Analytical Numerical Solutions Research Articles

Dynamical investigation of the perturbed Chen–Lee–Liu model with conformable fractional derivative

Abstract This study focuses on the investigation of the perturbed Chen–Lee–Liu model with conformable fractional derivative by the implementation of the generalized projective Riccati equations technique. The proposed method uses symbolic computations to provide a dynamic and powerful mathematical tool for addressing the governing model and yielding significant results. Numerous analytical solutions of the governing model, including bell-shaped soliton solutions, anti-kink soliton solutions, periodic solitary wave solutions and other solutions, have been constructed effectively utilizing this effective technique. The findings acquired from the governing model utilizing the suggested technique demonstrate that all results are novel and presented for the first time in this study. Solitons are of immense significance in the domain of nonlinear optics due to their inherent ability to preserve their shape and velocity during propagation. The study of the propagation and the dynamical behaviour of the derived results have been explored by representing them graphically through 3D, density, and contour plots with different selections of arbitrary parameter values. The solitons acquired from the proposed model can provide significant advantages in the field of fiber-optic transmission technology. The obtained results demonstrate that the suggested approach is extremely promising, straightforward, and efficient. Furthermore, this approach may be effectively used in numerous emerging nonlinear models found in the fields of applied sciences and engineering.

Numerical simulation of Alfvén dark envelope soliton in Hall-MHD plasmas

This study investigates the propagation of dark Alfvén solitons in low-β magnetized plasma using Hall-magnetohydrodynamic (Hall-MHD) numerical simulations. The rational solution of the nonlinear Schrödinger equation (NLSE) is presented, which is proposed as an effective tool for studying dark envelope solitons in plasma. Our results show a high degree of agreement between numerical simulations and analytical solutions derived from the NLSE via the reductive perturbation method. This agreement validates our modeling and computational approach. In addition, the simulations confirm the existence of dark Alfvén solitons in magnetized plasma. This work demonstrates the effectiveness of Hall-MHD simulations in studying complex plasma phenomena, contributing to the broader understanding of soliton propagation in plasma environments.

Symmetry-Optimized Dynamical Analysis of Optical Soliton Patterns in the Flexibly Supported Euler–Bernoulli Beam Equation: A Semi-Analytical Solution Approach

This study employs spatial optimization principles to investigate the nonlinear vibration of a flexibly supported Euler–Bernoulli beam, a (1 + 1)-dimensional system subjected to axial loads. The modified Khater method, a crucial tool in mechanical engineering, is utilized to analyze analytical solutions, which include a symmetric spatial representation of the waveform as an integral part of each solution. Notably, periodic soliton solutions for the nonlinear model closely align with numerical and approximate analytical solutions, demonstrating the accuracy of our modeling approach. Density diagrams, contour diagrams, and Poincaré maps depicting the obtained analytical solutions are presented to elucidate their accuracy and provide visual confirmation of the optimized engineering model’s physical significance. The planar dynamical system is derived through the Galilean transformation by employing mathematical models and appropriate parameter values, thereby further refining problem understanding. Sensitivity analysis is conducted, and phase portraits with equilibrium points are illustrated by analyzing a special case of the investigated dynamical system, emphasizing its symmetrical properties. Lastly, we perform a global analysis to identify periodic, quasi-periodic, and chaotic behaviors, with an extra weak forcing term confirmed by Poincaré maps and a two-dimensional symmetric basin of the largest Lyapunov exponent.

