Approximate Analytical Solution Research Articles

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.

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.

Effect of Soret diffusion on the growth of spherical crystals in supercooled alloy melts under oscillatory flow.

The growth of spherical crystals in binary alloy melts with thermal diffusion effects under oscillatory flow is investigated analytically. Using the multiple scale method, we derive approximate analytical solutions for both the crystal interface growth rate and the solute concentration. Our results demonstrate that the Soret effect significantly influences both the solute concentration near the crystal interface and the crystal growth rate. Specifically, with a positive Soret coefficient, the growth rate of spherical crystals in a binary dilute alloy melt decreases as the coefficient increases, while the solute concentration near the interface increases. In contrast, with a negative Soret coefficient, the growth rate of the spherical crystals increases as the coefficient decreases, and the solute concentration near the interface decreases. Additionally, the presence of oscillatory flow markedly promotes the grain refinement induced by the Soret effect.

Impact of temperature-reliant thermal conductivity and viscosity variations on the convection of Jeffrey fluid in a rotating cellular porous layer

In this analysis, the collective impact of temperature-dependent thermal conductivity and viscosity variations on the convective instability of a Jeffrey fluid in a rotating layer of cellular porous material is examined using an improved Jeffrey–Darcy model. This study has significant implications for cellular foams made from plastics, ceramics and metals, in which radiative heat transmission can be taken as a diffusion practice. Utilizing the linear stability concept and Galerkin method, approximate analytical and numerical solutions accurate to one decimal place are offered. The analysis reveals that the effect of the thermal conductivity variation factor and the rotation factor is to postpone the convective wave, whereas the viscosity variation factor and the Jeffrey factor have a dual effect in the form of rotation. The range of the convective cell is reduced with cumulating thermal conductivity variation factor, viscosity variation factor, Jeffrey factor and rotation factor. In the absence of rotation, the range of the convective cell is not dependent on the Jeffrey factor or the viscosity variation factor. Furthermore, the outcomes are matched with the existing literature for the specific case of this investigation.

Solving Viscous Burgers’ Equation: Hybrid Approach Combining Boundary Layer Theory and Physics-Informed Neural Networks

In this paper, we develop a hybrid approach to solve the viscous Burgers’ equation by combining classical boundary layer theory with modern Physics-Informed Neural Networks (PINNs). The boundary layer theory provides an approximate analytical solution to the equation, particularly in regimes where viscosity dominates. PINNs, on the other hand, offer a data-driven framework that can address complex boundary and initial conditions more flexibly. We demonstrate that PINNs capture the key dynamics of the Burgers’ equation, such as shock wave formation and the smoothing effects of viscosity, and show how the combination of these methods provides a powerful tool for solving nonlinear partial differential equations.

Employing the Sadik Residual Power Series Method to Analyze a System of Nonlinear Caputo Time-Fractional Partial Differential Equations

The present study proposes an approximate analytical solution to a nonlinear system of time-fractional partial differential equations. The Sadik residual power series method, which integrates the two-part Sadik integral transform with the residual power series technique, is employed to solve the fractional differential equation in the Caputo sense. Nonlinear problems with known and unknown solutions are examined to demonstrate the capacity of the technique. Numerical simulations and 3D visualizations are conducted for various values of the fractional order to further understand the solution’s behavior. Additionally, the results are validated against exact solutions or existing methodologies to ensure their reliability and accuracy. A key advantage of the proposed method is its ability to generate results without the need for Adomian polynomials, perturbation techniques, discretization, or linearization, enabling a more efficient.

