDF model. Key contributions include introducing the MKhat and UF methods for analytical solutions and employing the HVI scheme for numerical approximations, which are less represented in current literature. This work is positioned in mathematical physics, focusing on nonlinear evolution equations.

Analytical solutions for static longitudinal displacements of suspension bridges under a moving vertical concentrated load

The longitudinal displacement at the girder ends of long-span suspension bridges is a crucial issue that affects structural safety, durability, as well as traffic safety and ride comfort. The primary cause of these reciprocating displacements is the action of moving live loads, which induces quasi-static displacements in the longitudinal direction. This paper theoretically investigates the static longitudinal displacements of suspension bridges under a moving vertical concentrated load and elucidates the underlying deformation mechanisms. Considering geometrical nonlinearity, analytical equations are established for the longitudinal deformations of the main cable and stiffening girder under a vertical concentrated load. By analyzing the geometrical deformation conditions of the suspenders, the relationship between the longitudinal displacements of the stiffening girder and the main cable is derived. Due to the coupling between longitudinal and vertical displacements, the vertical displacement of the stiffening girder must be solved to obtain its longitudinal displacement. Based on deflection theory and considering the bending stiffness of the stiffening girder, this paper derives the vertical deformations of the stiffening girder for single-span suspension bridges and those with short continuous side spans under a vertical concentrated load, which is then used to calculate the longitudinal displacement. Finite element models are used to verify the accuracy of the analytical solutions. Differences between the proposed solutions and previous solutions neglecting the bending stiffness of the stiffening girder are compared and analyzed. The effects of neglecting the longitudinal displacement at the bridge tower top and the presence of short continuous side spans on the longitudinal displacement of stiffening girder are also investigated. The results demonstrate the high accuracy of the proposed analytical solution for the longitudinal displacement at the girder ends of suspension bridges. Importantly, the bending stiffness of the stiffening girder cannot be neglected in calculating the longitudinal displacement. The longitudinal displacement at the girder ends is primarily caused by the longitudinal deformation of the main cable, while the contribution from the flexural deformation of the girder is negligible. Furthermore, the longitudinal displacement at the bridge tower top has a negligible effect on the longitudinal displacement of the girder, and the presence of short continuous side spans is beneficial for reducing the longitudinal displacement of the stiffening girder. The proposed analytical solution method provides a rapid approach for calculating the static longitudinal displacement at the girder ends of suspension bridges under passing live loads and reveals the deformation mechanism. This is beneficial for preliminary design and structural optimization, as it avoids the complex finite element modeling and solution process.

Parametric Synthesis of Single-Stage Lattice-Type Acoustic Wave Filters and Extended Multi-Stage Design.

This study proposes a single-stage lattice-type acoustic filter using an analytical solution method for either a narrow passband filter or a wider passband filter using two kinds of parameter assignments in the Butterworth-Van Dyke (BVD) model. To achieve the goal of a large bandwidth or high return loss, two first-order all-pass conditions are used. For multi-stage lattice-type filters, the cost function is defined and design parameters are extracted by using pattern search, while the initial values are provided through single-stage design to shorten optimization time and allow convergence to a better solution. This method provides the S-parameter frequency response for the filter on the YX 42° cut angle of lithium tantalate (electromechanical coupling coefficient of about 6%) that can meet the system specifications as much as possible. Finally, the three-stage lattice-type was applied to various 5G bands with a fractional bandwidth of 2-5%, resulting in a passband return loss of 10 dB and an out-of-band rejection of 40 dB or more.

