Accurate Analytical Solution Research Articles

Heat and Mass Transfer Analysis for the Viscous Fluid Flow: Dual Approximate Solutions

The aim of this paper is to investigate effective and accurate dual analytic approximate solutions, while taking into account thermal effects. The heat and mass transfer problem in a viscous fluid flow are analytically explored by using the modified Optimal Homotopy Asymptotic Method (OHAM). By using similarity transformations, the motion equations are reduced to a set of nonlinear ordinary differential equations. Based on the numerical results, it was revealed that there are dual analytic approximate solutions within the mass transfer problem. The variation of the physical parameters (the Prandtl number and the temperature distribution parameter) over the temperature profile is analytically explored and graphically depicted for the first approximate and the corresponding dual solution, respectively. The advantage of the proposed method arises from using only one iteration for obtaining the dual analytical solutions. The presented results are effective, accurate and in good agreement with the corresponding numerical results with relevance for further engineering applications of heat and mass transfer problems.

Theoretical solution and failure analysis of water pressure on lining of deep-buried non-circular hydraulic tunnel based on the equivalent hydraulic radius method

The water pressure distribution acting on non-circular hydraulic tunnels have always been an important and crucial issue in deep-buried tunnels and caverns, due to its complexity, variability and potential threats to the stability of surrounding rock and the safety of lining structure. This research proposes an accurate axisymmetric analytical solution for water pressure and seepage discharge on the surface of surrounding rock, the lining and grouting circle under high external and internal water pressures, by introducing an equivalent hydraulic radius method in to seepage theory in fractured media. Then, the distribution law of seepage pressure within different distances from the tunnel wall is derived by theoretical analysis. The accuracy, superiority and applicability of the theoretical solution based on the equivalent hydraulic radius are verified sufficiently. The maximum errors of the theoretical solution are less than 15%, 18% and 5.98% respectively, when compared to the numerical solution, the in-situ water pressure monitoring data and the theoretical solution based on the conformal mapping. And the proposed theoretical solution is much closer to the numerical solution than the existing equivalent methods. Moreover, parameters of each part of the composite lining are optimized. Increasing 10% of the original of the permeability coefficient of the composite lining can maximally reduce the external water pressure, and the maximum principal stress and displacement of the composite lining can be effectively reduced by comparing the contour of the maximum principal stress and displacement before and after parameter optimization. Furthermore, by comparing the excavation damage zone distribution before and after parameter optimization, the shear failure occurring in the composite lining is significantly reduced, and the stability of the composite lining structure can be improved. The conclusions could provide important guiding theoretical value for the further study of water pressure distribution acting on non-circular hydraulic tunnels as well as for the correct grouting and drainage designs.

An Analytical Method of Beamwidth under Different Angles of Planar Array

It is widely recognized that the calculation of the beam width of the large-aperture planar array through beam pattern simulation is slow and inefficient, and obtaining the ability to distinguish the target from different angles within the observation range quickly is difficult. This paper uses methods such as the array product theorem, discrete array, continuous array beam width equivalence, and coordinate transformation to acquire a more accurate analytical solution of the beam width of the planar array. The analytical value of the beam width of the planar array at different angles and the number of array elements were compared with the simulated value. The results show that with the usage of the new method, the average deviation of the beam width analytical values in different main maxima directions can reach less than 1% when the number of array elements is less than 10 elements. Compared with the simulation method, the planar array beam width can be obtained quickly and accurately by the new method.

Machine learning algorithm for the structural design of MEMS resonators

MEMS resonators have become core devices in many fields, their geometric designs can profoundly affect performance. However, the theoretical modeling of MEMS resonators with complex structures is becoming more and more difficult, and the time consumption of using numerical simulation methods to solve the accurate analytical solution of performance is also increasing. In this paper, we report a working mode identification and a machine learning algorithm, which dramatically shorten the MEMS design cycle by constructing datasets and predicting physical properties with high speed and high accuracy. As an example, we apply the algorithms to the performance prediction of flower-like disk resonator. The typical structural parameters of MEMS resonators are used as the input layer of the neural network, and performance generated by finite element analysis methods are used as the output layer. For 50,000 modal shapes with 5000 different structural parameters, the accuracy of the working mode identification algorithm proposed in this paper to identify elliptical modes is 100%. After sufficient training, the obtained neural network calculators can predict the resonant frequency, thermoelastic quality factor, mechanical sensitivity and mechanical thermal noise of the MEMS resonator. Compared with traditional numerical simulation methods, the identification of resonant frequency and thermoelastic quality factor is 11,341 times faster, the identification of mechanical sensitivity and mechanical thermal noise is 1813 times faster, and the prediction regression accuracy is all greater than 96%. This high-speed and high-accuracy performance prediction method can effectively improve the design efficiency of MEMS resonators with complex structures, providing a promising tool for enhancing MEMS resonator performance.