Bernstein Polynomials for Solving Fractional Differential Equations with Two Parameters

This work presents a general framework for solving generalized fractional differential equations based on operational matrices of the generalized Bernstein polynomials. This method effectively obtains approximate analytical solutions of many fractional differential equations. The generalized fractional derivative of the Caputo type with two parameters and its properties are studied. Using orthonormal Bernstein polynomials has led to the development of fractional polynomials, which offer an approximate solution for ordinary fractional differential equations. The approach employs the generalized orthogonal Bernstein polynomials (FOBPs) and constructs their operational matrices for fractional integration and derivative in the generalized Caputo sense to achieve this objective. Operational matrices convert ordinary differential equations into a system of algebraic equations, which can be solved using Newton’s method. The convergence analysis and error estimate associated with the proposed problem have been investigated using the approximation of generalized orthogonal Bernstein polynomials (FOBPs). The effect of the new parameters of the fractional derivative is presented in several examples. Finally, several examples are included to clarify the proposed technique’s validity, efficiency, and applicability via generalized orthogonal Bernstein polynomials (FOBPs) approximation.

A Novel Analytical Approximate Solution for Strongly Nonlinear Third-Order Jerk Equations Using a Modified Iteration Method

Nonlinear jerk equations, characterized by a third-order time derivative, play a crucial role in modeling various physical phenomena across disciplines like mechanics, circuits, and biology. Accurately solving these equations is essential for understanding and predicting the behavior of such systems. However, obtaining analytical solutions for nonlinear jerk equations can be challenging, necessitating the development of robust and accurate approximation methods. This work explores, for the first time, the application of the modified iteration approach to solve third-order jerk equations. By comparing the obtained approximate solutions with both exact and existing analytical solutions for established engineering problems, we demonstrate the superior accuracy and rapid convergence of the proposed method. The significantly reduced error percentages highlight the effectiveness of the modified iteration approach in providing precise solutions for nonlinear jerk equations, paving the way for its application in a wide range of oscillation problems within nonlinear sciences and engineering.

Approximate analytic solution of the dissipative semiclassical Rabi model under parametric multi-tone modulations

Abstract We present an analytic method for obtaining the dynamics of the dissipative time-modulated
semiclassical Rabi model, which describes a two-level system (qubit) with time-dependent parameters, 
coupled to a single-mode bosonic field via the dipole-interaction and to
a thermal reservoir. We consider the simultaneous harmonic
modulations of the qubit transition frequency and the qubit-field coupling strength,
with arbitrary frequencies, and obtain closed analytic expressions for the
density operator under the coarse-graining approximation. Our
approximate results are in excellent ageement with exact numerical data, and
illustrate how the qubit state can be controlled in the dispersive regime by properly
adjusting the system parameters.

Darcy's law survival from no-slip to perfect-slip flow in porous media

A macroscopic model for perfect-slip flow in porous media is derived in this work, starting from the pore-scale flow problem and making use of an upscaling technique based on the adjoint method and Green's formula. It is shown that the averaged momentum equation has a Darcy form in which the permeability tensor, $\boldsymbol{\mathsf{K}}_{ps}$ , is obtained from an associated adjoint (closure) problem that is to be solved on a (periodic) unit cell representative of the structure. Similarly to the classical permeability tensor, $\boldsymbol{\mathsf{K}}$ , in the no-slip regime, $\boldsymbol{\mathsf{K}}_{ps}$ is intrinsic to the porous medium under consideration and is shown to be symmetric and positive. Integral relationships between $\boldsymbol{\mathsf{K}}_{ps}$ , the partial-slip flow permeability tensor, $\boldsymbol{\mathsf{K}}_{s}$ , and $\boldsymbol{\mathsf{K}}$ are derived. Numerical simulations carried out on two-dimensional model porous structures, together with an approximate analytical solution and an empirical correlation for a particular configuration, confirm the validity of the macroscopic model and the relationship between $\boldsymbol{\mathsf{K}}_{ps}$ and $\boldsymbol{\mathsf{K}}$ .

