Approximate Solution Research Articles

An Analysis of the Stress–Strain State of a Layer on Two Cylindrical Bearings

A spatial problem of elasticity theory is solved for a layer located on two bearings embedded in it. The bearings are represented as thick-walled pipes embedded in the layer parallel to its boundaries. The pipes are rigidly connected to the layer, and contact-type conditions (normal displacements and tangential stresses) are specified on the insides of the pipes. Stresses are set on the flat surfaces of the layer. The objective of this study is to obtain the stress–strain state of the body of the layer under different geometric characteristics of the model. The solution to the problem is presented in the form of the Lamé equation, whose terms are written in different coordinate systems. The generalized Fourier method is used to transfer the basic solutions between coordinate systems. By satisfying the boundary and conjugation conditions, the problem is reduced to a system of infinite linear algebraic equations of the second kind, to which the reduction method is applied. After finding the unknowns, using the generalized Fourier method, it is possible to find the stress–strain state at any point of the body. The numerical study of the stress state showed high convergence of the approximate solutions to the exact one. The stress–strain state of the composite body was analyzed for different geometric parameters and different pipe materials. The results obtained can be used for the preliminary determination of the geometric parameters of the model and the materials of the joints. The proposed solution method can be used not only to calculate the stress state of bearing joints, but also of bushings (under specified conditions of rigid contact without friction on the internal surfaces).

Two-timescale joint service caching and resource allocation for task offloading with edge–cloud cooperation

Task offloading with edge–cloud cooperation has emerged as a pivotal solution for meeting the intricate array of application coupled with dynamically evolving business demand in 6G business scenarios, such as traffic sensing, environmental monitoring, and video surveillance in smart cities. Nonetheless, effectively leveraging heterogeneous edge–cloud network resources for effective task offloading presents substantial challenges. Additionally, the inherent differences in system decision cycles escalate the complexity of the task offloading problem to a new dimension. In this study, we delve into a two-timescale joint service caching and resource allocation optimization for task offloading within edge–cloud cooperation aiming to maximize long-term network performance while adhering to energy constraints. We propose a novel edge–cloud cooperation task offloading scheme that supports both edge–cloud and edge–edge cooperation to effectively balance the edge–cloud and edge–edge loads, promoting the efficient co-utilization of all edge–cloud system resources. Furthermore, we devise an online two-timescale Lyapunov-based joint optimization framework for service caching, task offloading, and computing resource allocation. Our two-timescale decision-making framework can flexibly accommodate the inherent differences in the sensitive decision optimization periods, thereby mitigating the degradation of task offloading performance caused by frequent service caching updates. Finally, theoretical analysis confirms that our proposed algorithm can converge to an approximate optimal solution in polynomial time, and the superiority of our scheme is validated by extensive simulation experiments.

A Scalable Multi-FPGA Platform for Hybrid Intelligent Optimization Algorithms

The Intelligent Optimization Algorithm (IOA) is widely focused due to its ability to search for approximate solutions to the NP-Hard problem. To enhance applicability to practical scenarios and leverage advantages from diverse intelligent optimization algorithms, the Hybrid Intelligent Optimization Algorithm (H-IOA) is employed. However, IOA typically requires numerous iterations and substantial computing resources, resulting in poor execution efficiency. In complex optimization scenarios, IOA traditionally relies on population partitioning and periodic communication, highlighting the feasibility and necessity of parallelization. To address the challenges above, this paper proposes a general hardware design approach for H-IOA based on multi-FPGA. The approach includes the hardware architecture of multi-FPGA, inter-board communication protocols, population storage strategies, complex hardware functions, and parallelization methodologies, which enhance the computing capabilities of H-IOA. To validate the proposed approach, a case study is conducted, in which an H-IOA integrating genetic algorithm (GA), a simulated annealing algorithm (SA), and a pigeon-inspired optimization algorithm (PIO) are implemented on a multi-FPGA platform. Specifically, the flexible job-shop scheduling problem (FJSP) is employed to verify the potential in industrial applications. Two Xilinx XC6SLX16 FPGA chips are used for hardware implementation, encoded in VHDL, and an AMD Ryzen 7 5800U was used for the software implementation of Python programs (version 3.12.4). The results indicate that hardware implementation is 13.4 times faster than software, which illustrates that the proposed approach effectively improves the execution performance of H-IOA.

