Semi-inverse Method Research Articles

Exploring new traveling wave solutions by solving the nonlinear space–time fractal Fornberg−Whitham equation

Complex and nonlinear fractal equations are ubiquitous in natural phenomena. This research employs the fractal Euler−Lagrange and semi-inverse methods to derive the nonlinear space–time fractal Fornberg–Whitham equation. This derivation provides an in-depth comprehension of traveling wave propagation. Consequently, the nonlinear space–time fractal Fornberg–Whitham equation is pivotal in elucidating fundamental phenomena across applied sciences. A novel analytical technique, the generalized Kudryashov method, is presented to address the space–time fractal Fornberg–Whitham equation. This method combines the fractional complex approach with the modified Kudryashov method to enhance its effectiveness. We derive an analytical solution for the space–time fractal Fornberg–Whitham equation to elucidate how various parameters influence the propagation of new traveling wave solutions. Furthermore, Figures 1 through 6 analyze the impact of parameters α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha$$\\end{document}, β,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\upbeta ,$$\\end{document}b1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$b_{1}$$\\end{document}, and k\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$k$$\\end{document} on these new traveling wave solutions. Our results show that the solitary wave solutions remain intact for both case 1 and case 2, regardless of the time fractional orders β\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${ }\\left( \\upbeta \\right)$$\\end{document}. At the end, the manuscript discusses the implications of these findings for understanding complex wave phenomena, paving the way for further exploration and applications in wave propagation studies.

Задача нелинейной теории упругости о растяжении и кручении составного цилиндра с предварительно напряжённым включением, содержащим винтовую дислокацию

The problem of large torsional and tension-compression deformations of a composite circular cylinder of incompressible Bartenev-Khazanovich material is considered. The cylinder contains a central circular cylindrical inclusion in which a concentrated screw dislocation is formed. It is pre-twisted and stretched (or compressed) along the axis and bonded to an unstressed external hollow cylinder. A single reference configuration is used when solving a problem for a composite body. This configuration is natural (unstressed) for the outer hollow cylinder and prestressed for the inner solid cylinder. The large deformation superposition theory is used to write the determining ratios of the material of the inner cylinder. The nonlinear theory of torsion of elastic cylinders containing a screw dislocation is used to solve the problem of the prestressed state of the internal inclusion. This theory is physically correct not for any models of isotropic elastic materials, but only for those for which the screw dislocation in the cylinder has a finite linear energy and creates a longitudinal force of finite value. The Bartenev-Khazanovich model belongs to this class. The problem for a composite cylinder is solved by a semi-inverse method. Using this method, it is reduced to nonlinear ordinary differential equations. The assumption of isotropy and incompressibility of the material makes it possible to find an exact solution to the problem.

Generalized variational structures of the (3+1)-dimensional Zakharov–Kuznetsov–Burgers equation in dusty plasma

The center of this paper is to establish the generalized variational structure (GVS) of the [Formula: see text]-dimensional Zakharov–Kuznetsov–Burgers equation (ZKBe) by taking advantage of the Semi-inverse method (SIM). Two different GVSs are extracted and the derivation process is presented in detail. The extracted GVSs reveal the energy conservation law and can offer some new insights on the study of the variational method.

Galerkin-Kantorovich variational method for solving saint venant torsion problems of rectangular bars

The unrestrained torsional analysis of bars is an important theme in elasticity theory, first solved by Saint-Venant using semi-inverse methods. It has been considered and solved by several others using analytical methods and numerical procedures due to the importance in the design of machine parts under torsional moments. In this paper, the Saint Venant torsion problem is solved for rectangular prismatic bars using Galerkin-Kantorovich variational method (GKVM). The work presents a detailed theoretical framework of the problem, deriving using first principles considerations the stress compatibility equation in terms of the Prandtl stress function ϕ(x,y). The derived domain equation which is required to be satisfied over the rectangular cross-sectional domain is a partial differential equation of the Poisson type. GKVM is adopted as the solution method for finding the solution to the domain equation. The unknown Prandtl stress function ϕ(x,y) is assumed, following Kantorovich method to be a product of an unknown function for fx sought to minimize the Galerkin-Kantorovich variational functional (integral) (GKVF) and a known function (y2-b2) which satisfies the boundary conditions at all boundary points in the y-direction, that is, at y=±b. The resulting GKVF is a simplified functional whose integral is a second order inhomogeneous ordinary differential equation (ODE) in fx. The integrand is solved to find fx leading in a full determination of the Prandtl stress function. The expression for stresses, torsional moments and torsional parameters are then found and they satisfy the boundary conditions and the domain equation. The results for the torsional moments and torsional parameters are identical to previous results obtained using double finite sine transform method (DFSTM), and analytical methods. The merit of GKVM is that it has led to the exact solution of the unrestrained torsion problems.