A New Method for Free Vibration Analysis of Triangular Isotropic and Orthotropic Plates of Isosceles Type Using an Accurate Series Solution

In this paper, a new method based on an accurate analytical series solution for free vibration of triangular isotropic and orthotropic plates is presented. The proposed solution is expressed in terms of undetermined arbitrary coefficients, which are exactly satisfied by the governing differential equation in free vibration. The approach used is based on an innovative extension of the superposition method through the application of a modified system of trigonometric functions. The boundary conditions for bending displacements and bending rotations on the sides of the triangular plate led to an infinite system of linear algebraic equations in terms of the undetermined coefficients. Following this development, the paper then presents an algorithm to solve the boundary value problem for isotropic and orthotropic triangular plates for any kinematic boundary conditions. Of course, the boundary conditions with zero displacements and zero rotations on all sides correspond to the case when the plate is fully clamped all around. The convergence of the proposed method is examined by numerical simulation applying stringent accuracy requirements to fulfill the prescribed boundary conditions. Some of the computed numerical results are compared with published results and finally, the paper draws significant conclusions.

Highly Accurate Analytical Approximate Solutions of the Electrostatically Actuated Parallel Plates

Highly Accurate Analytical Approximate Solutions of the Electrostatically Actuated Parallel Plates

Nonlinear analysis of Euler beams resting on a tensionless soil with arbitrary configurations

BackgroundThe nonlinear interaction between an elastic Euler beam and a tensionless soil foundation is studied. The exact analytical solutions of the nonlinear problem are rather complicated. The main difficulty is imposing compatibility conditions at lift-off points. These points are determined as a part of the solution, although being needed to get the solution itself. In the current work, semi-analytical solutions are derived using the Rayleigh–Ritz method. The principle of vanishing variation of potential energy is adopted. The solution is approximated using a set of suitable trial functions. Accurate high-order approximate analytical solutions are obtained using MAXIMA symbolic manipulator. Lift-off points are identified through an iterative procedure and compatibility conditions are satisfied automatically. The methodology is designed to accommodate arbitrary configurations for the load distribution and the beam properties.ResultsExact solutions are revised briefly to verify the semi-analytical solutions in terms of deflection, bending moment, and shear. Semi-analytical solutions for constant beam properties including various support conditions and load distributions are verified. Convergence of high-order semi-analytical solutions is illustrated for cases including one and two contact points. A parametric study is provided to illustrate the effect of soil stiffness on the contact length. The case of a finite beam with free ends is considered. The semi-analytical solutions for variable beam moment of inertia are provided and verified.ConclusionsHighly accurate semi-analytical solutions can be obtained for the problem considered using the Rayleigh–Ritz method along with a symbolic manipulator. Arbitrary load and support configurations can be modeled, and the locations of lift-off points are well predicted. The semi-analytical solutions are extremely valuable for cases of variable moment inertia since exact solutions are rather rare.

Nonlinear Vibration of Double-Walled Carbon Nanotubes Subjected to Mechanical Impact and Embedded on Winkler-Pasternak Foundation.

This study was devoted to an investigation on the dynamics of double-walled carbon nanotubes (DWCNTs) under the influence of Winkler-Pasternak foundation near the primary resonance. Two Euler-Bernoulli beams embedded on nonlinear foundation, interacting through van der Waals forces, subjected to mechanical impact are considered. By means of Hamilton's principle, Eringen's nonlocal elastic theory, and taking into account the moving nanoparticles, the Galerkin-Bubnov method is applied and accordingly, governing partial differential equations are reduced to two differential equations with variable coefficients. The nonlinear damped and forced vibration is studied using the optimal auxiliary functions method (OAFM). An explicit and very accurate analytical solution is obtained by means of OAFM without considering simplifying hypotheses. An accurate analysis is for the first time reported considering the cumulated effects of nonlinearities simultaneously induced by the Winkler-Pasternak foundation, the curvature of beams and van der Waals force, and also the effect of discontinuities marked by the presence of the Dirac function. Finally, a stability analysis of the considered model is developed by means of the homotopy perturbation method (HPM) using the condition of existence of the two frequencies. It was shown that an increasing of some constitutive parameters substantially reduces the area of stability, all these being of much help in guiding the design of advanced nanoelectromechanical devices, in which nanotubes act as basic elements.

