Approximate Solution Research Articles

Random flutter analysis of a novel binary airfoil with fractional order viscoelastic constitutive relationship

The fractional order model emerges naturally when the integer order model fails to meet practical requirements and finds extensive applications in engineering practice. However, there is still limited research on the random flutter of fractional order viscoelastic binary airfoils. This study aims to explore the random flutter of such airfoils under the influence of Gaussian white noise excitation by utilizing the pth moment Lyapunov exponent. Firstly, a two-degree-of-freedom fractional order stochastic dynamic equation is established for the viscoelastic binary airfoil through the introduction of fractional order damping. The singular perturbation method is then employed to derive the pth moment Lyapunov exponent of the system. An approximate analytical solution of the pth moment Lyapunov exponent is obtained through Fourier series expansion, which exhibits good agreement with numerical results obtained from Monte Carlo simulations. Subsequently, a comprehensive analysis is conducted to examine the impacts of fractional order, airflow velocity, noise intensity, and key structural parameters on the stochastic flutter of the fractional order viscoelastic binary airfoil. The findings indicate that compared with the typical binary airfoil model, the flutter speed of the fractional-order viscoelastic binary airfoil is significantly increased, the stability of the system is highly sensitive to the natural frequency of the system, and the fractional-order factor has a positive effect on the stochastic stability of the binary airfoil before the critical value. The above research results can guide the design optimization of the fractional-order viscoelastic airfoil.

The dynamics around the collinear points of the elliptic three-body problem: A normal form approach

We study the dynamics of the collinear points in the planar, restricted three-body problem, assuming that the primaries move on an elliptic orbit around a common barycenter. The equations of motion can be conveniently written in a rotating-pulsating barycentric frame, taking the true anomaly as independent variable. We consider the Hamiltonian modeling this problem in the extended phase space and we implement a normal form to make a center manifold reduction. The normal form provides an approximate solution for the Cartesian coordinates, which allows us to construct several kinds of orbits, most notably planar and vertical Lyapunov orbits, and halo orbits. We compare the analytical results with a numerical simulation, which requires special care in the selection of the initial conditions.

On sparsity of approximate solutions to max-plus linear systems

On sparsity of approximate solutions to max-plus linear systems

Analysis of the COVID-19 pandemic and prediction with numerical methods

Coronavirus spreads worldwide with various symptoms and strains such as COVID-19, SARS, MERS. Outbreak and prevention of the coronavirus can be studied by many mathematical models like SIR, SEIR and SEIQDR. In this paper, we employed Euler’s method based on the SEIQDR mathematical model of COVID-19. Also, after certain iterations of the population categories, we obtained the solution range of the SEIQDR model. In the end, we studied the impact of [Formula: see text] for integer 1 and fractional order 0.9 and 0.85 on the obtained numerical solutions. The graphical solutions are also analyzed for integer and fractional values of [Formula: see text]. The advantage of this proposed method is that it reduces the mathematical and numerical computations significantly and the proposed method is effective and efficient in extracting the approximate solution of various population categories for COVID-19 mathematical model.

A case study of displacement, stress condition and failure criterion of surrounding rock in deep super-large section double chambers in the Longgu Coal Mine, Shandong, China

To investigate the stability of a super-large section chamber group, an analysis was conducted based on the in-situ geological conditions of a super-large section chamber group utilized as a coal gangue separation system at the Longgu Coal Mine. Field measurement and numerical simulation were employed to analyze the failure area and stress state of the surrounding rock under varying chamber spacings. The results indicate that the extent of the plastic zone significantly expands when the spacing between chambers is less than 2.0 times the chamber width. When the distance between chambers is 1.5 times the chamber width, it results in the rock pillars being entirely within the plastic zone. As the chamber spacing decreases, the tangential stress within the rock pillar range increases. When the chamber spacing is 2.5, 2.0, and 1.5 times the chamber width, the maximum tangential stress is 1.19, 1.46, and 1.18 times that in the case of a single chamber, respectively. Based on the displacement analysis, it was observed that as the distance between the chambers decreases, there is a notable increase in the displacement of the pillar sides and chamber top, indicating a higher risk of collapse. Integrating the plastic area and stress analysis allows for the categorization of the rock pillar area into four sections: the broken area, loose area, stable area, and firm area. Drawing upon the theoretical solution of the plastic zone of a circular chamber and the equivalent radius method, an approximate solution for the plastic zone of a non-circular chamber has been provided. Furthermore, the minimum reasonable spacing between chambers in a super-large section chamber group is provided as a distance criterion for the failure of a double chamber.

Trapping of particles diffusing in cylindrical cavity of arbitrary length and radius by two small absorbing disks on the cavity side wall: Narrow escape theory and beyond.