Simulation and modeling of grinding surface topography based on fractional derivatives

The grinding surface’s 3-D topography has self-affinity. To more accurately simulate this characteristic, this paper proposes for the first time to construct a grinding surface simulation model based on fractional derivatives, including: utilizing the Weierstrass-Mandelbrot function to simulate the point cloud of the grinding wheel surface; further deriving the trajectory equation based on the He’s fractal derivative; the semi-inverse method is employed to construct the fractional-order variational principle of the vibration equation. Experiments show that the simulation model designed in this paper can predict the surface height more accurately, and is better than the traditional model in the calculation of roughness parameters Sa and Sq. The model is helpful to thoroughly understand the formation mechanism of grinding surface topography, and provides simulation data guidance for machine vision inspection of grinding surface in micron scale.

Comparison of free vibration behaviors for simply supported and clamped T-shaped thin plate resting on Winkler elastic foundation

The analysis of free vibration problems for thin plates is essential for the design of various structural systems. However, it is difficult to find analytical solutions due to the complexity of mathematical computing. Based on the symplectic superposition method, the T-shaped thin plate on the Winkler elastic foundation is divided into four sub-plates and are solved by using the symplectic eigen expansion method, and the modes and frequencies are studied. The method begins directly with the fundamental equations and undergoes a rigorous mathematical derivation without assuming the form of the solution beforehand. This approach helps circumvent the drawbacks associated with traditional semi-inverse solution methods. In addition, the theoretical calculation model and finite element analysis model of T-shaped thin plates on elastic foundation are established by using Mathematic software and ABAQUS software in present paper. It proves that the symplectic superposition method converges very fast and has a good consistency with the finite element simulation results. Results show that the boundary condition, foundation stiffness and aspect ratio have great influences on vibration frequency and mode shape for T-shaped structures.

Hamiltonian system-based analytic thermal buckling solutions of orthotropic rectangular plates

New analytic solutions for the thermal buckling of orthotropic rectangular plates are presented using the symplectic superposition method (SSM). The SSM deals with higher-order partial differential equations within the Hamiltonian framework, distinguishing it from conventional methods in the Lagrangian framework. The solution procedure involves dividing the original problem into two subproblems, utilizing the techniques in the symplectic space, like separating variables and symplectic eigen expansion, to achieve analytic solutions, and using superposition to obtain the final solutions. One notable advantage of the SSM is the absence of predetermined assumptions, which is a departure from conventional semi-inverse methods. The new analytic solutions under challenging non-Lévy-type boundary conditions (BCs) were rarely addressed in prior research due to their complex mathematical characteristics, which reveals the main novelty. Comprehensive thermal buckling results are provided for isotropic/orthotropic plates under various BCs, validated by those from the finite element method with a maximum difference being 2.7374% and literature with a maximum difference being 5.9201%. Parameter analyses quantify the effects of BCs, aspect ratio, and thickness-to-length ratio on critical buckling temperatures and mode shapes.

VARIATIONAL PRINCIPLE FOR A FRACTAL LUBRICATION PROBLEM

Micro/nanoscale lubrication must take into account the fractal profile of the shaft and bearing surfaces. A new fractal rheological model is proposed to describe the properties of the non-Newtonian fluid, and a fractal variational principle is established by the semi-inverse method, and finally the Lagrange multipliers can be found in the obtained variational formulation. This work provides a new fractal approach to nano/micro lubrication.

Exact solitary wave solutions for non-linear optic model by variational perspective

A variational principle for the non-linear optic model is established by semi-inverse method. Two new exact solitary wave solutions are obtained by using the variational transform method. Numerical examples show the novel method is efficient and simple, and can be applied to find solitary wave solutions for different types of wave equations. The physical properties of solitary wave solutions are illustrated by some figures.

A FRACTAL MODIFICATION OF THE PSEUDO-PARABOLIC EQUATION AND ITS GENERALIZED FRACTAL VARIATIONAL PRINCIPLE

In this work, a new fractal pseudo-parabolic equation is derived by means of He’s fractal derivative. The semi-inverse method (SIM) is employed to develop the generalized fractal variational principle (GFVP), which can reveal the energy conservation law in the fractal space and provide some new insights on the study of the variational method.

