Perturbation Terms Research Articles

Creeping thermocapillary motion of a Newtonian droplet suspended in a viscoelastic fluid

In this work we consider theoretically the problem of a Newtonian droplet moving in an otherwise quiescent infinite viscoelastic fluid under the influence of an externally applied temperature gradient. The outer fluid is modelled by the Oldroyd-B equation, and the problem is solved for small Weissenberg and Capillary numbers in terms of a double perturbation expansion. We assume microgravity conditions and neglect the convective transport of energy and momentum. We derive expressions for the droplet migration speed and its shape in terms of the properties of both fluids. In the absence of shape deformation, the droplet speed decreases monotonically for sufficiently viscous inner fluids, while for fluids with a smaller inner-to-outer viscosity ratio, the droplet speed first increases and then decreases as a function of the Weissenberg number. For small but finite values of the Capillary number, the droplet speed behaves monotonically as a function of the applied temperature gradient for a fixed ratio of the Capillary and Weissenberg numbers. We demonstrate that this behaviour is related to the polymeric stresses deforming the droplet in the direction of its migration, while the associated changes in its speed are Newtonian in nature, being related to a change in the droplet’s hydrodynamic resistance and its internal temperature distribution. When compared to the results of numerical simulations, our theory exhibits a good predictive power for sufficiently small values of the Capillary and Weissenberg numbers.

Analytical quasibound states of black holes emerging from modified theories of gravity

The Rayleigh–Schrodinger method or more commonly called time independent perturbation theory is a powerful method to solve a Hamiltonian system with a relatively small perturbation [W. Nolting, Theoretical Physics 7: Quantum Mechanics–Methods and Applications (Springer International, 2017); F. Schwabl, Quantum Mechanics, 4th edn. (Springer, 2007)]. In this work, we make use of the Rayleigh–Schrodinger method to formulate a general method to calculate quasibound states of static spherically symmetric black hole solutions arising from modified theories of gravity. We discover that the Schwarzschild-like term corresponds to the main Hamiltonian while the modified theory of gravity’s contribution of the spacetime metric corresponds to the perturbation terms. At the end, formula to calculate main Hamiltonian and perturbed Hamiltonian are discovered and presented.

Study on dynamic features and traveling wave solutions of perturbed nonlinear Biswas–Milovic equation combining Kudryashov's law of refractive index

This article mainly explores the Biswas–Milovic equation containing Kudryashov's refractive index law and nonlinear perturbation terms. We take the complete discrimination system for polynomial method and the trial equation method to implement qualitative and quantitative studies for the equation. We verify that the Hamiltonian of the dynamic system is conserved and apply the complete discrimination system of polynomial method to give the associated global phase diagrams, which are qualitatively analyzed to illustrate the presence of periodic solutions and solitons. In addition, we gain a classification of all single traveling wave solutions, provide optical wave patterns with specific parameters and validate the above findings.

Peano and Osgood theorems via effective infinitesimals

We provide effective proofs using infinitesimals of the globalversions of Peano's existence theorem and Osgood's theorem on maximalsolutions. We characterize all solutions in terms of infinitesimal perturbations. Our proofs are more effective than traditionalnon-infinitesimal proofs found in the literature. The backgroundlogical structure is the internal set theory SPOT, conservativeover ZF.

Zero-field splitting parameters within exact two-component theory and modern density functional theory using seminumerical integration.

An efficient implementation of zero-field splitting parameters based on the work of Schmitt et al. [J. Chem. Phys. 134, 194113 (2011)] is presented. Seminumerical integration techniques are used for the two-electron spin-dipole contribution and the response equations of the spin-orbit perturbation. The original formulation is further generalized. First, it is extended to meta-generalized gradient approximations and local hybrid functionals. For these functional classes, the response of the paramagnetic current density is considered in the coupled-perturbed Kohn-Sham equations for the spin-orbit perturbation term. Second, the spin-orbit perturbation is formulated within relativistic exact two-component theory and the screened nuclear spin-orbit (SNSO) approximation. The accuracy of the implementation is demonstrated for transition-metal and diatomic main-group compounds. The efficiency is assessed for Mn and Mo complexes. Here, it is found that coarse integration grids for the seminumerical schemes lead to drastic speedups while introducing clearly negligible errors. In addition, the SNSO approximation substantially reduces the computational demands and leads to very similar results as the spin-orbit mean field Ansatz.