Narrow escape theory deals with the first passage of a particle diffusing in a cavity with small circular windows on the cavity wall to one of the windows. Assuming that (i) the cavity has no size anisotropy and (ii) all windows are sufficiently far away from each other, the theory provides an analytical expression for the particle mean first-passage time (MFPT) to one of the windows. This expression shows that the MFPT depends on the only global parameter of the cavity, its volume, independent of the cavity shape, and is inversely proportional to the product of the particle diffusivity and the sum of the window radii. Amazing simplicity and universality of this result raises the question of the range of its applicability. To shed some light on this issue, we study the narrow escape problem in a cylindrical cavity of arbitrary size anisotropy with two small windows arbitrarily located on the cavity side wall. We derive an approximate analytical solution for the MFPT, which smoothly goes from the conventional narrow escape solution in an isotropic cavity when the windows are sufficiently far away from each other to a qualitatively different solution in a long cylindrical cavity (the cavity length significantly exceeds its radius). Our solution demonstrates the mutual influence of the windows on the MFPT and shows how it depends on the inter-window distance. A key step in finding the solution is an approximate replacement of the initial three-dimensional problem by an equivalent one-dimensional one, where the particle diffuses along the cavity axis and the small absorbing windows are modeled by delta-function sinks. Brownian dynamics simulations are used to establish the range of applicability of our approximate approach and to learn what it means that the two windows are far away from each other.

Novel Approach by Shifted Fibonacci Polynomials for Solving the Fractional Burgers Equation

This paper analyzes a novel use of the shifted Fibonacci polynomials (SFPs) to treat the time-fractional Burgers equation (TFBE). We first develop the fundamental formulas of these polynomials, which include their power series representation and the inversion formula. We establish other new formulas for the SFPs, including integer and fractional derivatives, in order to design the collocation approach for treating the TFBE. These derivative formulas serve as tools that aid in constructing the operational metrics for the integer and fractional derivatives of the SFPs. We use these matrices to transform the problem and its underlying conditions into a system of nonlinear equations that can be treated numerically. An error analysis is analyzed in detail. We also present three illustrative numerical examples and comparisons to test our proposed algorithm. These results showed that the proposed algorithm is advantageous since highly accurate approximate solutions can be obtained by choosing a few terms of retained modes of SFPs.

A new approximation to the first order fractional derivative in the Caputo-Fabrizio sense using Haar Wavelet integration formula

Decades ago, fractional calculus arose to generalize ordinary derivation and integration, and then became a means of modeling and interpreting many phenomena in various fields such as engineering, physics, chemistry, biology and signal processing. The definition of the fractional derivative began with a derivative with a singular kernel, such as the Riemann-Liouville and Caputo derivative. Due to the singularity of the kernel, the definition of Caputo-Fabrizio appeared, which has a non-singular kernel and mathematical properties similar to the derivative of the integer order. This last definition attracted many mathematicians and researchers to use it in modeling phenomena and obtaining historical information about the development of the studied phenomena, but usually the analytical solution does not exist, which necessitated numerical methods to find an approximate solution. These approximate methods depend on finding an approximate formula for the fractional derivative, and then the problem is transformed into a system of algebraic equations that is easy to solve. In fact, all the numerical methods that have been used have a polynomial rate of convergence, which calls for thinking about a new method that is more effective and has a better rate of convergence. For this reason, in this paper, we propose an efficient numerical method to approximate the first order fractional derivative in the Caputo-Fabrizio sense. This method develops a new quadratic formula using Haar wavelet integration method. Error analysis of our proposed method gives an exponential convergence rate of . To check the effectiveness of the proposed method, we examine some examples with different fractional orders. The quantative results demonstrated the stability and efficiency of the proposed method.

Imaging method of surface defect using leaky Rayleigh wave with ultrasonic phased array

ABSTRACT Leaky Rayleigh wave testing has the advantages of non-contact, high sensitivity, and good beam directivity. However, current leaky Rayleigh wave testing mostly uses monolithic transducers and is seriously affected by acoustic wave attenuation and diffusion, resulting in poor image quality, limited detection range and low imaging efficiency. In this paper, a novel leaky Rayleigh wave testing method using an ultrasonic phased array is developed, and the total focusing method (TFM) is used to achieve high-quality imaging of defects. To improve imaging efficiency, the root-mean-square velocity (RMSV) model is utilised to rapidly acquire an approximate analytical solution for the time of flight of the acoustic wave. To obtain more accurate defect images further, the amplitude of the total focusing imaging signal is calibrated. The experimental results demonstrate that the proposed method achieves a 48% improvement in the imaging accuracy of a 1 mm hole, compared to the traditional Synthetic Aperture Focusing Technique (SAFT) imaging method. The proposed method achieves an 92% reduction in imaging time compared to the conventional TFM. The proposed method provides a new non-contact approach for high-accuracy and high-efficiency defect imaging using leaky Rayleigh wave.