Homotopy Perturbation Method for Pneumonia–HIV Co-Infection

It is well known that HIV (human immunodeficiency virus) weakens the immune system of individuals, resulting in risk of other infections, such as pneumonia. The most frequent viral pneumonia seen in individuals infected with HIV is cytomegalovirus (CMV). In this paper, pneumonia–HIV co-infection is modeled through the formulation of a mathematical compartmental model consisting of nine compartments. Some of the basic properties of the model are established, such as the positivity, boundedness of the system, equilibrium points, and computation of the basic reproduction number. After obtaining the solution, the homotopy perturbation method (HPM) is applied, as it is known for its convergence properties. It is observed that the HPM gives an accurate analytical solution that indicates various important factors that are responsible for the spread of cytomegalovirus pneumonia in HIV-infected populations, and this is justified through a plot made by using MATLAB 2020a.

Static Magnetic Shielding of a Perfectly Electric Conducting Circular Disk: Exact Analytical Solution Considering Radius of Emitting Loop

This letter proposes an accurate analytical formulation for the shielding problem of a perfectly electric conducting disk against the static magnetic field of a circular current loop placed coaxially with the disk. The key is to convert this problem into an electric field problem: the circular disk is grounded and is excited by nonuniform line charges distributed on the loop. The latter can be solved through the known Green function. Then, we obtain the accurate solution in integral form for magnetic vector potential at any field point and the accurate analytical solution for the magnetic field on the central axis of the disk. Also, the solution for the magnetic field is extended to the whole space when the loop radius is approaching zero.

The Analytic Solutions of Nonlinear Generalized Pantograph Differential Equations of Higher Order Via Coupled Adomian-Homotopy Technique

In this study, an efficient novel technique is presented to obtain a more accurate analytical solution to nonlinear pantograph differential equations. This technique combines the Adomian decomposition method (ADM) with the homotopy analysis method concepts (HAM). The whole integral part of HAM is used instead of an integral part of ADM approach to get higher accurate results. The main advantage of this technique is that it gives a large and more extended convergent region of iterative approximate solutions for long time intervals that rapidly converge to the exact solution. Another advantage is capable of providing a continuous representation of the approximate solutions, which gives better information over whole time interval. Finally, selected examples are given to show the accuracy, efficiency and effectiveness of this technique. This technique can be addressed and applied to other non-linear problems.

Hall Current and Soret Effects on Unsteady MHD Rotating Flow of Second-Grade Fluid through Porous Media under the Influences of Thermal Radiation and Chemical Reactions

The unsteady MHD free convection heat and mass transfer flow of a viscous, incompressible, and electrically conducting fluid passing through a vertical plate embedded in a porous medium in the presence of chemical reactions and thermal radiation is investigated. The effects of the Hall current, rotation and Soret are studied. Using the perturbation approach, one can obtain an accurate analytical solution to the governing equations for the fluid velocity, fluid temperature, and species concentration, provided that the initial and boundary conditions are acceptable. It is possible to obtain expressions for the shear stress, rate of heat transfer, and rate of mass transfer for both plates with the ramping temperature and isothermal conditions. On the one hand, the numerical values of the primary and secondary fluid velocities, fluid temperature, and species concentration are presented graphically. On the other hand, the numerical values of the shear stress and rate of mass transfer for the plate are presented in tabular form for various values of the relevant flow parameters. These values are given for a range of pertinent flow parameters. It was determined that an increase in the Hall and Soret parameters over the whole fluid area leads to a corresponding increase in the resulting velocity. The resultant velocity continually climbs to a high level due to the contributions of the thermal and solute buoyancy forces. Lowering the heat source parameter reduces the temperature distribution, resulting in a lower overall temperature. When there is a rise in the chemical reaction parameter over the whole fluid area, there is a corresponding decrease in the concentration. The concentration buoyancy force, Hall current, and Prandtl number reduce the skin friction. On the other hand, the permeability of the porous medium, rotation, chemical reaction, the Soret number, thermal buoyancy force, and mass diffusion all have the opposite effects on the skin friction.