Unconventional magnon blockade in a dissipative photon–magnon coupling system

We investigate unconventional magnon blockade (UMB) in a dissipative photon–magnon coupling cavity, where the coherent and dissipative coupling between photon and magnon can be adjusted by the position of the yttrium iron garnet sphere in the cavity. An approximate analytical solution for the statistical property of magnon is provided, which is in good agreement with the results obtained by numerically solving the Lindblad master equation. Our results indicate that UMB can be achieved in the case of dissipative photon–magnon coupling, but not in the case of coherent photon–magnon coupling. Secondly, by adjusting the amplitude of the external laser field, dynamic control of UMB can be achieved. Finally, by solving the Lindblad master equation containing the occupation number of thermal magnon and photon, the statistical properties of magnons will transition from nonclassical to classical. It is shown that the above phenomena originate from the interference between different quantum paths. Our research may provide theoretical possibilities for quantum manipulation schemes at the single-magnon level and open up a new avenue for designing single-magnon sources in a dissipative photon–magnon coupling system.

A Study on the Channel Holes’ Diameter Effects of High-Performance Vertical-Channel Flash Memory Cells

Abstract High-performance vertical-channel flash (HVF) memory cells were fabricated on the single crystalline Si (c-Si) sidewalls of the cylindrical deep wells in c-Si substrate. To investigate the diameter effects of the cylindrical deep wells, namely channel holes, on HVF cells, the channel holes with different diameters, ranging from 65 nm to 260 nm, were made. Memory gate stacks of SiO2/ Al2O3/HfO2/Al2O3/TiN/W were formed by ozone oxidation and then ALD with the deposition thicknesses of 1/5/7/8/2/150 nm, respectively. For the devices with their diameters equal to or greater than 150 nm, their electrical properties, such as Vt, SS, DIBL, and program/erase characteristics, are close. As expected, DIBL and SS become better as the diameter increasing due to better gate control with larger diameter. However, large changes were occurred for the devices with the diameters of 90 nm and 65 nm. A simple model based on cylinder bulk for vertical flash memory devices was presented to obtain an approximate analytical solution for depletion-width and explain our experimental data. For the devices with the diameters of 150 nm, the high On/Off current ratio of 107 and relatively large memory window of 4.5 V were achieved. However, programming/erasing efficiency were degraded with hole diameter decreasing.

Exponential series approximation of the SIR epidemiological model

IntroductionThe SIR (Susceptible-Infected-Recovered) model is one of the simplest and most widely used frameworks for understanding epidemic outbreaks.MethodsA second-order dynamical system for the R variable is formulated using an infinite exponential series expansion, and a recursion relation is established between the series coefficients. A numerical approximation scheme for the R variable is also developed.ResultsThe proposed numerical method is compared to a double exponential (DE) nonlinear approximate analytic solution, which reveals two coupled timescales: a relaxation timescale, determined by the ratio of the model’s time constants, and an excitation timescale, dictated by the population size. The DE solution is applied to estimate model parameters for a well-known epidemiological dataset—the boarding school flu outbreak.DiscussionFrom a theoretical standpoint, the primary contribution of this work is the derivation of an infinite exponential, Dirichlet, series for the model variables. Truncating the series yields a finite approximation, known as a Prony series, which can be interpreted as a sequence of coupled exponential relaxation processes, each with a distinct timescale. This apparent complexity can be approximated well by the DE solution, which appears to be of main practical interest.

Analysis of the Impact of Clusters on Productivity of Multi-Fracturing Horizontal Well in Shale Gas