Prediction method of thermal expansion coefficient for 2D composite fabric

In this article, an analytical method for predicting the thermal expansion coefficient of 2D composite fabrics is established. First of all, different from the traditional prediction of thermal expansion coefficient by classical laminate theory, this paper extends the self-consistent method of micromechanics to 2D composite fabrics to improve the accuracy of thickness direction prediction. Analytical Prediction method of Thermal expansion coefficient of 2D Composite fabric (APTC)was established. In the x section of Representative Volume Element(RVE) model, the stress balance equation and deformation coordination equation are applied to derive the internal thermal expansion coefficient. In the z section, the coefficient of thermal expansion is derived from the Poisson effect. Secondly, the formula of thermal expansion coefficient is derived for plain fabric, twill fabric and satin fabric respectively, which is consistent with the APTC method, indicating that the method is representative. Finally, RVE finite element model is established according to the braided structure parameters of 5284/T800 composite fabric. Comparing analytical solution, numerical solution and experimental solution, the α 1 and α 2 of the numerical and experimental solutions is close enough to be considered equal, which is consistent with the transverse isotropy of the composite fabric in xOy plane predicted by theoretical analysis.When predicting α 1 and α 2 , the maximum error between numerical solution and analytical solution is 1.79%, the maximum error between numerical solution and experimental solution is 22.29%, and the error remains at about 4% during predicting α 3 . In general, the APTC method for predicting the thermal expansion coefficient of composite fabrics is accurate and effective, and the simplified formula is more convenient to popularize and apply.

Numerical and Experimental Analysis of Buckling and Post-Buckling Behaviour of TWCFS Lipped Channel Section Members Subjected to Eccentric Compression.

The paper presents a static analysis of the buckling and post-buckling state of thin-walled cold-formed steel (TWCFS) lipped channel section beam-columns subjected to eccentric compression. Eccentricity is taken into consideration relative to both major and minor principal axes. An analytical-numerical solution to the buckling and post-buckling problems is described. The solution is based on the theory of thin plates. Equations of equilibrium of section walls are derived from the principle of stationary energy. Then, to solve the problem, the finite difference (FDM) and Newton-Raphson methods are applied. Linear (buckling) and nonlinear (post-buckling) analyses are performed. As a result, pre- and post-buckling equilibrium paths are determined. Comparisons of the obtained numerical results, FE simulation results, and experimental test results are carried out and presented in comparative load-shortening diagrams. Additionally, a comparison of the buckling forces and buckling modes obtained from theoretical analysis and experiments is presented.

A fresh perspective on solution to fragmentation problems

The population balances offer a continuum framework to model fragmentation of discrete size drops, bubbles, particles, crystals, polymer chains, etc. We have examined the available approaches to solve the resulting population balances and the older discrete kinetic equations. The analytical solutions have enhanced our understanding of breakup processes. We offer in this work new analytical solutions using a fresh and counter-intuitive approach. The approximate numerical solutions approach analytical solutions as the fineness of discretization increases. We looked for scaling patterns in coarse numerical solutions and harnessed the observed scaling to obtain closed-form analytical solutions in the limit of infinitely fine discretization. The approach provides analytical solutions for a family of breakup functions, including evolving populations of infinite number of particles. The success of the new approach is contingent on the discretization step having internal consistency concerning particle number and mass. The approach holds promise for empirical breakage functions.

Numerical solution, conservation laws, and analytical solution for the 2D time-fractional chiral nonlinear Schrödinger equation in physical media

Numerical solution, conservation laws, and analytical solution for the 2D time-fractional chiral nonlinear Schrödinger equation in physical media

Micro-PIV study on the influence of viscosity on the dynamics of droplet impact onto a thin film

This work presents a systematic experimental study of droplet impact onto a wet substrate. Four different silicone oils are used, covering a range of Reynolds number between 116<Re<1106\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$116< \ ext{Re} <1106$$\\end{document} at two different initial wall film heights. The objective is to characterize the temporal and radial evolution of the velocity field within the crown crater by means of micro-PIV. Our findings show that the velocity field has the structure of an axisymmetric stagnation point flow with decaying strength a(t). The latter exhibits an exponential decay and can be explained in terms of the exponential decay of the pressure force exerted by the impacting droplet onto the wall film. In this context, the commonly accepted functional dependence a(t)∝t-1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$a(t) \\propto t^{-1}$$\\end{document} represents only the first-order Taylor approximation of the exponential decay and has therefore only a limited temporal validity. The analysis also corroborates the existence of an inertial regime concerning the velocity field for Re>270\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\ ext{Re}} > 270$$\\end{document}. This is not observed at lower Re numbers due to the increased pressure losses caused by the extensional (normal) strain during the radial spreading of the lamella. To validate these findings a holistic approach is chosen, which combines numerical results, analytical solutions and experimental data from literature. In particular, by using the continuity equation, it is shown that the experimental decay of the wall film height can be reconstructed from the velocity measurements. Consilience of results from different approaches provides a robust validation of the micro-PIV data obtained in this work.Graphical abstract