Extension of Analytical Solution Methods to the Stability of Exposed Stratified Cemented Backfills

Extension of Analytical Solution Methods to the Stability of Exposed Stratified Cemented Backfills

Анализ частотных характеристик токовых датчиков цифрового трансформатора тока и напряжения

Digital measuring current and voltage transformers (DCVT), which incorporate various current and voltage sensors are the key sources of information about processes at modern digital substations. Classic small-sized current transformers, Rogowski coils, magnetotransistor sensors, Hall effect sensors, fiber-optic Faraday effect sensors, as well as shunts can be used as current sensors in DCVTs. For redundancy purposes, current channels are performed in several copies and on different sensors. Sensors that are different in nature have their own set of advantages, which together makes it possible to obtain the effect of eliminating the shortcomings of specific sensors. For example, they may not have a saturation effect or transmit the direct current component quite accurately, which is a significant advantage over classic current transformers. At the mains frequency, the behavior of these sensors is predictable, but their behavior is not well studied at higher and lower frequencies. Thus, it is decided to study the frequency response and phase response of these sensors in a wide frequency range. To solve the problems within the framework of this study, physical experiment, analytical and empirical solution methods have been used. As a result of the study, the frequency response and phase response of current sensors of DCVT in a wide frequency range have been obtained. Analog filter has been designed to correct the signal of the magnetotransistor sensor. The results obtained coincide with theoretical data and can be considered when developing digital relay protection and automation algorithms.

A Robust Numerical Simulation of a Fractional Black–Scholes Equation for Pricing American Options

After the discovery of fractal structures of financial markets, fractional partial differential equations (fPDEs) became very popular in studying dynamics of financial markets. Available research results involves two key modelling aspects; firstly, derivation of tractable asset pricing models, those that closely reflects the actual dynamics of financial markets. Secondly, the development of robust numerical solution methods. Often times, most the effective models are of a nonlinear nature, and as such, reliable analytical solution methods are seldomly available. On the other hand, the accurate value of American options strongly lies on the unknown free boundaries associated with these types of derivative contracts. The free boundaries emanates from the flexibility of the early exercise features with American options. To the best of our knowledge, the approach of pricing American options under the fractional calculus framework has not been extensively explored in literature, and an obvious wider research gap still exist on the design of robust solution methods for pricing American option problems formulated under the fractional calculus framework. Therefore, this paper serve to propose a robust numerical scheme for solving time-fractional Black–Scholes PDEs for pricing American put option problems. The proposed scheme is based on the front-fixing algorithm, under which the early exercise boundaries are transformed into fixed boundaries, allowing for a simultaneous computation of optimal exercise boundaries and their corresponding fair premiums. Results herein indicate that, the proposed numerical scheme is consistent, stable, convergent with order O(h2,k)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {O}}(h^2,k)$$\\end{document}, and also does guarantee positivity of solutions under all possible market conditions.

Investigation of accelerated moving load on dynamic response of FG Timoshenko nanobeam in thermal environment based on nonlocal strain gradient theory

For the first time, this paper investigates the forced vibrations of a functionally graded (FG) nanobeam with an accelerating moving load in a thermal environment. There is no exact coupling solution for the vibrations of nanobeam with an accelerated moving load, so this paper aims to provide a method to obtain an accurate solution for nanoscale structures with an accelerated moving load. The equations of motion are derived using Timoshenko's beam theory and the nonlocal strain gradient theory (NSGT). The Laplace method has been used to solve the coupling and exact differential equations. Then, by inverting Laplace from the coupled equations, an exact solution of the temporal response for FG nanobeam with constant acceleration and initial velocity in a thermal environment was obtained. The natural frequency was compared with previous works for validity and had acceptable results. Finally, the effect of parameters such as changes in acceleration and velocity of moving force, negative acceleration, power law index of FG material, different temperatures, nonlocal parameter, and longitudinal scale parameter on the maximum dynamic displacement of nanobeam is investigated. These results can be used better to design FG nanostructures with accelerated moving loads. Considering the sensitivity of data analysis in nano dimensions, it is necessary to provide an analytical solution method in these dimensions to reduce the error percentage in nano dimensions to zero. which is presented in this work for the first time.

