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This paper presents an analytical solution for a complex mechanical system consisting of a pendulum with a rolling wheel, which combines translational and rotational motion. The system's dynamics are described by a set of coupled differential equations that are challenging to solve analytically using traditional methods. To overcome this challenge, the variation iteration method (VIM) is employed to derive an analytical solution. VIM is a powerful technique that allows for the approximate solution of differential equations by constructing a series solution iteratively. The Lagrange multiplier, a crucial component in the solution process, is determined for the first time using the Elzaki transformation. Interestingly, the Lagrange multiplier obtained through the Elzaki transformation matches the result obtained from Laplace transformation, which is a fundamental finding of this paper. Also we compared its results with the by employing the VIM and the derived Lagrange multiplier, a comprehensive analytical solution for the complex pendulum with a rolling wheel system is obtained. The solution provides insights into the system's behavior, such as the oscillation amplitudes, angular velocities and other relevant dynamic parameters. The proposed approach demonstrates the efficacy of the VIM in tackling complex mechanical systems and showcases the equivalence between the Lagrange multiplier derived through Elzaki transformation and the well-established Laplace transformation. The results obtained from this study contribute to the understanding and analysis of coupled translational and rotational systems, providing a valuable tool for researchers and engineers working in the field of mechanical dynamics. Sketches are made of the phase portraits close to the equilibrium points. [ABSTRACT FROM AUTHOR]

In this paper, a numerical framework for modelling multiphase flows with interface correction is proposed. In original Volume-of-Fluid method, the interface smearing may appear since the interface profile deviates from its equilibrium state, especially in the simulations of complex interface deformation. To solve this problem, a novel correction step is introduced into the procedure after solving the volume fraction transport equation, and this step just works on the region of phase interface. The difference between the current method and the previous work is that the correction coefficient can be adjusted adaptively with the gradient of normal velocity at the interface, which is strongly correlated to the intensity of interface smearing and changes with the computational time and interface position. The validation computations are performed for bubble deformation (initial shapes of two-dimensional four-lobed-star and three-dimensional (3D) box with holes), bubble rising in a channel and Rayleigh–Taylor instability problems (Reynolds number of 100 and 1000). The obtained results show that the unphysical phenomenon of interface smearing is suppressed effectively, and the interface sharpness is improved greatly by the present method. In addition, the mass of bubble deformation by the original method will decrease by 13.3%. While these results obtained by the present method are in good agreement with the analytical solutions or published data. [ABSTRACT FROM AUTHOR]

In this paper, inscribed non-Newtonian Williamson fluid relaxation and contraction phenomena due to the effect of magnetohydrodynamics (MHD) is taken into account. This issue is modeled, and precise solutions are handled using a low Reynolds number and long wavelength approximation. Graphical depiction of the consequences of various parameters on temperature and velocity are provided. It becomes clear that the temperature reactions to increases in Biot numbers are decreased. Considering a cylindrical coordinate system, the impact of various physical parameters on the temperature distribution, velocity, the pressure gradient, and frictional force are secured from the flow for non-Newtonian Williamson fluid exposed from MATLAB. Additionally, it has been discovered that the velocity profile at the tube's center increases for shear-thinning a fluid and decreases for thickening a fluid. Furthermore, the peristaltic walls show the opposite behavior. [ABSTRACT FROM AUTHOR]

This paper deals with the numerical solution of the time-fractional Gilson–Pickering equation using the Kansa method, in which the multiquadrics were utilized as the radial basis function. To achieve this, a meshless numerical scheme based on the finite difference along with the Kansa method has been presented. First, the finite difference approach has been utilized to discretize the temporal derivative, and subsequently, the Kansa method is employed to discretize the spatial derivatives. The stability and convergence analysis of the numerical scheme are also elucidated in this paper. Furthermore, the soliton solutions have been acquired by implementing the Kudryashov method for comparison with the numerical results. Finally, numerical simulations are performed to confirm the applicability and accuracy of the proposed scheme. [ABSTRACT FROM AUTHOR]

In this research paper, we scrutinize the novel traveling wave solutions and other solutions with conformable, M-truncated and beta fractional derivatives for the nonlinear fractional Hirota–Maccari system. In order to acquire the analytical solutions, the Riccati–Bernoulli sub-ODE technique is implemented. Presented method is the very powerful technique to get the novel exact soliton and other solutions for nonlinear partial equations in sense of both integer and fractional-order derivatives. Mathematical properties of different kinds of fractional derivatives are given in this paper. A comparative approach is presented between the solutions with the fractional derivatives. For the validity of the solutions, the constraints conditions are determined. To illustrate the physical meaning of the presented equation, the 2D and 3D graphs of the acquired solutions are successfully charted by selecting appropriate values of parameters. [ABSTRACT FROM AUTHOR]