Three-Dimensional Analytical Solutions for Acoustic Transverse Modes in a Cylindrical Duct with Axial Temperature Gradient and Non-Zero Mach Number

Cylindrical ducts with axial mean temperature gradient and mean flows are typical elements in rocket engines, can combustors, and afterburners. Accurate analytical solutions for the acoustic waves of the longitudinal and transverse modes within these ducts can significantly improve the performance of low order acoustic network models for analyses of acoustic behaviours and combustion instabilities in these kinds of ducts. Here, we derive an acoustic wave equation as a function of pressure perturbation based on the linearised Euler equations (LEEs), and the modified WKB approximation method is applied to derive analytical solutions based on very few assumptions. The eigenvalue system is built based on the proposed solutions and applied to predict the resonant frequencies and growth rate for transverse modes. Validations of the proposed solutions are performed by comparing them to the numerical results directly calculated from the LEEs. Good agreements are found between analytical reconstruction and numerical results of three-dimensional transverse modes. The system with both mean temperature profile and mean flow presents a larger absolute value of the growth rate than the condition of either uniform mean temperature or no mean flow.

Analytical solutions of inerter-added sliding isolation structures to ground motions

• Contribute unfulfilled analytical solutions of inerter-added sliding structures. • Reveal explicitly the role of additional inerter played in a sliding isolation system. • Derive explicit conditions for motion modes of inerter-added sliding isolation structures. • Recommend the trade-off between reduction of isolation displacement and acceleration. Previous studies have proven that adding an inerter to a sliding isolation structure can reduce the isolation displacement without compromising the control effect of the superstructure. However, due to the strong nonlinearity of the sliding isolator , approximate methods or numerical analyses are adopted to investigate inerter-added sliding isolation structures, which hinders the understanding of the inherent working mechanisms of such structures. This study derives exact analytical solutions during both sliding and stick motion states of inerter-added sliding isolation structures under harmonic ground motions and investigates the influence of an additional inerter on the motion states and responses of such structures. The main novelties of this paper are the employment of the inerter technique in a sliding system for performance improvement and explicit expressions of the dynamic responses and motion modes for inerter-added sliding isolation structures that assist in understanding the role of the additional inerter. Differential equations are established for both the stick and sliding motions of inerter-added sliding isolation structures subjected to harmonic ground excitations. The dynamic characteristics of the structure are analyzed, and accurate analytical solutions are derived for structural responses. The explicit forms for the occurrence conditions of three fundamental modes (i.e., the stick-stick, stick-slip and slip-slip modes) for the motion of the inerter-added sliding isolation structures are presented. Based on the derived analytical solutions, extensive parametric analyses are conducted to investigate the influences of the added inerter on the motion modes and responses of sliding isolation structures. By adding an inerter, a decreased natural frequency, a decreased damping ratio and a reduction in ground motion excitation are observed for the vibration of the superstructure in the inerter-added sliding isolation structure compared to that in the sliding isolation structure without an inerter. Furthermore, the inerter causes the slip-slip mode to occur more easily, which is preferred for sliding isolation structures when sliding occurs. An increasing inertance of the inerter is beneficial for reducing isolation displacement in general but contrary to reducing maximum acceleration in some frequency ranges and excitations with large amplitudes. The trade-off between the reduction of isolation displacement and acceleration is recommended for determining the inertance of the inerter in sliding isolation structures.

Dynamics of the Vibro-Impact Nonlinear Damped and Forced Oscillator under the Influence of the Electromagnetic Actuation

The main objective of the present work is to find an approximate analytical solution for the nonlinear differential equation of the vibro-impact oscillator under the influence of the electromagnetic actuation near the primary resonance. The trigger of vibro-impact regime is due to Hertzian contact. The optimal auxiliary functions method (OAFM) is utilized to give an analytical approximate solution of the problem. The influences of static normal load and electromagnetic actuation near the primary resonance are completely studied. The main novelties of the proposed procedure are the presence of some new adequate auxiliary functions, the introduction of the convergence-control parameters, the original construction of the initial and of the first iteration, and the freedom to choose the method for determining the optimal values of the convergence-control parameters. All these led to an explicit and accurate analytical solution, which is another novelty proposed in the paper. This technique is very accurate, simple, effective, and easy to apply using only the first iteration. A second objective was to perform an analysis of stability of the model using the multiple scales method and the eigenvalues of the Jacobian matrix.