Biophysical phenotyping of single-cell based on impedance and application for individualized precision medicine

Single-cell biophysical characterization based on impedance measurement is an advantageous approach due to its label-free, high-efficiency, cost-effective and real-time capability. Biophysical phenotyping can yield timely and rich information on physiological and pathological state of cells for disease diagnosis, drug screening, precision medicine, etc. However, precise measurement on single-cell impedance is challenging, particularly hard to figure out the detailed biophysical parameters of single cell due to coupling and complexity of impedance model. Here, we propose an analytic determination method to decode single-cell electrophysiological parameters (including cell-substrate interface capacitance, cell membrane capacitance, cell membrane conductivity, and cytoplasm conductivity) from the impedances measured at optimized frequencies by using analytic solution rather than spectrum fitting. With this simple and fast analytic solution method, the physiological parameters of single cell in natural adhesion state can be accurately determined in real time. We validate this cell parameter determination method in monitoring the change of cell adhesion under hydraulic effects and exploring electrophysiological differences among MCF-7, HeLa, Huh7, and MDA-MB-231 cell lines. Particularly, we apply the approach to optimize tumor treating fields (TTFields) therapy, realizing individualized precision medicine. Our work provides an accurate and efficient approach for characterizing single-cell biophysical properties with real-time, in-situ, label-free, and less invasive advantages.

Разработка и исследование подхода к обработке сигналов цифровых измерительных трансформаторов тока и напряжения

During the development of secondary systems of stations and substations, including relay protection, automation, control, electricity metering, etc., qualitative changes have recently occurred due to the introduction of digital technologies. The introduction of digital technologies is supported by innovative development programs such as “Digital Transformation 2030”. The introduction of digital technologies is associated with the use of digital data transmission channels according to uniform standards. Projects of “digital substations” are being actively implemented with the transmission of digital information at all levels, including the primary equipment, the transmission of information both from current and voltage measuring transformers, and from switching devices. Currently, digital instrument transformers are actively being developed and implemented, in which information about measured currents and voltages is presented in digital code. New approaches to measure primary currents and voltages, and transmit information pose new challenges, and open prospects for improving signal processing methods used for relay protection purposes. Thus, the relevant aim of the study is the development and research of approaches to digital processing of signals of digital instrument current and voltage transformers. To solve the problems posed within the framework of this study, analytical and numerical solution methods have been used. Numerical methods are applied using software products such as MathCAD and the Python programming language to generate signals and display them on graphs. To substantiate the reliability, verification of the results obtained by different methods is carried out. The authors have formulated an approach to digital processing of signals obtained using current sensors of a digital instrument transformer. The authors have studied the approach to determine synchronized current vector values by processing information received from primary current converters. The characteristics of the algorithm for obtaining a vector have been analyzed and the dependences of errors on the influence of distorting factors have been plotted. The proposed approach is a development of the existing approach to obtain a vector by getting information from current sensors operating on different physical principles. The results show the advantages of the proposed method. The proposed digital signal processing algorithm is more technically expensive since it contains at least two metrologically calibrated measuring channels instead of one. However, all other things being equal, it allows to increase the accuracy and speed of the digital processing algorithms taken as a prototype. The results obtained can be used to formulate algorithms for measuring relay protection devices.

Free vibration analysis of tall buildings using a generalized continuous model: simplified approach to soil-structure interaction