Heat transfer simulation of reline flowing in an elliptic shaped duct: A deep eutectic solvent

Deep Eutectic solvents have emerged as promising alternatives to conventional solvents due to their unique properties and applications. The flow of deep eutectic solvents in various industrial processes has garnered significant attention due to their versatile applications in fields ranging from chemical engineering to energy storage. This study presents a comprehensive mathematical model aimed at elucidating the intricate behavior of eutectic solvent flow within an elliptic duct, a geometric configuration relevant to many real-world systems. In this article, the deep eutectic solvent is composed of choline chloride–urea and is also called Reline. The proposed mathematical model accounts for the complex interplay of fluid dynamics, thermodynamics, and elliptic duct geometry. Key components of the model include the Navier-Stokes equations, which describe the fluid flow, coupled with heat transfer equations to account for temperature variations within the system. The model also considers the phase change behavior of the eutectic solvent, which may exhibit solidification or crystallization phenomena under certain conditions. Numerical simulations and analytical solutions are employed to investigate various aspects of eutectic solvent flow within elliptic ducts, such as velocity profiles, pressure distributions, temperature gradients, and phase transition phenomena. The study explores the influence of key parameters, including the Reynolds number, the aspect ratio of the duct, and the thermophysical properties of the eutectic solvent, on the system’s behavior. From the results, it was clearly observed that the velocity at the narrow region decreased as the pressure raised and the Reynold’s number profile indicated the presence of turbulent flow behavior.

Integral Transform Solution for Natural Convection Within Horizontal Concentric Annular Cavities

Abstract Natural convection inside horizontal concentric annular cavities is dealt with through the generalized integral transform technique (GITT), offering a hybrid numerical-analytical solution of the continuity, Navier–Stokes, and energy equations in cylindrical coordinates. The flow is in steady-state, laminar regime, two-dimensional, buoyancy-induced, and the governing equations are written in the streamfunction-only formulation. Two strategies of integral transformation are adopted to verify the best computational performance, namely, the usual one with eigenvalue problems for both streamfunction and temperature defined in the radial variable, and a novel alternative with eigenvalue problems defined in the azimuthal angular coordinate. First, the eigenfunction expansions convergence behavior is analyzed to critically compare the two integral transform solution strategies. Then, test cases for different aspect ratios and Rayleigh numbers are validated with experimental data from the classical work of Kuehn and Goldstein. A maximum relative deviation of 5% is found comparing the GITT results for the average Nusselt number against the experimental data, while an 8% maximum relative deviation is found comparing against an empirical correlation by the same authors. It is concluded that the GITT solution with the eigenvalue problem in the angular coordinate yields better convergence rates than the more usual eigenfunction expansion in the radial variable. This is due to the originally homogeneous boundary conditions in the angular direction, which do not require filtering for convergence enhancement, as opposed to the required filter in the radial direction that introduces a source term in the filtered equation for the streamfunction.

Accurate non-linear bending analytical solutions setting new benchmarks for rectangular thin plates with four simply supported edges

In the field of nonlinear structural analysis, the ability to provide accurate and reliable analytical solutions is of paramount importance. This paper presents more accurate nonlinear analytical solutions for rectangular plates with four simply supported edges, establishing a new benchmark that replaces and complements Lévy’s manual analytical solutions. These accurate solutions can serve as a reference for validating various nonlinear numerical and approximate analytical solutions, thereby providing a robust basis for engineering design. Notably, the central deflection error in this study is less than 0.005%, while the central stress error remains below 0.4%. The accuracy of the present solutions provided significantly exceeds that found in the existing literature.