Contact stress induced by the rotation of a rigid bolt bonded to an elastic body

Torsional deformation of an elastic half-space containing a vertical cylindrical cavity caused by a rotation of a bonded rigid bolt with a long shank is studied in this article by employing Weber integral transforms and integral equations. In the limiting case of the infinite shank, we suggest two different techniques reducing the problem to the regular integral equations of the second kind, which permit efficient solutions in distinct ranges of geometric parameters and allow us to derive a highly accurate approximate analytic solution of the problem for any values of the parameters. Almost exact approximations are suggested for the stress concentration at the edge of the bolt head. In the case of a bolt with a long finite shank, relationships between the torque and the angle of twist of the bolt are derived by utilizing solutions obtained for the bolt with the infinite shank as asymptotic approximations. The effect of the geometric parameters is investigated providing insight into the behavior of the bolt and contact stress in dependence on the bolt head radius and the length of the shank.

An Analytical Approximation for the Ionomer Film Model in PEMFC

The ionomer film and its transport resistances for oxygen are considered to be an important aspect for PEMFC performance. Ionomer film sub-models are therefore frequently used in PEMFC modeling to account for this effect. Mathematically these are expressed by a non-linear equation for the oxygen concentration, which depending on the reaction order cannot be solved analytically. Typically, a numerical solution of this equation, e.g., using the Newton-method is needed. Here, we derive a highly accurate approximate analytical solution for the ionomer film model. This enables faster computation, which is particularly important for computationally demanding higher dimensional PEMFC models.

Improving the Learning and Teaching of Mathematical Logic Elements Using Maple

The study of the elements of mathematical logic and its sections, especially in the process of teaching in high school in general and specialized courses of mathematics or higher educational institutions, is a rather tricky task. The paper demonstrates how one can design and develop a package of interactive applications to help learn, understand, and apply the essential elements of mathematical logic. The research aims to develop and describe an interactive application for studying the order of constructing truth tables and studying and demonstrating the production of principal normal disjunctive and conjunctive forms. The development is based on the methods of the Maple computer algebra system of the Maplet library. In addition, other methods are used to solve the research issues – theoretical and accurate analytical solutions to mathematical logic issues. All constructed material (Maplet) will serve as a good simulator for students, allowing one to perform an infinite number of attempts to analyze, synthesize, and build truth tables.

Використання багатопозиційної інтерферометрії для визначення положення об’єктів

An accurate analytical solution for positioning technologies based on both the difference of distances from the object to reference points (TDOA) and the distances themselves (TOA) is considered. The bijection of the obtained coordinate transformation allows reducing the problem of hyperbolic positioning to the Cartesian coordinate system. It is shown that all localization systems of the same rank with different numbers of sensors reduce to a single canonical form with a fixed number of (virtual) sensors corresponding to the dimension of space plus one. The resulting solution allows us the simultaneous observation of many objects, both close and distant, with determination of the distance to them. The possibilities of using positioning systems with a reduced rank have been analyzed. The synthesis of a sensor system with a higher rank from several separate systems is considered. Algorithms for solving the problem are linear and allow direct reconstruction of the image of objects

An Analytic Solution for Riccati Matrix Delay Differential Equation using Coupled Homotopy-Adomian Approach

An efficient modification and a novel technique combining the homotopy concept with Adomian decomposition method (ADM) to obtain an accurate analytical solution for Riccati matrix delay differential equation (RMDDE) is introduced in this paper . Both methods are very efficient and effective. The whole integral part of ADM is used instead of the integral part of homotopy technique. The major feature in current technique gives us a large convergence region of iterative approximate solutions .The results acquired by this technique give better approximations for a larger region as well as previously. Finally, the results conducted via suggesting an efficient and easy technique, and may be addressed to other non-linear problems.