Solving Interval Type-2 Fuzzy Pentagonal Polynomial Equation Using Horner-Muller's Method

The recent boom in interval type-2 fuzzy sets has attracted many researchers to the fields of fuzzy numbers, algebraic expressions, and iterative methods. Interval type-2 fuzzy pentagonal polynomial (IT2 FPP) is one of the algebraic expressions that combine the knowledge of interval type-2 pentagonal fuzzy numbers and polynomial equations. Solving IT2 FPP in the absence of an iterative method is becoming a common encounter as the solution is often difficult, expensive, and inexact. This paper defines IT2 FPP and the iterative method Horner-Muller’s used to find the approximate solution of the IT2 FPP. The IT2 FPP is introduced by combining the generalization of interval type-2 pentagonal fuzzy numbers and fuzzy polynomials, whereas the Horner-Muller’s method is a combination of Horner’s method and Muller’s method. A numerical example is presented to illustrate the computational procedures for finding the approximate solution. The approximate solution is obtained after two iterations with zero error. A Comparative analysis is also presented to validate the consistency and efficiency of the proposed method compared to benchmark iterative methods.

Numerical Comparisons Between Some Recent Modifications of Fractional Euler Methods

For dealing with systems of Fractional Differential Equations (FDEs), the primary goal of this work is to offer several graphical comparisons performed with the use of specific recent adjustments of the so-called Fractional Euler Method (FEM). These numerical adjustments are the Modified FEM (or MFEM) and Improved Modified FEM (or IMFEM) coupled with the FEM itself. This would reveal the best approximate solution to the fractional-order system consisting of FDEs in comparison with its exact solution.

Vector rogue waves in spin-1 Bose–Einstein condensates with spin–orbit coupling

Abstract We analytically and numerically study three-component rogue waves (RWs) in spin-1 Bose–Einstein condensates with Raman-induced spin–orbit coupling (SOC). Using the multiscale perturbative method, we obtain approximate analytical solutions for RWs with positive and negative effective masses, determined by the effective dispersion of the system. The solutions include RWs with smooth and striped shapes, as well as higher-order RWs. The analytical solutions demonstrate that the RWs in the three components of the system exhibit different velocities and their maximum peaks appear at the same spatiotemporal position, which is caused by SOC and interactions. The accuracy of the approximate analytical solutions is corroborated by comparison with direct numerical simulations of the underlying system. Additionally, we systematically explore existence domains for the RWs determined by the baseband modulational instability (BMI). Numerical simulations corroborate that, under the action of BMI, plane waves with random initial perturbations excite RWs, as predicted by the approximate analytical solutions.

Bivariate Jacobi polynomials depending on four parameters and their effect on solutions of time-fractional Burgers’ equations

The utilization of time-fractional Burgers’ equations is widespread, employed in modeling various phenomena such as heat conduction, acoustic wave propagation, gas turbulence, and the propagation of chaos in non-linear Markov processes. This study introduces a novel pseudo-operational collocation method, leveraging two-variable Jacobi polynomials. These polynomials are obtained through the Kronecker product of their one-variable counterparts, concerning both spatial (x) and temporal (t) domains. The study explores the impact of four parameters (θ,ϑ,σ,ς>−1) on the accuracy of resulting approximate solutions, marking the first examination of such influence. Collocation nodes in a tensor approach are constructed employing the roots of one-variable Jacobi polynomials of varying degrees in x and t. The study delves into analyzing how the distribution of these roots affects the outcomes. Consequently, pseudo-operational matrices are devised to integrate both integer and fractional orders, presenting a novel methodological advancement. By employing these matrices and appropriate approximations, the governing equations transform into an algebraic system, facilitating computational analysis. Furthermore, the existence and uniqueness of the equations under study are investigated and the study estimates error bounds within a Jacobi-weighted space for the obtained approximate solutions. Numerical simulations underscore the simplicity, applicability, and efficiency of the proposed matrix spectral scheme.