Analytical and Numerical Conceptions of Flame Propagation in Porous Media with a Generalization in Diffusion and Reaction Modeling

ABSTRACT In this paper, we expand on previous studies to further describe the flame propagation dynamics in porous media. We develop a model that incorporates a p-Laplacian operator to represent both fast and slow diffusion effects. Additionally, we include an advection term and a modified reaction term to address a broader range of chemical reactions related to flame spread. Based on analytical assessments, we first ascertain fundamental properties of the model’s solutions, such as regularity, boundedness, and uniqueness. We also examine stationary solutions, deriving their approximate analytical forms using regular perturbation methods and analyzing them within a Hamiltonian framework. Finally, we compare these analytical solutions with numerical ones to determine the spatial region where the approximate analytical solutions are valid within acceptable levels of error.

Fast hybrid method for solving variational inequalities beyond monotonicity in real Banach spaces

In this work, we introduce a new inertial hybrid algorithm that combines the subgradient extragradient, Halpern, and S-iteration methods. We provide an approximate solution to variational inequality and common fixed point problem of relatively non-expansive mappings in real Banach spaces, where our cost operator is Lipschitz continuous and quasimonotone, and our step size is self-adaptive, non-monotone, and non-reliant on any line-search principle. We provide two strong convergence results with (and without) recourse to the monotonicity property, and further conduct several relevant numerical experiments to reveal the efficiency of our method over others in the literature.

Approximate analytical solution of a soil water movement equation under different ponding radii on the basis of numerical simulation

AbstractThis study addresses the problem of 2D soil water movement under ponding radii of 1, 2, and 3 cm. The soil water movement characteristics (shape parameters of the water content profile, ratio of horizontal wetting front to vertical wetting front, relationship between infiltration time and horizontal wetting front, and relationship between infiltration time and cumulative infiltration) under the above three kinds of water ponding radius were analyzed. On the basis of the assumption that the soil wetting body is a semi‐ellipse and the analytical solution of the 1D soil water movement equation at any angle, the approximate analytical solution of the 2D soil water movement equation under ponding conditions is optimized. The function relationships between infiltration time, wetting front, and cumulative infiltration are established. We applied the numerical data simulated by HYDRUS‐3D to validate the parameters in proposed analytical solutions and evaluated the relationships between the wetting front and hydraulic parameters. The results indicate that as the water ponding radius increases, the wetting body and 2D water content distribution becomes larger. When the water ponding radius was 2 cm, the numerical and analytical solution of 1D soil water distribution showed the best comparison results, and the model error was the smallest. The ratio of wetting fronts was linearly increased with the increase of air‐entry suction with R2 = 0.9969.

Analytical and numerical study of light-induced optical microcavity generation by half-cycle light pulses in the resonant medium

An optical Bragg-like microcavity can be formed when extremely short light pulses interacting coherently with the medium collide in the resonant medium. Here we present an analytical approach, based on the approximate solution of the time-dependent Schrödinger equation, that reveals both the cavity formation and its control by the half-cycle pulses colliding in the medium. This approach allows the calculation of cavity properties in the weak-field regime without the need for complex numerical simulations. Beyond the approximate analytical solutions, our analytical results are confirmed by numerical simulations of the Maxwell-Bloch equations. Grating formation is also studied in the strong-field regime, where collisions of self-induced transparency half-cycle pulses occur. The results presented here reveal, to our knowledge, a new type of spatio-temporal photonic crystal in the form of micro-cavities in a simple atomic medium. The results obtained demonstrate the feasibility of attosecond optical switching in a simple atomic medium with half-cycle pulses.

Pseudo-supersymmetric approach to the Dirac operator in the Schwarzschild spacetime

Abstract We have discussed the Dirac equation in Schwarzschild spacetime using pseudo-supersymmetric quantum mechanics and have obtained the partner Hamiltonian of the initial Hamiltonian operator. We demonstrate that the partner metric tensors, corresponding to these Hamiltonians, can be derived using the intertwining relations inherent in pseudo-supersymmetric approaches. We have seen that the Schwarzschild metric may be expanded into Schwarzschild–Tangherlini metric using the aspects of pseudo-supersymmetry. Furthermore, we have derived approximate solutions for the radial component within the framework of N = 2 supersymmetry with the corresponding radial potential graphs presented. Subsequently, our discussion extends to the quasinormal modes, focusing particularly on the asymptotic limits as r approaches ± ∞ .