THE FRACTAL ZAKHAROV–KUZNETSOV–BENJAMIN–BONA–MAHONY EQUATION: GENERALIZED VARIATIONAL PRINCIPLE AND THE SEMI-DOMAIN SOLUTIONS

By means of He’s fractal derivative, a new fractal (2 + 1)-dimensional Zakharov–Kuznetsov–Benjamin–Bona–Mahony equation is extracted in this paper. The semi-inverse method is employed to establish the generalized fractal variational principle. The generalized fractal variational principle can show the conservation laws through the energy form in the fractal space. Moreover, some semi-domain solutions are also explored by applying the variational approach and the one-step method namely Wang’s direct mapping method-II. The dynamics of the extracted solutions on the Cantor set are unveiled graphically. The findings of this study are expected to provide some new insights into the exploration of the fractal PDEs.

Fractal solitary waves of the (3+1)-dimensional fractal modified KdV-Zakharov-Kuznetsov

In this work, the fractal (3+1)-D modified KdV-Zakharov-Kuznetsov (MKdV-ZK) model is studied, which can represent weakly non-linear waves under the unsmooth boundary. With the help of the fractal traveling wave transformation and the semi-inverse method, a fractal variational principle is obtained, which is a strong minimum one according to the He-Weierstrass function. From the variational principle, a fractal solitary wave solution is obtained, and the influence of un-smooth boundary on solitary waves is studied and the behaviors of the solutions are presented via 3-D plots. This paper shows that the fractal dimensions can affect the wave pattern, but cannot influence its crest value.

VARIATIONAL FORMULATIONS FOR A COUPLED FRACTAL–FRACTIONAL KdV SYSTEM

Every shallow-water wave propagates along a fractal boundary, and its mathematical model cannot be precisely represented by integer dimensions. In this study, we investigate a coupled fractal–fractional KdV system moving along an irregular boundary within the framework of variational theory, which is commonly employed to derive governing equations. However, not every fractal–fractional differential equation can be formulated using variational principles. The semi-inverse method proves to be challenging in finding an appropriate variational principle for nonlinear problems and eliminating extraneous components from the studied model. We consider the coupled fractal–fractional KdV system with arbitrary coefficients and establish its variational formulation to unveil the remarkable insights into the energy structure of the model and interrelationships among coefficients. Encouraging results are obtained for this coupled KdV system.

THE FRACTAL MODIFICATION OF THE ROSENAU–BURGERS EQUATION AND ITS FRACTAL VARIATIONAL PRINCIPLE

A new fractal Rosenau–Burgers equation with He’s fractal derivative is proposed in this work. Exerting the semi-inverse method, two different kinds of the generalized fractal variational principles (GFVPs) of the fractal Rosenau–Burgers equation are constructed. The GFVPs extracted in this paper are expected to bring some new inspiration for the study and application of the variational method in the fractal space.

Study on stress distribution and extrusion load threshold of compressed filled rock joints

Abstract The distribution of stress and the normal extrusion load threshold in weak interlayer are crucial for direct shear test of filled rock joints, but there is a lack of theoretical research in this area. First, an analytical solution for stress distribution was derived using a semi-inverse method. Then, it is compared by the numerical simulation method. Finally, the influence of the width and thickness of weak interlayer on the extreme values of stress components was analyzed, and the distribution pattern of the normal extrusion load was discussed. The results show that under the same conditions, the analytical solution and the numerical simulation results are in good agreement. The maximum horizontal stress in the weak interlayer decreases with increasing width and increases with increasing thickness, while the change of the minimum is opposite. The normal extrusion load increases first and then decreases along the width direction of the weak interlayer. By comparing the normal extrusion load with the empirical value, the mechanism of extrusion failure in the weak interlayer is revealed.

Calculation and Selection of Airfoil for Flapping-Wing Aircraft Based on Integral Boundary Layer Equations

The flight of a migratory bird-like flapping-wing aircraft is characterized by a low Reynolds number and unsteadiness. The selection of airfoil profiles is critical to designing an efficient flapping-wing aircraft. To choose the suitable airfoil for various wing sections, it is necessary to calculate the aerodynamic forces of the unsteady two-dimensional airfoil with a Reynolds number in the range of 105. While accurate, calculating this by solving the Navier–Stokes equations is impractical for early design stages due to its high consumption of computing resources and time. The computational demands for extending it to 3D aerodynamic calculations are even more prohibitive. In this paper, a relatively simple method is proposed. The two-dimensional unsteady panel method is utilized to derive the inviscid flow field, the unsteady integral boundary layer method is utilized to solve the boundary layer viscous flow, and the eN transition model is adopted to predict the position of the transition. These models are coupled with the semi-inverse interaction method to solve the aerodynamics of the unsteady low-Reynolds-number two-dimensional airfoil. The unsteady aerodynamics of the symmetric and cambered airfoils at different wing sections are calculated respectively by the proposed method. Mechanism analysis of the calculation results is conducted, and a symmetrical airfoil or a slightly cambered airfoil is recommended for the wing tip, a moderately cambered airfoil is suggested for the outer-wing section, and a highly cambered airfoil is suggested for the inner-wing section.