The Approximate Solutions of 2D- Burger’s Equations

This paper investigates the use of the fractional variational iteration method (FVIM) to obtain approximate analytical solutions to two dimensional Burger’s Equations with the Atangana-Baleanu fractional operator (ABFO). This study provides insight on the fractional variational iteration method's accuracy and reliability while approximating fractional differential equation solutions.

Time-delay feedback control of nonlinear forced vibration of graphene/piezoelectric sandwich nanoplates under a mechanical shock

In this study, the nonlinear forced vibration of graphene/piezoelectric sandwich nanosheets under mechanical impact was investigated. For the first time, linear (displacement and velocity) and nonlinear (displacement) time-delay proportional derivative (PD) controllers were applied to graphene/piezoelectric sandwich nanosheets, and the primary/secondary resonance under different time-delay parameters were analyzed. An approximate analytical solution for the nonlinear forced vibration control equation was obtained using the multiple scale perturbation method, and the stability of graphene/piezoelectric sandwich nanosheets under primary/secondary resonance conditions was discussed using the Ross Herwitz theorem. Finally, the results of this study were compared with those of other studies. The feasibility of the research methodology was demonstrated. On this basis, the research results indicate that linear displacement delay, nonlinear displacement delay, and velocity delay parameters can suppress the unsteady vibration characteristics of the main system. And comparing the linear and nonlinear displacement time-delay, it is found that the nonlinear displacement time-delay has a greater effect on the system. Specifically, the presence of nonlinear displacement time-delay changes the vibration characteristics of the curve from hard spring characteristics to soft spring characteristics. Comparison of displacement time-delay and velocity time-delay reveals that velocity time-delay gives better control over the system. In addition, the results show that the excitation amplitude, nonlocal parameters, and external voltage enhance the nonlinear characteristics of the system under the main and superharmonic resonance conditions before the time-delay control. Subharmonic resonance conditions, the system parameters make the vibration range wider. After the time-delay control, the influence of these three factors on the system vibration is significantly reduced for the three resonance conditions, especially for the non-local parameters. The results of the study can provide guidance for active control of time delay in different types of nanodevices for engineering applications.

VIBRATIONS OF A VERTICAL BEAM ROTATING WITH VARIABLE ANGULAR VELOCITY

An Euler-Bernoulli beam in vertical position rotating about its symmetry axis along its length is considered. The angular velocity is assumed to have small fluctuations about a constant mean velocity. The partial differential equation of motion is derived first. The equation is cast into a non-dimensional form. The natural frequencies are calculated for the pinned-pinned case. Principle parametric resonances such that the fluctuation frequency being close to two times one of the natural frequencies are considered. By employment of the Method of Multiple Scales, an approximate perturbation solution is found. The frequency response diagrams are drawn and the bifurcation points for transition from the trivial solution to the non-trivial solution are calculated. The conditions for which such resonances occur are exploited in the numerical results.

Hyers–Ulam stability of Nipah virus model using Atangana–Baleanu–Caputo fractional derivative with fixed point method

In this study, we present a novel investigation into the dynamics of the Nipah virus through the lens of fractional differential equations (FDEs), employing the Atangana–Baleanu–Caputo fractional derivative (ABCFD) and the fixed-point approach (FPA). The core contribution of this work lies in establishing the existence and uniqueness of solutions to the proposed FDEs, a critical step for validating the model. Furthermore, we explore the Hyers–Ulam (HU) stability of these generalized FDEs, providing a rigorous mathematical foundation for the stability analysis within the context of viral dynamics. By leveraging the ABCFD, our work extends the classical stability criteria, offering new insights into the role of memory effects in disease modeling. Additionally, we present approximate solutions across various compartments and fractional orders, highlighting the sensitivity of the system to key parameters. Numerical simulations, conducted using the Cullis method, illustrate the impact of fractional orders and validate the theoretical findings, positioning this work as a significant advancement in the application of fractional calculus to epidemiological models.