In this paper, we study the time evolution of three-states spin-1 non-Hermitian systems. The state vector of the wave function for Schrödinger equation is obtained analytically by using the Cayley–Hamilton theorem. We note that the behavior of the non-Hermitian three-states system corresponds to the Hermitian three-state system. Furthermore, the fluctuation between three-state spin-1 is periodically oscillating when the eigenvalues of the non-Hermitian system are real. Moreover, if the eigenvalues are imaginary, we explore that from our analysis and analytical solution that the probabilities' amplitudes always grow with time for all physical parameters in the three-state, i.e. including the -symmetry non-Hermitian system. It means that the Schrödinger equation becomes the growth exponential. [ABSTRACT FROM AUTHOR]

The coupled nonlinear Schrödinger (CNLS) system is often used to describe physical problems such as wave propagation in a birefringent optical fiber. Under investigation in this paper is the nonlocal -symmetric reverse-space type of CNLS system, through constructing two types of Darboux transformation (DT), we will derive a new series of nonlocal analytic solutions: (1) Single-periodic and double-periodic solutions; (2) Four different combinations about dark and anti-dark solitons, as well as the bright and dark breathers on a periodic background. Moreover, the fundamental properties and dynamical behaviors of those solutions will be discussed. [ABSTRACT FROM AUTHOR]

In this paper, the generalized Kudryashov (GK) approach and the sine-Gordon expansion approach are used for constructing new specific analytical solutions of the deoxyribonucleic acid model, which include the well-known bell-shaped soliton, kink, singular kink, periodic soliton, contracted bell-shaped soliton and anti-bell-shaped soliton. The efficacy of these strategies demonstrates their utility and efficiency in addressing a wide range of integer and fractional-order nonlinear evolution problems. The physical relevance of the demonstrated results has been proven using three-dimensional forms. It is interesting to mention that the solutions achieved here using the provided methods are extra-extensive and may be used to explain the internal interaction of the deoxyribonucleic acid model originating in mathematical biology. The suggested approach was utilized to get exact traveling wave solutions for fractional nonlinear partial differential equations appearing in nonlinear science. [ABSTRACT FROM AUTHOR]

In this paper, the double-chain model for deoxyribonucleic acid (DNA) is studied analytically by taking advantage of two reliable methodologies: the new auxiliary equation method (NAEM) and the expa function method. The deoxyribonucleic acid (DNA) contains the genetic information that creatures require for their survival and reproduction. The used approaches produce exact explicit solutions to the double chain model of DNA in mathematical biology, which are presented in this paper. Many novel solutions such as trigonometric, exponential, singular, and hyperbolic solutions, are obtained by using the aforementioned techniques. Numerical simulations are employed to supplement the obtained solutions, making this research biologically relevant. Eventually, kink wave, nonlinear wave, plane wave-front and doubly soliton profiles of solutions are presented in analysis and discussion section. It is indeed important to note that the solutions produced using these techniques are more generalized and can be more useful to demonstrate the internal interactions of the double chain DNA model. [ABSTRACT FROM AUTHOR]

This paper reveals optical soliton solutions to fiber Bragg gratings (FBGs) with dispersive reflectivity having Kerr law of nonlinear refractive index. Bragg gratings are no doubt a technological spectacle that sustained balance between dispersion and the nonlinear effects that leads to a stable transmission of solitons across intercontinental distances. FBGs as sensor elements are used for measuring numerous engineering parameters such as temperature, strain, pressure, tilt, displacement, acceleration, load. Two recently developed mechanisms such as the unified method and extended sinh-Gordon equation expansion method (ShGEEM) are successfully employed to secure optical pulses in the shapes of the bright, dark, singular, complex combo, periodic, and plane wave solutions. The achieved solutions contain key applications in engineering and physics. These solutions define the wave performance of the governing models. By selecting suitable parametric values, the dynamics of the evaluated results are exemplified by sketching their 2-dimensional, 3-dimensional, and contour profiles to understand the real phenomena for such sort of nonlinear models. The novelty of the gained outcomes is manifested by a detailed comparison with the results that already exist. [ABSTRACT FROM AUTHOR]