Mathematical Evaluation and Dynamic Transmissions of a Cervical Cancer Model Using a Fractional Operator

This scientific research investigates the fractional-order compartmental model of cervical cancer. The human papillomavirus (HPV) is responsible for the development of cervical cancer. The Caputo-Fabrizio fractional operator, which includes a section for treatment with antiretroviral drugs, is employed to examine this worldwide phenomenon. The mathematical model of cervical cancer that has been proposed recently is an extension of the integer-order model. The fractional derivative technique is a novel approach for these types of biological models. This paper employs fixed point theory to provide requirements for the existence, uniqueness, and stability of the fractional order cervical cancer model, utilizing the Caputo- Fabrizio operator. We have initially found an approximation solution to a proposed model via the iterative Laplace transform method, which is easily accessible. This approach integrates the Laplace transform technique with a reliable, novel iterative method. We evaluated parameters indicating the advancement of the illness and presented the numeric simulations. The observed outcomes indicate that a fractional factor has a crucial effect on controlling cervical cancer.

Numerical analysis of a time discretized method for nonlinear filtering problem with Lévy process observations

In this paper, we consider a nonlinear filtering model with observations driven by correlated Wiener processes and point processes. We first derive a Zakai equation whose solution is an unnormalized probability density function of the filter solution. Then, we apply a splitting-up technique to decompose the Zakai equation into three stochastic differential equations, based on which we construct a splitting-up approximate solution and prove its half-order convergence. Furthermore, we apply a finite difference method to construct a time semi-discrete approximate solution to the splitting-up system and prove its half-order convergence to the exact solution of the Zakai equation. Finally, we present some numerical experiments to demonstrate the theoretical analysis.

Performance of surge-induced synchronized switch harvesting on inductor as an efficient technique under strong electromechanical coupling and low amplitudes of vibration

Abstract The process of harvesting energy from ambient sources is key for various applications. This study examined the performance of two representative techniques from the surge-induced synchronized switch harvesting on inductor (S3HI) family under strong electromechanical coupling and compared this performance with that of some established techniques. S3HI techniques exploit the surge voltage to overcome the voltage barrier of diodes for rectifiers and storage capacitor voltages. One of their features, revealed in a previous study on weakly coupled systems, is that they can harvest substantial energy from low-level vibrations. This feature is desirable for certain use cases. This study aimed to clarify whether this feature is applicable even when the coupling is strong. The performance of various established techniques and two representatives from the S3HI group was studied by formulating an approximate analytical solution and performing numerical simulations and experiments. The theoretical results were confirmed to be consistent, and the discrepancy between experimental and theoretical results was minor. These results clearly demonstrate that the techniques from the S3HI family can effectively harvest substantial energy from small-amplitude vibrations, even when the coupling is strong. Moreover, the performance advantage of S3HI methods is even greater when the coupling is strong.

Parameter Tuning of Agent-Based Models: Metaheuristic Algorithms

When it comes to modelling complex systems using an agent-based approach, there is a problem of choosing the appropriate parameter optimisation technique. This problem is further aggravated by the fact that the parameter space in complex agent-based systems can have a large dimension, and the time required to perform numerical experiments can be large. An alternative approach to traditional optimisation methods are the so-called metaheuristic algorithms, which provide an approximate solution in an acceptable time. The purpose of this study is to compare various metaheuristic algorithms for parameter tuning and to analyse their effectiveness applied to two agent-based models with different complexities. In this study, we considered commonly used metaheuristic algorithms for agent-based model optimisation: the Markov chain Monte Carlo method, the surrogate modelling approach, the particle swarm optimisation algorithm, and the genetic algorithm, as well as the more novel chaos game optimisation algorithm. The proposed algorithms were tested on two agent-based models, one of which was a simple toy model of the spread of contagious disease, and the other was a more complex model of the circulation of respiratory viruses in a city with 10 million agents and 26 calibrated parameters.

Numerical Study on the Impact Pressure of Droplets on Wind Turbine Blades Using a Whirling Arm Rain Erosion Tester

The leading-edge erosion of a wind turbine blade was tested using a whirling arm rain erosion tester, whose rotation rate is considerably higher than that of a full-scale wind turbine owing to the scale effect. In this study, we assessed the impact pressure of droplets on a wet surface of wind turbine blades using numerical simulation of liquid droplet impact by solving the Navier–Stokes equations combined with the volume-of-fluid method. This was conducted in combination with an estimation of liquid film thickness on the rotating blade using an approximate solution of Navier–Stokes equations considering the centrifugal and Coriolis forces. Our study revealed that the impact pressure on the rain erosion tester exceeded that on the wind turbine blade, attributed to the thinner liquid film on the rain erosion tester than on the wind turbine blade caused by the influence of centrifugal and Coriolis forces. This indicates the importance of correcting the influence of liquid-film thickness in estimating the impact velocity of droplets on the wind turbine blade. Furthermore, we demonstrated the correction procedure when estimating the impact velocity of droplets on the wind turbine blade.

Application of the Continuous Method for Solving Operator Equations to the Approximate Solution to the Amplitude–Phase Problem

Application of the Continuous Method for Solving Operator Equations to the Approximate Solution to the Amplitude–Phase Problem