Solving One-Dimensional Non-Linear Klein-Gordon Equations via Combining Adomian Polynomials and Rohit Transform Method

In this study, one-dimensional non-linear Klein-Gordon equations are solved by applying the integral transform known as the Rohit transform method. The approximate solutions of one-dimensional non-linear Klein- Gordon equations are obtained by combining the Adomian polynomials with the Rohit transform. To show the effectiveness and performance of the Rohit transform method, five one-dimensional non-linear Klein- Gordon type equations are considered and solved. The graphs of the solutions obtained are plotted to indicate the generality and clarity of the proposed method. It can be easily verified that the proposed method yielded the results that satisfy their corresponding non-linear Klein-Gordon equations. The integral Rohit transform combined with Adomian polynomials brought progressive methodologies that offer new insights on the problems (i.e., one-dimensional non-linear Klein-Gordon) examined in the paper, distinguishing itself from existing methods and doubtlessly beginning up new research instructions. Moreover, it reduces the complexity that occurs when non-linear Klein-Gordon are solved by other methods available in the literature. It provided precise results for the specific problems discussed in the paper, surpassing the capabilities of different methods in terms of decision, constancy, or robustness to noise and disturbances.

Numerical discretization of initial–boundary value problems for PDEs with integer and fractional order time derivatives

This paper is mainly concerned with introducing a numerical method for solving initial–boundary value problems with integer and fractional order time derivatives. The method is based on discretizing the considered problems with respect to spatial and temporal domains. With the help of finite difference methods, we transformed the studied problem into a set of fractional differential equations. Then, we implemented the fractional Adams method to solve this set in order to provide approximate solutions to the main problem. This combination results in an algorithm that can efficiently and accurately solve a general class of integer and fractional order initial–boundary value problems, such that it does not need to solve large systems of linear equations. In addition, we discussed the stability of the proposed scheme. Three illustrative examples are numerically solved to reveal the effectiveness and validity of the proposed technique.

Computing two actuarial quantities under multilayer dividend strategy with a constant interest rate: Based on Sinc methods

As risk processes with some kind of dividend strategies receive remarkable attention in the financial market, the analysis of risk quantities in the presence of interest (or return) has become an important issue in insurance risk theory, and the corresponding research should be of concern to actuaries. In this paper, we study a perturbed dual risk model with a constant interest rate and a multilayer threshold strategy. We derive respectively the integral differential equations satisfied by the expected present value of the total dividend until ruin and the Laplace transform at the time of ruin. By applying the Sinc method developed in this paper, we derive the approximate solutions of the integral differential equations satisfied by these two risk quantities. Finally, various numerical examples are provided to demonstrate the feasibility of the Sinc method.

A new approximation solutions for Fractional Order Biological Population Model

The homotopy permutation approach is used in this article to solve the fractional-order biological population model (FOBPM). The fractional derivative is defined using the Atangana-Baleanu operator (ABO). The suggested technique provides a number of solutions for FOBPM. The HPM technique is regarded as one of the finest analytical processes for solving fractional-order, nonlinear PDEs, notably the FOBPM, since it requires less computations and has a greater rate of convergence than other analytical approaches.

Poisson noise removal based on non-convex hybrid regularizers

The presence of TV regularizer always induces an unsatisfactory staircase effect. To overcome the staircase while better sustaining edge information, this work proposes a novel model for Poisson noise removal. The model is based on non-convex mixed regularizers, which involves introducing a non-convex penalty into a composition of the total variation and the higher-order total variation. The iterative reweighted l1 algorithm was used to convert the non-convex model into a convex one. The classic alternating direction method of multipliers was then employed to obtain approximate solutions of the model. When applying this model to degraded images contaminated by Poisson noise of medium to high intensity, its performance in noise suppression was tested. The regularizer was compared with others in the terms of the visual effect of the picture, time cost, and several commonly accepted quantitative indicators for evaluation, such as peak signal-to-noise ratio, feature similarity index and structural similarity index. Numerical experiments showed that the present model not only eliminates block artifacts but also retains sharp edges.