Currently, due to the simplifications imposed on continuous models, there is an absence of a generalized approximate solution to investigate all possible dynamic behaviors of a tall building. In an attempt to address this deficiency, this paper proposes a new generalized continuous model and its solution methods to generalize all types of behavior that may be encountered in structural systems and tall buildings. The continuous model results from a composite beam that serially couples the classical sandwich beam and a pure shear beam. Unlike existing continuous models, the serial coupling, inclusion of new kinematic fields, and inclusion of rotational inertia allow for the resolution of complex interaction among different types of bending and shear behaviors. Specifically, it is shown how local shear behavior simultaneously interacts with global bending behavior, global shear, and local bending. Two solutions to the dynamic problem are presented: a closed-form analytical solution method for structures with uniformity along their height, and a numerical method that combines the advantages of the continuous method and the transfer matrix method to resolve structural and geometric variability in height. The transfer matrix is analytically solved by maintaining a fixed 6 × 6 matrix range that is independent of the number of levels of the structure, drastically reducing computational cost by eliminating the classical need to calculate the inverse of the zero matrix. Numerical applications investigate the effect of rotational inertia and local shear on the accuracy of the proposed results compared to results from the finite element method. To this end, a parametric analysis of 1560 different cases covering a wide range of typical behaviors in individual structural systems and tall buildings is conducted. The results exhibit promising accuracy, showing that the qualities of the proposed continuous model significantly improve precision compared to classical continuous models and therefore justify its practical application by academics and practicing engineers. Furthermore, the proposed mathematical approach is easily applicable in various domains of composite materials engineering.

Comparative study and validation of a new analytical method for hydraulic modelling of bidirectional low temperature networks

Bidirectional Low Temperature Networks (BLTNs) are an innovative technology of district heating and cooling systems. Because of their intrinsic feature of supplying simultaneous heating and cooling either directly from an energy source/sink or through distributed heat pumps and chillers, circulation pumps installed at each building extract water from the network and a distinct pressure cone is subsequently created. Variations in the amount (and direction) of the extracted water call for accurate hydraulic design and analysis to ensure safe and adequate network operation. This paper proposes a new analytical solution method for fast hydraulic modelling of BLTNs. The analytical method was compared against three numerical solution methods and validated using different current and future operating scenarios of a real-world bidirectional network. In all modelled cases, the calculated mass flows and pressure drops using the new analytical method yielded very small Normalised Mean Bias Error (NMBE) and Coefficient of Variation of the Root Mean Square Error (CV(RMSE)), suggesting excellent agreement between the analytical and numerical solution methods. The new analytical method offers better performance over its numerical counterparts due to its explicit nature, ease of implementation, and quick modelling time.

ANALYTICAL SOLUTION OF THE DIFFERENTIAL EQUATION OF HEAT CONDUCTIVITY FOR DAMAGED THERMAL INSULATION OF PIPELINES

For heat supply systems of Ukraine, the determination and forecasting of thermal energy losses during the transportation of heat carrier is an urgent problem. The thermal lines of the heating networks are long and its insulation is damaged. Installation of heat energy metering devices at all sources and all consumers (i.e. buildings) without exception allows determining and forecasting real heat losses in the heat network, but this is a difficult task, not all consumers and sources are actually covered by heat consumption accounting. The task of determining heat losses is also relevant for energy management systems of heat supply systems, energy companies and industry. The solution to the problem is proposed by a separate mathematical modeling of the temperature state of the section of the damaged insulating layer with the determination of the heat flow through it. It is proposed to solve the problem by the method of analytical solution of the differential equation of heat conduction with boundary conditions of the third kind. With the uniform distribution of characteristic damages along the total length of the pipeline, knowing the limits of damage influence, the share of insulation damage and the number of damages on the pipeline, it is possible to determine the real heat flow from the outer surface of the pipelines, including coefficient of increase of heat losses on the section of the heat pipe in relation to those determined by regulatory documents. Traditionally, one-dimensional models or determinations of the two-dimensional temperature field and actual heat fluxes in a cross-section to the axis of a characteristic damaged section of insulation are considered. But this does not take into account the two-dimensionality of the temperature field of the damaged layer of insulation along the length (that is, along the axis) of the pipeline. Therefore, the purpose of this work is to develop a methodology for determining heat losses through pipelines, taking into account the damage to their insulation and the distribution of characteristic damages along the length. The following simulation was carried out for one of the areas. Also, experimental and numerical studies using the finite difference method in combination with the method of running variable directions for similar models confirmed the coincidence of the analytical solution of the proposed model and the finite difference model within the permissible error.