In this paper, a (2+1)-dimensional Kadomtsev–Petviashvili (KP) equation is studied from the governing equations of dusty plasma by means of multiscale analysis and reduced perturbation method. We utilize the generalized exponential rational function method to give various forms of analytical solutions with free parameters of the corresponding equation by using Mathematica software calculation. Then, appropriate parameter values are selected to draw the three-dimensional (3D) diagram and contour diagram of the analytic solutions. We also give the influence of these physical parameters on wave amplitude, waves profile, and stability of waves. The results show the generalized method is a reliable mathematical method to solve similar nonlinear equations in mathematical physics and these solutions can enrich the physical behavior of the equation in dusty plasma. [ABSTRACT FROM AUTHOR]

In this paper, the soliton solutions and dynamical wave structures for the generalized (3+1)-dimensional shallow water wave (SWW) equation, which is an important physical property in ocean engineering and hydrodynamics, are presented. The generalized exponential rational function (GERF) method is used to investigate the closed-form wave solutions of the generalized SWW equation, which is used to describe the evolutionary dynamics of SWW. We successfully archive a variety of soliton solutions such as exponential solutions, kink wave solutions, non-topological solutions, periodic singular solutions, and topological solutions. These newly established results are also important for understanding the wave-propagation and dynamics of exact solutions of the equation, which is of great significance in physical oceanography and chemical oceanography. Eventually, it is shown that the proposed GERF technique is effective, robust, and straightforward and is also used to solve other types of higher-dimensional nonlinear evolution equations. In our work, we have used Mathematica extensively for such complicated algebraic calculations. [ABSTRACT FROM AUTHOR]

In this paper, the (2+1)-dimensional resonant Davey–Stewartson equations are solved by using two methods; namely, (m + 1 G ′ ) -expansion and (m + G ′ G) -expansion methods. A wave transform is used to convert the (2+1)-dimensional resonant Davey–Stewartson (RDS) equations with M-derivative into a system of nonlinear ordinary differential equations. Different forms of solutions, such as dark, bright, singular and periodic singular solutions are successfully constructed. The obtained solutions are plotted in 3D for both M- derivative and classical derivative to more understand the effect of M-derivative on the studied equation. [ABSTRACT FROM AUTHOR]

This paper investigates the high-frequency waves' dynamical behavior in the relaxation medium through two recent analytical schemes. This study depends on the Vakhnenko–Parkes (VP) equation that has been reduced from the well-known Ostrovsky equation. The modified Khater (MKhat) and the extended simplest equation (ESE) methods are used to handle the considered model. As a result, many novel solitary wave solutions have been obtained to construct the initial and boundary conditions. These conditions allow employing the variational iteration (VI) method to study the semi-analytical solutions of the considered model. The accuracy of solutions is explained along with showing the matching between analytical and semi-analytical solutions and comparing our obtained solutions with the previous results that have been obtained in published research papers. Moreover, the high-frequency waves' behavior relaxation medium is illustrated through some distinct sketches. The methods' performance shows their effectiveness, direct, easy, and consequential for studying many nonlinear evolution equations. [ABSTRACT FROM AUTHOR]

In this paper, the stable analytical solutions' accuracy of the nonlinear fractional nonlinear time–space telegraph (FNLTST) equation is investigated along with applying the trigonometric-quantic-B-spline (TQBS) method. This investigation depends on using the obtained analytical solutions to get the initial and boundary conditions that allow applying the numerical scheme in an easy and smooth way. Additionally, this paper aims to investigate the accuracy of the obtained analytical solutions after checking their stable property through using the properties of the Hamiltonian system. The considered model for this study is formulated by Oliver Heaviside in 1880 to define the advanced or voltage spectrum of electrified transmission, with day-to-day distances from the electrified communication or the application of electromagnetic waves. The matching between the analytical and numerical solutions is explained by some distinct sketches such as two-dimensional, scatter matrix, distribution, spline connected, bar normal, filling with two colors plots. [ABSTRACT FROM AUTHOR]