A Numerical and Closed-Form Analytical Solution of the Global Buckling Critical Load of Tall Buildings Including Soil Flexibility: Timoshenko Beam

A Numerical and Closed-Form Analytical Solution of the Global Buckling Critical Load of Tall Buildings Including Soil Flexibility: Timoshenko Beam

Numerical solution of two-dimensional fractional differential equations using Laplace transform with residual power series method

Abstract One of the efficient and reliable methods for resolving fractional order linear as well as non-linear differential equations is the Laplace transform with residual power series method. This approach is used in the current research to obtain the numerical solutions of the two-dimensional fractional differential equations, namely, the temporal fractional order diffusion equation and the fractional biological population equation. The unknown coefficients of the series solutions to these equations are determined using the proposed approach. The difference between exact and analytical-numerical solutions is presented for these equations in the form of errors. The advantage of the suggested method over alternative approaches is that it requires less computation to solve these two-dimensional differential equations of time-fractional order.

Viscoelastic solution of optimal reserved deformation for deep soft rock tunnels with large deformation

In the construction process of soft rock tunnels, determining a reasonable amount of reserved deformation is important to ensure the tunnel stability. This article presents the viscoelastic solution of reserved deformation for deep soft rock tunnels considering the support effects. Based on the analytical solution of the Burgers model, the expression of surrounding rock displacement was derived by considering reserved deformation and optimal reserved deformation. Subsequently, based on numerical simulation experiments, the variation laws and errors of the numerical and analytical solutions of the expressions of reserved deformation and surrounding rock displacement were analyzed. To gain a better understanding of the factors that affect reserved deformation, the factors influencing the expression of optimal reserved deformation were analyzed. The errors in the numerical simulation and analytical solution results were within 10%. This study could provide a theoretical basis for determining the amount of reserved deformation and analyzing the variation law of surrounding rock affected by the amount of reserved deformation.

Modeling of the Combined Effect of the Surface Roughness and Coatings in Contact Interaction

The model of indentation of a spherical punch with a rough surface into a thin elastic layer lying on an elastic half-space has been developed. A numerical-analytical solution is suggested based on the two-scales approach. At macroscale, the integral equation of the second kind is reduced to calculate the nominal contact characteristics, taking into account the surface layer thickness and its mechanical characteristics, as well as additional compliance due to surface roughness calculated from the microscale analysis. The influence of the punch roughness and the surface layer mechanical and geometrical characteristics on the nominal contact pressure distribution, on the nominal contact area and the indentation depth, as well as on the real contact area and pressures at the individual contact spots, is analyzed. The developed contact model can be used to analyze the indentation of the punch into the layered elastic base, taking into account the roughness of the punch surface, and can also be used to give a complete analysis of the roughness effect on the contact process at both scale levels. The results can be used to control the indentation of the rough-coated bodies.

Salt intrusion dynamics in a well-mixed sub-estuary connected to a partially to well-mixed main estuary

Abstract. Salt intrusion in estuaries has been exacerbated by climate change and human activities. Previous studies have primarily focused on salt intrusion in the mainstem of estuaries, whereas those in sub-estuaries (those that branch off their main estuaries) have received less attention. During an extended La Niña event from 2021 to 2022, a sub-estuary (the East River estuary) alongside the Pearl River estuary, China, experienced severe salt intrusions, posing a threat to the freshwater supply in the surrounding area. Observations revealed that maximum salinities in the main estuary typically preceded spring tides, exhibiting significant asymmetry in salinity rise and fall over a fortnightly timescale. In contrast, in the upstream region of the sub-estuary, the variation in salinity was in phase with that of the tidal range, and the rise and fall of the salinity were more symmetrical. Inspired by these observations, we employed idealized numerical models and analytical solutions to investigate the underlying physics behind these behaviors. It was discovered that under normal dry conditions (with a river discharge of 1500 m3 s−1 at the head of the main estuary), the river–tide interaction and change in horizontal dispersion accounted for the in-phase relationship between the salinity and tidal range in the upstream region of the sub-estuary. Under extremely dry conditions (i.e., a river discharge of 500 m3 s−1 at the head of the main estuary), salinity variations were in phase with those of the tidal range in the middle as well as the upstream region of the sub-estuary. The variation in salinity in the main estuary along with those in salt dispersion and freshwater influx inside the sub-estuary collectively influenced salinity variation in the well-mixed sub-estuary. These findings have important implications for water resource management and salt intrusion prevention in the catchment area.