A new kind of double phase elliptic inclusions with logarithmic perturbation terms I: Existence and extremality results

This paper is devoted to introduce a new double phase elliptic inclusion problem (DPEI) involving nonlinear and nonhomogeneous partial differential operator which has unbalanced growth and logarithmic perturbation terms, and two multivalued functions which are defined in the domain and its boundary. The main goal of this paper is to establish the existence and extremality results to the elliptic inclusion problem under consideration. More exactly, we give the definitions of weak solutions, subsolutions and supersolutions to (DPEI). Then, under the coercive setting, an existence theorem of weak solutions to (DPEI) is obtained by employing a surjectivity theorem for pseudomonotone operators. Moreover, in the noncoercive framework, we apply the method of sub-supersolution combined with the nonsmooth calculus analysis and truncation techniques to prove that (DPEI) has at least a weak solution within an ordered interval of sub-supersolution. Finally, when the constraint set K satisfies a lattice condition, the existence of smallest and greatest elements of solution set to (DPEI) is established.

Bifurcation Analysis and Chaotic Behavior of the Concatenation Model with Power-Law Nonlinearity

This paper presents a comprehensive analysis of the concatenation model with power-law nonlinearity. The research encompasses multiple key aspects, providing a detailed exploration of the model's behavior and implications within the context of nonlinear dynamics and optics. The study commences with an in-depth bifurcation analysis, aiming to unravel the intricate dynamics and transitions within the system. This analysis not only uncovers the system's behavior under varying conditions but also sheds light on its stability and the emergence of bifurcation phenomena. Our research delves into the retrieval of soliton solutions within the model. The exploration of solitons is of paramount significance, offering insights into localized, self-sustaining waveforms that often play a crucial role in nonlinear systems. These soliton solutions are identified, characterized, and their relevance to the model is established. The paper addresses the complex dynamics of the system in the presence of perturbation terms. By incorporating perturbations into the analysis, we elucidate how external influences impact the system's behavior and lead to chaotic phenomena. This analysis helps uncover the system's sensitivity to external factors and provides a deeper understanding of chaotic behavior.

The KG oscillator in the background of external magnetic field with a Cornell interaction in non-commutative quantum mechanics

In this paper, we investigated the two-dimensional Klein–Gordon oscillator in non-commutative quantum mechanics (NCQM). We also studied the case of a spin-0 particle moving in a background magnetic field with the Cornell potential in commutative space, non-commutative space, and non-commutative space by using a quasi-exact methodology. The Hamiltonian was modified by the non-commutative parameter θ. We observed that the terms related to the deformation parameter can be taken as perturbation terms in QM. It was demonstrated that the non-commutative Hamiltonian was derived from the Moyal–Weyl multiplication and the Bopp shift method. We numerically calculated the energy spectrum in both commutative and non-commutative spaces. The behavior of all energies (the first, second, third, and fourth states) for the magnetic field was shown graphically. Furthermore, we derive the non-relativistic limit of the energy eigenvalues, which were comparable to the energy eigenvalues in the presence of the magnetic field in commutative space, known as the Zeeman effect.

Uncovering diverse soliton solutions in the modified Schrödinger’s equation via innovative approaches

The topic of soliton propagation in optical fibers is explored in our research paper, with a focus on the utilization of the modified nonlinear Schrödinger’s equation with perturbation terms. Two effective techniques have been employed in this model, resulting in the generation of wave structures in a closed-form manner and the production of a wide range of distinct solutions. These solutions hold potential applications in various fields such as optical fibers, plasma fluids, and biomolecular dynamics. The proposed approaches are characterized by their simplicity, robustness, and ability to generate new solutions for nonlinear partial differential equations in the field of mathematical physics. Captivating figures depicting the propagation of traveling wave solutions for carefully chosen parameter values are included in the paper. Overall, valuable insights into the behavior of soliton propagation in optical fibers are provided by this research, and new avenues for future research in this field are offered.