Intrinsic squeezing of mechanical motion of a harmonically trapped atom with spin-orbit coupling

In this paper we investigated one-dimensional harmonically trapped atom with Raman-induced spin-orbit coupling by mapping to quantum-like model. Using displacement Fock states in quantum optics and variational methods we found the odd-parity superposition state of left- and right-displaced oscillator state is a good approximate solution to the ground state, which captures the essential physics of the system. The ground state wave function owes nodes which is remarkably different from the conventional no-node theory in quantum mechanics. The results show the atomic motion of center of mass exhibits intrinsic squeezing properties with relation to the Raman and spin-orbit coupling strength characterized by the variance in position and momentum, information entropy, as well as the Wigner function of the ground state. The dynamics of center of mass are also studied which demonstrates intuitive Zitterbewegung oscillations in momentum space and real space. Our results could be used for observing Zitterbewegung oscillations in neutral atomic gases and engineering nonclassic states.

Analytical approximate solutions of some fractional nonlinear evolution equations through AFVI method

In this article, we construct and investigate a new fractional variational iteration technique which is named as AFVI method. After that, we formulate some IVPs corresponding to the fractional nonlinear evolution equations NLTFFWE, mNLTFFWE, TFmCHE and TFmDPE. The Caputo fractional order derivative has been used to fractionalize the considered NLEEs. Then, we solve the considered IVPs by applying the formulated AFVI method. Lastly, we check the accuracy of the attained AASs by making some graphical and numerical comparisons with their corresponding exact solutions and other existing equivalent AASs. The result of this article confirm that the efficiency, appropriateness and time spent capability of the newly constructed AFVI method are better than that of the other existing analogous fractional analytical approximation methods. Here, we apply the Maple 2021 programming software to obtain the AASs of the formulated IVPs and drawing the 3D graphs of the attained AASs. Finally, in this article we validate the applicability of the Caputo fractional order derivative to form a novel analytical approximation method as well as to fractionalize some essential NLEEs of Mathematical physics.

Innovative binary nanofluid approach with copper (Cu - EO) and Magnetite ([formula omitted] - EO) for enhanced thermal performance

In this study, we present an innovative and detailed analysis of the motion of a binary nanofluid, specifically copper (Cu - EO) and Magnetite (Fe3O4 - EO), in a porous medium under solar radiation. Our research is driven by the potential to significantly enhance the thermal performance of a Parabolic Trough Solar Collector (PTSC) using a binary nanofluid. The boundary conditions, partial differential restrictions, and the governing PDEs are transformed into the ODEs system with appropriate similarity transformations. The finite element method (FEM) and semi-analytical technique memad Akbari-Ganji's Method (AGM) are used to solve the approximate solutions of ODEs. The findings of evaluating Copper (Cu - EO) and Magnetite (Fe3O4 - EO) were the two different nanofluids based on engine oil that were presented. Furthermore, the model's overall entropy variations are enhanced using Reynolds. Moreover, our study contributes to understanding fluidity and viscosity alterations in such systems. We use the Brinkman number to observe these alterations. We reveal that the thermal efficiency of (Cu - EO) is lower than (Fe3O4 - EO) by a range of 0.005–0.6. This indicates a substantial enhancement in the thermal performance of the PTSC.

An improved analytical solution on viscous dissipation effect in extended Stokes’ second problem in microchannel with isothermal boundaries

In this analytical study, the fluid motion within a microchannel is induced by the oscillation of one surface parallel to the other stationary surface, termed the extended Stokes’ problem. The novelty and research gap are acquiring the thermal effect of such motion due to the viscous dissipation or fluid friction, subject to symmetric isothermal boundary conditions. The study may shed light on the role of viscous dissipation in temperature rise in the synovial fluid of an artificial hip joint, or in the fluid layer of a mechanical bearing. The full exact analytical temperature field, until now, has been unsolved, as it involves unsteady flow with manipulation of a complicated velocity field. The assumptions in the model are one-dimensional, incompressible, laminar, Newtonian flow with constant properties in a microchannel. Through the methodology of partial differential equation analysis, the temperature field is obtained in terms of Brinkman number, Prandtl number and a dimensionless angular frequency, and results are verified with a reported numerical solution, for specified range of the variables. Results complement recent approximate solutions which are valid only for the limited condition of the dimensionless angular frequency being less than or equal to unity, whereby suggesting a new Stokes number.