Shale gas reservoirs with nanoporous media have become one of the primary resources for natural gas development. The nanopore diameters of shale reservoirs range from 5 to 200 nm, with permeability ranging from 1 × 10−9 to 1 × 10−6 μm2. The natural gas production from shale gas reservoirs is low, necessitating the use of multi-stage hydraulic fracturing in horizontal wells. Segmented multi-cluster perforation fracturing is an effective method for shale gas extraction in these wells. The number of clusters significantly impacts the productivity of horizontal wells. Therefore, it is essential to analyze the impact of cluster numbers on fracture productivity in shale gas reservoir development. In this study, the equivalent flow resistance method was applied to establish a productivity model for multi-stage hydraulic fracturing horizontal wells in shale gas reservoirs considering diffusion and slip. An approximate analytical solution was obtained, and the effects of cluster length, diffusion coefficient, and fracture network permeability on productivity were analyzed. The results show that gas production gradually increases with the increase in the number of clusters and cluster length. However, as the number of clusters increases, the interference between clusters leads to a decrease in the productivity of individual clusters. As the fracture permeability, fracture network permeability, and diffusion coefficient increase, shale gas production also gradually increases. The permeability of the fracture network has the greatest impact on productivity. These research results are beneficial for the design of clusters in horizontal well fracturing and are of great importance for the development and production of shale gas reservoirs.

Design and kinematics of a novel rigid-flexible coupling hybrid robot for aeroengine blades in situ repair

PurposeThis paper aims to present the mechanical design and kinematics of a novel rigid-flexible coupling hybrid robot to develop a promising aeroengine blades in situ repair technology.Design/methodology/approachAccording to requirements analysis, a novel rigid-flexible coupling hybrid robot is proposed by combining a three degrees of freedom (DOF) parallel mechanism with a flexible continuum section. Then the kinematics models of both parallel mechanism and flexible continuum section are derived respectively. Finally, based on equivalent joint method, a two-step numerical iterative inverse kinematics algorithm is proposed for the whole robot: (1) the flexible continuum section is equivalently transformed to a 2-DOF spherical joint, thus the approximate analytical inverse kinematic solution can be obtained; (2) the accurate solution is derived by an iterative derivation of both parallel mechanism and flexible continuum section.FindingsTo verify structure scheme and the proposed kinematics modeling method, numerical simulations and prototype experiments are implemented. The results show that the proposed kinematics algorithm has sufficient accuracy and computational efficiency in the whole available workspace, that is end-effector position error and orientation error are less than 0.2 mm and 0.01° respectively, and computation time is less than 0.22s.Originality/valueA novel rigid-flexible coupling hybrid robot for aeroengine blades in situ repair is designed. A two-step numerical iterative inverse kinematics algorithm is proposed for this unique hybrid robots, which has good accuracy and computational efficiency.

Approximate solution of a zero-sum linear-quadratic differential game by the auxiliary parameter method: analytical/numerical study

ABSTRACT A zero-sum finite horizon linear-quadratic differential game is considered. First, the terminal-value problem for the game-theoretic Riccati matrix differential equation, associated with the considered game by the solvability conditions, is analysed. This problem may not have the solution in the entire time-interval of the game's duration. Using the method of auxiliary parameter, a sufficient condition (in the terms of the game's data) for the existence of the solution to the aforementioned terminal-value problem in the entire time-interval of the game's duration is derived. An approximate analytical solution to this problem also is obtained as a partial sum of some infinite series of functions arising in the method of auxiliary parameter. Based on this approximate solution, suboptimal state-feedback controls of the players are formally designed and justified. It is shown that the pair of these controls constitutes an approximate-saddle point in the considered game. The theoretical results of the article are illustrated by their application to approximate solution of the problem of pursuit-evasion engagement between two flying vehicles.

An Approach to Estimate the Temperature of an Induction Motor under Nonlinear Parameter Perturbations Using a Data-Driven Digital Twin Technique

To monitor temperature as a function of varying inductance and resistance, we propose a data-driven digital twin approach for the rapid and efficient real-time estimation of the rotor temperature in an induction motor. By integrating differential equations with online signal processing, the proposed data-driven digital twin approach is structured into three key stages: (1) transforming the nonlinear differential equations into discrete algebraic equations by substituting the differential operator with the difference quotient based on the sampled voltage and current; (2) deriving approximate analytical solutions for rotor resistance and stator inductance, which can be utilized to estimate the rotor temperature; and (3) developing a general procedure for obtaining approximate analytical solutions to nonlinear differential equations. The feasibility and validity of the proposed method were demonstrated by comparing the test results with a 1.5 kW AC motor. The experimental results indicate that our method achieves a minimum estimation error that falls within the standards set by IEC 60034-2-1. This work provides a valuable reference for the overheating protection of induction motors where direct temperature measurement is challenging.