Application of homotopy perturbation method to solve a nonlinear mathematical model of depletion of forest resources

Reduction in forest resources due to increasing global warming and population growth is a critical situation the World faces today. As these reserves decrease, it alarms new challenges that require urgent attention. In this paper, we provide a semi-analytical solution to a nonlinear mathematical model that studies the depletion of forest resources due to population growth and its pressure. With the help of the homotopy perturbation method (HPM), we determine an approximate series solution with few perturbation terms, which is one of the essential power of the HPM method. We compare our semi-analytical results with numerical solutions obtained using the Runge-Kutta 4th-order (RK-4) method. Furthermore, we analyze the model’s behaviour and dynamics by changing the parametric coefficients that represent the depletion rate of forest resources and the growth rate of population pressure and present these findings using various graphs.

Periodic Perturbations of Codimension-Two Bifurcations with a Double Zero Eigenvalue in Symmetrical Dynamical Systems

We study bifurcation behavior in periodic perturbations of two-dimensional symmetric systems exhibiting codimension-two bifurcations with a double zero eigenvalue when the frequencies of the perturbation terms are small. We transform the periodically perturbed systems to simpler ones which are periodic perturbations of the normal forms for the codimension-two bifurcations, and apply the subharmonic and homoclinic Melnikov methods to analyze bifurcations occurring there. In particular, we show that there exist transverse homoclinic or heteroclinic orbits, which yield chaotic dynamics, in wide parameter regions. These results can be applied to three or higher-dimensional systems and even to infinite-dimensional systems with the assistance of center manifold reduction and the invariant manifold theory. We illustrate our theory for a pendulum subjected to position and velocity feedback control when the desired position is periodic in time. We also give numerical computations by the computer tool AUTO to demonstrate the theoretical results.

Dynamical properties, modulation instability analysis and chaotic behaviors to the nonlinear coupled Schrödinger equation in fiber Bragg gratings

The nonlinear coupled Schrödinger equation in fiber Bragg gratings is studied in this paper. The existence of soliton solutions and periodic solutions are proved by qualitative analysis, and exact solutions are given, as well as the parameter condition of each solution is described. Then the modulation instability (MI) analysis is carried out and the linear stability criterion is given. In particular, external perturbation terms are introduced to prove that the equation exists chaotic behaviors.

Robust Nonlinear Control of a Wind Turbine with a Permanent Magnet Synchronous Generator

This paper addresses the design of a robust nonlinear dynamic controller for a wind turbine. The turbine is equipped with a permanent magnet synchronous generator. The control problem involves tracking a suitable reference value for the turbine’s angular velocity, which corresponds to the wind speed. This issue is tackled by compensating for variations in the electrical and mechanical parameters present in the mathematical model. Additionally, the problem is approached under the assumption that wind speed cannot be directly measured, a fact verified in practical scenarios. This situation is particularly relevant for real-world applications, where only nominal parameter values are accessible and accurate wind speed measurement is challenging due to disturbances caused by the turbine or other factors, despite the use of appropriate sensors. To achieve precise tracking of the angular velocity reference, effective compensation of perturbation terms arising from parameter uncertainties and errors in wind estimation becomes crucial. To address this problem, a wind velocity estimator is employed in conjunction with high-order sliding mode parameter estimators, ensuring the turbine’s operation attains a high level of performance.

Dynamical profile and multi-stability of ion-acoustic waves with soliton solution arising in plasma physics

The present exploration studies the dynamical profile, multistability and soliton solution of ion-acoustic waves (IAWs) in plasma physics. The new auxiliary equation approach is implemented to compute a wide range of solitary wave structures of different types. The dynamical profile is studied by using the bifurcation theory. This system is converted into a planar dynamical system via the Galilean transformation. Taking different cases of its coefficients, the phase plane analysis of this resultant dynamical system is performed. The phase diagrams show the different characteristics of the equilibrium points of the planar dynamical system. The quasi-periodic behaviors, multistability of this planar dynamical system are done by adding the perturbation term. The sensitivity assessment of this perturbed dynamical system is performed by taking different initial constraints.

Qualitative analysis and optical solitons of the coupled Biswas–Arshed equation in single mode polarization-preserving birefringence fibers

In this article, the topological phase portrait and dynamical behavior of the coupled Biswas–Arshed equation in single mode polarization-preserving birefringence fibers are discussed via qualitative analysis of planar dynamic system. When assigning values to the parameters of the system with perturbation terms, the system exhibits chaotic behavior. By the polynomial complete discrimination method, the optical solitons are obtained such as dark solitons, bright solitons, dark-bright solitons, and bright-dark solitons. Moreover, the physical properties of optical solitons are studied through two-dimensional and three-dimensional diagrams, density maps and contour maps of optical solitons. Finally, the modulation instability of the system is analyzed via standard linear stability.