Generation a solution to the equations of elasticity theory for a layered strip basing on the principle of compressed mappings

A systematic presentation of the modified classical semi-inverse SaintVenant method as an iterative one is given on the example of generating a solution to the differential equations of elasticity theory for a long layered strip. The firstorder differential equations of the plane problem are reduced to the dimensionless form and replaced by integral equations with respect to the transverse coordinate, just as it is done in the Picard method of simple iterations. In this case, a small parameter appears in the integral equations before the integral sign as a multiplying factor, which is used to ensure convergence of solutions in accordance with the Banach’s principle of compressed mappings. The equations and elasticity relations are converted to a form that enables to calculate the unknowns consecutively, so that the unknowns being calculated in one equation are the inputs for the next equation, and etc. Fulfillment of the boundary conditions at the long edges leads to ordinary differential equations for slowly and rapidly changing singular components of the solution with sixteen effective stiffness coefficients that are defined by integrals from the given ones as a stepped function of Young's moduli for each layer. Integrating of these ordinary differential equations makes it possible to obtain the formulas for all the required unknowns of the problem, including transverse stresses that are not defined in the classical theory of the beam and solutions of the edge effect type, and to fulfill all the boundary conditions for the elasticity theory problem. The solution of three boundary value problems of the strip elasticity theory is provided such as for a two-layer strip with layers of the same thickness and different thicknesses, and a strip with an arbitrary number of layers. Formulas for all unknowns of the problem are obtained.

The soliton solutions for stochastic Calogero–Bogoyavlenskii Schiff equation in plasma physics/fluid mechanics

Abstract The generalized (2+1)-dimensional stochastic Calogero–Bogoyavlenskii Schiff equation (SCBSE) driven by a multiplicative Brownian motion is taken into consideration. The Riccati equation mapping and He’s semi-inverse methods are utilized to obtain the rational function, hyperbolic function, and trigonometric function for SCBSE. We expand some solution from previous studies. The acquired solutions of SCBSE may explain many exciting physical phenomena because it is widely used in plasma physics and fluid dynamics. Also, it explains the relationship between the Riemann y-axis propagating wave and the long x-axis propagating wave. Using a variety of 2D and 3D graphs, we illustrate how the Brownian motion influences the exact solutions of SCBSE.

Fractional-stochastic shallow water equations and its analytical solutions

In this paper, the stochastic shallow water wave equation with M-Truncated derivative is taken into account. New rational, trigonometric and hyperbolic solutions are produced using the He’s semi-inverse method and modified extended tanh-function method. The solutions found can be used to simulate a wide range of fascinating physical phenomena since the shallow water equation has numerous applications including weather simulations, tsunami prediction, tidal waves, and river irrigation flows. Also, we generalize some previous results. To show how M-Truncated Derivative and white noise affect the analytical solutions of the stochastic shallow water wave equation with M-Truncated derivative, we create different two dimension and three dimension graphs.

Dynamic response and dynamic buckling of general laminated plates: A semi-inverse method

In this paper, a simplified way of deriving the equation of motion is presented for the general laminated composite plates under time-dependent compression. The considered plates with simply supported boundary conditions are subjected to the in-plane harmonic loads. By employing a novel semi-inverse method (SIM) based on least square (LS) fitting, the equation of motion is derived from the static equilibrium path. Dynamic responses of glass fiber reinforced polymer (GFRP) laminated plates with different layer arrangements are analyzed by such equations and the dynamic buckling loads are obtained based on Budiansky–Hutchinson and Volmir criteria. Static equilibrium paths are obtained based on commercial finite element method (FEM), i.e., ANSYS software. Again, the results of SIM are compared with those of FEM in ANSYS software and the verifications of dynamic responses are proved considering full types of material couplings. The presented results show that the proposed simplified way of the derivation of the equation of motion allows for a very low time consumption analysis.