A numerical and closed-form analytical solution for the global buckling critical load of tall buildings considering soil flexibility: A coupled shear-flexural model

The critical global buckling load is a fundamental parameter governing the performance of tall buildings, however, the existing literature lacks an efficient solution that comprehensively captures the behavior of tall structures with uniform or variable properties. In response to this gap, this study introduces a novel and generalized solution, employing the coupled shear-flexural model. This solution is designed for determining the critical global buckling load in tall buildings subjected to vertical loads with diverse profiles, while also incorporating soil flexibility. The analytical solution, designed for the uniform continuous model, offers a breakthrough by providing graphical representations facilitating the direct determination of eigenvalues based on a single dimensionless parameter. Addressing the unique challenge posed by tall buildings with variable properties, a numerical transfer matrix method based on a Laplacian approach is proposed. This method enables the direct calculation of the transfer matrix for each level, displaying exceptional convergence in just three iterations. The study sheds light on a crucial finding that emphasizes the adverse impact of rotational flexibility on soil, leading to a decrease in eigenvalues as soil flexibility increases. Importantly, the proposed solution methods not only show excellent accuracy but also serve as invaluable tools for performing parametric analyses. Furthermore, they present cost-effective alternatives to exact methods, providing a solid foundation for studying the overall stability of tall buildings. This research contributes significantly to the existing literature by introducing efficient methodologies that improve our understanding and analysis of the behavior of tall buildings under vertical loads.

Deformations of Single-Crystal Silicon Circular Plate: Theory and Experiment

In this paper, the experimental methodology for the single-crystal circular plate deformation measurement and subsequent procedure for the quantitation of its mechanical properties are developed. The procedure is based on a new numerical-analytical solution of non-linear boundary-value problem for finite deformations of a circular anisotropic plate. Using the developed method, a study of the deformation of single-crystal circular plates formed on the basis of a silicon-on-insulator structure was carried out. The values of residual stresses are determined and it is shown that the presence of these stresses increases the flexural rigidity of the plate by several times.

Axisymmetric Stokes flow of a spherical droplet or slip particle in the presence of a circular disk

This article deals with the axisymmetric flow of a Newtonian fluid droplet moving along the axis of a thin circular disk under the conditions of a low Reynolds number in the quasi-steady limit. The system is immersed in a second Newtonian fluid. With the same geometry, the motion of a slip spherical particle is treated also. Basic solutions are constructed in cylindrical and spherical coordinates according to the appropriate flow region. A combined analytic-numerical solution procedure in conjunction with a dual integral equation is used. Numerical results show that the normalized drag force coefficient acting on the droplet/ slip particle is obtained with good convergence for various values of viscosity ratio, slip parameter, and clearance parameter. The findings demonstrate that the results of the drag coefficient are in good agreement with the limiting cases available in the literature.

Laboratory experiments and theoretical study of pressure and fluid influences on acoustic response in tight rocks with pore microstructure

AbstractWave‐induced fluid flow is considered to be a major source of seismic attenuation and dispersion in porous rocks. From the physical description of partially saturated reservoirs, numerous analytical solutions based on upscaling homogenization theories have been employed to calculate equivalent frequency‐dependent poroelastic media. Nevertheless, dispersion and attenuation predictions are often not reasonably consistent with laboratory and field measurements in a broad frequency range, particularly due to influences of biphasic fluids and their distribution, presence of heterogeneities on various length scales, and pore microstructure. We investigate the role of pore microstructure on pressure and fluid saturation dependence of elastic velocities in tight sandstones. Previous work points out that differentiating the impacts of heterogeneities at various scales on dispersion within seismic exploration and sonic frequencies can be very difficult. In practice, this is because fluid‐related dispersion mechanisms are impossible to be independent. Thus, it is important for a theoretical and more quantitative analysis of the relative contribution of interrelated energy dissipation processes through a better understanding of combined influences due to the presence of microscopic and mesoscopic heterogeneities. Based on microscopic squirt flow and mesoscopic flow in a partially saturated medium, we develop a poroelastic model that allows evaluating the overall frequency‐dependent dispersion via considering a random distribution of fluid heterogeneities as well as the broadly distributed aspect ratio of compliant pores. Experimental validation of the model is accomplished via a comprehensive comparison of predictions with measurements of partially saturated velocities versus pressure and fluid for sandstones with specific pore microstructures.