Performance enhancement energy harvesting from aeroelastic vibrations using a van der Pol circuit

This paper explores galloping vibration-based electromagnetic energy harvesting (EH) in a Duffing-Rayleigh oscillator coupled to a storage RLC resonator circuit with a van der Pol model. Aeroelastic instability induces the mechanical system’s self-induced vibration. The electromechanical coupling mechanism between the resonant van der Pol circuit and the mechanical structure is obtained by the permanent interaction of the magnet and the coil. We examine two cases in a comparative study. In the first case, we use an RLC resonator circuit without a van der Pol model. The second case considers the RLC resonator circuit with a van der Pol model. Firstly, the modulation equations of slow flow of the vibration and the output voltage responses are obtained using a complexification-averaging (CX-A) method. Furthermore, the analytical approximate solutions of the steady-state responses are derived. The obtained analytical results have been confirmed using numerical simulations. Results show that the Duffing-Rayleigh mechanical oscillator can provide higher electrical energy by introducing a van der Pol circuit instead of without a van der Pol circuit. Moreover, the presence of the van der Pol model with the RLC circuit can significantly reduce the critical wind speed required to harvest energy. This result may provide an optimization process that can be used to supply some guidelines in the design of galloping vibrations-based electromagnetic energy harvesters.

Complex Multi-nonlinearity for piezoelectric energy harvesting systems based on an accurate higher-order perturbation method

A theoretical model of a complex nonlinear coupling energy harvester under hybrid excitations is established with analytical and numerical solutions. Different orders of the method of multiple scales are derived to approximate and analyze the equation of motion for the electromechanical coupling system, thus obtaining higher-order approximate analytical solutions for its operating status and output performance indicators. The significant effects of external excitation, linear and nonlinear damping coefficients, as well as impedance on the transverse displacement, voltage, and power of higher-order and lower-order methods are analyzed. The results show that higher-order solution methodology are more sensitive to parameter excitation. For a large damping, the higher-order analysis has more excellent performance in capturing energy. The higher order method shows more pronounced nonlinear softening phenomenon and it is more suitable for low frequency environments. The higher order methods improves solution accuracy of piezoelectric energy harvesting systems. Consequently, it is easier to derive the nonlinear performance of the harvesting systems, and thus improves the average output power.

Energy harvesting from transverse galloping using a two-degree-of-freedom flow energy harvester

Abstract A two degree-of-freedom (2-DOF) galloping piezoelectric energy harvesting system where both oscillators are excited by the incoming flow is studied in this paper. A coupled lumped-parameter model is developed to simulate the electro-fluid-structural coupling behaviour assuming a quasi-steady aerodynamic flow field. The differential equations governing the dynamics of the lumped system are converted into nonlinear algebraic equations employing the harmonic balance method (HBM) and then solved using Newton’s method. The approximate analytical solutions are compared to the solutions obtained by numerical integration, and the results agree, both qualitatively and quantitatively. The solutions were subsequently used to investigate the effects of the bluff body dimension, mechanical, electrical, aerodynamic, and electromechanical parameters on the performance of the harvester and to illustrate how to optimise some of these parameters to reduce the cut-in wind speed and maximise the power output of the energy harvester. It will be shown that the energy harvesting performance could be significantly improved by introducing piezoelectric transducers on both oscillators and including both masses in the incoming flow. A comparative study between the proposed 2-DOF flow energy harvesting system, a single-degree-of-freedom (SDOF) galloping piezoelectric energy harvester (GPEH), and other configurations of 2-DOF GPEH shows that the present 2-DOF GPEH generates the largest power peak.