Biological, social and man-made systems often harbor various types of higher-order interaction, whose presence may substantially impact the dynamics on these systems. As a special random walk, paradigmatic trapping problem is still scarcely explored on networks encoding higher-order interactions. In this paper, applying simplicial networks produced by edge corona product as test bed, we explore how the higher-order interaction impact trapping process. Specifically, an immobile trap is first deployed at an initial node of the networks and average trapping time (ATT) is derived analytically to measure trapping efficiency of standard weight-dependent random walk. Then, delayed random walk incorporating delay phenomenon into random walk process is introduced into the networks and closed form solution of ATT is still obtained for quantifying trapping efficiency. The obtained analytical solutions of ATT in both scenarios show that ATT grows sub-linearly with network size but its leading scaling is quantitatively dependent on the parameter specifying building block of networks. Besides, analytical expression of ATT obtained in the latter scenario shows that the parameter p manipulating delayed random walk does not affect the leading scaling of ATT but can quantitatively modify its prefactor. Therefore, introduction of delay phenomenon still can modulate trapping efficiency quantitatively in the presence of higher-order interaction. This work may pave the way for modulating trapping process and related dynamical processes on general networks modeling higher-order interactions. [ABSTRACT FROM AUTHOR]

In this paper, the leaderless coherence and leader–follower coherence defined by the Laplacian spectrum are applied to noisy trees with given parameters. The asymmetry of the trees is controlled by the degrees of hub nodes. Based on the tree's structures, analytical solutions are obtained for these two types of consensus algorithms. In order to compare the leaderless and leader–follower coherence, two trees with the same network size are selected and then explicit expressions of the coherence are given. Obtained results reveal that bigger difference of the degrees of the hubs leads to higher coherence and the consensus of a tree without a leader becomes better. Finally, it is found that the leader–follower coherence changes higher with more leaders and larger network parameters. [ABSTRACT FROM AUTHOR]

This paper explores accurate, stable, and novel soliton wave solutions of the nonlinear Qiao model. This model, which was derived in 2007, possesses a Lax representation and bi-Hamiltonian structure, where it is a second positive member in the utterly integrable hierarchy. The well-known generalized extended tanh-function method is employed to construct novel soliton wave solutions. The stable property of the obtained solutions is examined along with the Hamiltonian system's characterizations. Furthermore, the accuracy of the obtained solutions is checked by comparing it with the model's semi-analytical solutions that have been obtained by employing the variational iteration (VI) method. The obtained analytical and semi-analytical solutions are demonstrated through some distinct graphs to show more physical and dynamical behavior of the investigated model. The used analytical and semi-analytical schemes' performance is checked to show if it is effective and powerful. [ABSTRACT FROM AUTHOR]

In this paper, we check and scan the (3+1)-dimensional variable-coefficient nonlinear wave equation which is considered in soliton theory and generated by considering the Hirota bilinear operators. We retrieve some novel exact analytical solutions, including cross-kink soliton solutions, breather wave solutions, interaction between stripe and periodic, multi-wave solutions, periodic wave solutions and solitary wave solutions for the (3+1)-dimensional variable-coefficient nonlinear wave equation in liquid with gas bubbles by Maple symbolic computations. The required conditions of the analyticity and positivity of the solutions can be easily achieved by taking special choices of the involved parameters. The main ingredients for this scheme are to recover the Hirota bilinear forms and their generalized equivalences. Lastly, the graphical simulations of the exact solutions are depicted. [ABSTRACT FROM AUTHOR]

In this paper, we put forward a class of weighted extended tree-like fractals and further use them as test bed to unveil the impact of weight heterogeneity on random walks. Specifically, a family of weighted extended tree-like fractals are first proposed, which are parameterized by a growth parameter m and weight parameter w. Then, we explore standard weight-dependent walk on the networks by deploying three traps at initial three nodes. To this end, we derive analytically the average trapping time (ATT) to measure the trapping efficiency and the obtained results show that depending on values of w , ATT may grow sub-linearly, linearly and super-linearly with the network size. Besides, it can also quantitatively impact the leading behavior and pre-factor of ATT simultaneously. Finally, more challenging mixed weight-dependent random walk that takes non-nearest-neighbor hopping is addressed. Analytical solutions of ATT derived under this new scenario imply that weight parameter w still can qualitatively, quantitatively steer leading behavior and quantitatively affect pre-factor of ATT. As to the stochastic parameter controlling mixed random walk, it could only impact the pre-factor of ATT and only have negligible effect on the leading behavior of ATT. In summary, this work could further augment our understanding of random walks on networks. [ABSTRACT FROM AUTHOR]

This key purpose of this study is to investigate soliton solution of the fifth-order Sawada–Kotera and Caudrey–Dodd–Gibbon equations in the sense of time fractional local M -derivatives. This important goal is achieved by employing the unified method. As a result, a number of dark and rational soliton solutions to the nonlinear model are retrieved. Some of the achieved solutions are illustrated graphically in order to fully understand their physical behavior. The results demonstrate that the presented approach is more effective in solving issues in mathematical physics and other fields. [ABSTRACT FROM AUTHOR]