Analysis of perturbation factors and fractional order derivatives for the novel singular model using the fractional Meyer wavelet neural networks

In this study, an analysis of the perturbation factors and fractional order derivatives is performed for the novel singular model. The design of the perturbed fractional order singular model is presented by using the traditional form of the Lane-Emden along with the detail of singular points, fractional order, shape, and perturbed factors. The analysis of the perturbation factors and fractional order terms for the singular model is provided in two steps by taking three different values of the perturbed term as well as fractional order derivatives. The numerical analysis of the perturbation and fractional order terms for the novel fractional Meyer wavelet neural network (FMWNN) along with the global and local search effectiveness of the genetic algorithm (GA) and active-set algorithm (ASA) called as FMWNN-GAASA. The modeling of the FMWNN is presented in terms of mean square error, while the optimization is performed through the GAASA. The authentication, validation, excellence, and correctness of the singular model are observed by using the comparative performances of the obtained and the reference solutions. The stability of the proposed stochastic scheme is observed through the statistical performances for taking large datasets to present the analysis of the perturbation and fractional order terms.

Exponential stability and boundedness of nonlinear perturbed systems by unbounded perturbation terms

We study the exponential stability and boundedness problem for perturbed nonlinear time-varying systems using Lyapunov Functions with indefinite derivatives. As the limiting function for the perturbation term, we use different forms and give stability and boundedness conditions in terms of the coefficients in these bounds. Contrary to most of the available conditions, we allow the coefficients to be unbounded. But instead, we put forward a condition that requires a series produced by coefficients to be limited and exponentially decaying. We perform our results on Linear time-varying systems and generalize many of the available results.

Numerical Simulation of Cubic-quartic Optical Soliton Perturbation by the Laplace–Adomian Decomposition

The current paper is on the numerical analysis of bright and dark cubic-quartic solitons that emerges from the Fokas–Lenells equation with Hamiltonian perturbation terms. The Kerr law of nonlinearity is the source of self-phase modulation. The adopted numerical scheme is the Laplace–Adomian decomposition.

Colorized visual aesthetics of interior design based on the VAR model

Abstract Color matching is an extremely critical aspect of interior design, and the development of the times has put forward higher requirements for the science of color matching. In this paper, the framework of the structural vector autoregressive model is established by introducing a structural equation in the vector autoregressive model, associating the equation for the lagged value of each endogenous variable and the deterministic time trend term, and solving the equation according to the contemporaneous relationship between data variables. Meanwhile, the structural equation is transformed into the induced form, and the parameters of the SVAR model are estimated using the perturbation terms of the induced form. Then, the underlying principle of color matching effect is analyzed in terms of hue, lightness, and purity based on the SVAR model, and the design is optimized by combining the needs in different scenes. For the home’s interior design, the color matching based on the SVAR model improved the spatial sense effect by 24.05% on average. For office interior design, the SVAR approach improved the emotional expression effect by an average of 18.96%. For commercial interior design, SVAR-based color matching effectively improved coordination by 11.25%. This study helps to break the traditional interior design color application, emphasizes the role of ambiance played by color, and provides an effective addition to the theory of color application in interior design.

Asset Portfolio Investment Strategies Based on Econometric Models

In order to provide traders with a daily trading strategy for gold and bitcoin, a price forecasting model was first developed. After performing ADF stationarity tests and LM tests on returns, the ARMA (3,7) and ARMA (4,5) models are estimated using THE PERTUR (3,7) and ALM (5) models with perturbation terms GARCH (1,1) and vt following a t-distribution. The second step is to build a risk judgement model that considers the impact of the special bull and bear market quotes, sets the bull score and determines its weight as 0.333. The third step is to build a decision model that introduces a purchase score and builds a planning equation based on the fact that gold is only traded on trading days to obtain the final decision. With this model, a principal of $1,000 could end up with an asset of $164,080. MCM/ICM: Procedures and techniques for a great experience. Proving that the model used is the optimal strategy is done in two parts: an accuracy test and an analysis of investment behavior. The accuracy test introduces the regression evaluation metrics MSE and RMSE and the results show that the portfolio investment model is the most accurate. The analysis of investment behavior demonstrates the failure of one-way investments by comparing complete bitcoin investments or gold investments with the information reviewed.