Modified Perturbation Method Research Articles

The role of temperature-dependent solubility in the stability of thermohaline convection within a Voigt-fluid layer

The study focuses on both linear and weakly nonlinear stability analyses of thermohaline convection within a Voigt-fluid layer, considering the effects of temperature-dependent solubility. Analytical methods are employed to derive conditions for the onset of both stationary and oscillatory convection. The linear stability analysis is performed using normal mode analysis to explore how various physical parameters influence the initiation of convection. Notably, unlike Newtonian fluids, the Navier-Stokes-Voigt fluid suppresses the typical quasiperiodic bifurcation from a steady state. To gain deeper insight into the onset of convection, a weakly nonlinear stability analysis is conducted using a modified perturbation method, leading to the Ginzburg-Landau equation. This reveals the potential for subcritical instability, a phenomenon not fully captured by linear analysis alone. Additionally, the study examines how different physical parameters affect convective heat and mass transfer, with findings that align with previous studies in specific limiting cases.

A modified perturbation method for global dynamic analysis of generalized mixed Rayleigh–Liénard oscillator with cubic and quintic nonlinearities

Deriving analytical relationship between the system parameters and amplitude of the limit cycle is a meaningful and challenging task. Currently, numerous existing analytical approximate methods struggle to achieve this goal when expressions of restoring force or nonlinear damping is complicated. To overcome this shortcoming, this study proposes a modified generalized harmonic function perturbation method. Using the proposed method, a generalized mixed RayleighLiénard oscillator with cubic and quintic nonlinearities was investigated. The analytical relationships between the system parameters and amplitude of the limit cycle, as well as the expression of its characteristic quantity, were derived. By employing these analytical relationships, the existence, stability, number, position, and amplitude of each limit cycle are quantitatively analysed. The homoclinic and heteroclinic bifurcations were also predicted using the above analytical relationships. Additionally, analytical approximate solutions for this oscillator were calculated using the proposed method. All results obtained in this study were subsequently confirmed numerically to demonstrate their feasibility and validity. Consequently, the proposed method can be considered an effective supplement to perturbation-based methods. This also implies that the work presented in this paper has a certain theoretical significance and application value in the research area of quantitative analysis methods for strongly nonlinear oscillators.

Triangular intuitionistic fuzzy linear system of equations with applications: an analytical approach

This study extended an existing semi-analytical technique, the Homotopy Perturbation Method, to the Block Homotopy Modified Perturbation Method by solving two n × n crisp triangular intuitionistic fuzzy (TIF) systems of linear equations. In the original system, the coefficient matrix is considered as real crisp, while the unknown variable vector and right hand side vector are regarded as triangular intuitionistic fuzzy numbers. The Block Homotopy Modified Perturbation Method is found to be efficient and practical to solve n × n TIF linear systems as it only requires the non-singularity of the n × n TIF linear system's coefficient matrix, whereas the point Homotopy Perturbation Method and other classical numerical iterative methods typically require non-zero diagonal entries in the coefficient matrix. A set of theorems relevant to this study are presented and demonstrated. We solve an engineering application, i.e. a current flow circuit problem that is represented in terms of a triangular intuitionistic fuzzy environment, using the suggested method. The unknown current is then obtained as a triangle intuitionistic fuzzy number. The proposed semi-analytic method is used to solve some numerical test problems in order to validate their performance and efficiency in comparison to other existing techniques. The numerical results of the example are displayed on graphs with different degrees of uncertainty. The efficiency and accuracy of the proposed method are further demonstrated by comparisons to block Jacobi, Adomain Decomposition method, Successive Over-Relaxation method and the classical Gauss-Seidel numerical method.

On the Solutions of the Fractional-Order Sawada–Kotera–Ito Equation and Modeling Nonlinear Structures in Fluid Mediums

This study investigates the wave solutions of the time-fractional Sawada–Kotera–Ito equation (SKIE) that arise in shallow water and many other fluid mediums by utilizing some of the most flexible and high-precision methods. The SKIE is a nonlinear integrable partial differential equation (PDE) with significant applications in shallow water dynamics and fluid mechanics. However, the traditional numerical methods used for analyzing this equation are often plagued by difficulties in handling the fractional derivatives (FDs), which lead to finding other techniques to overcome these difficulties. To address this challenge, the Adomian decomposition (AD) transform method (ADTM) and homotopy perturbation transform method (HPTM) are employed to obtain exact and numerical solutions for the time-fractional SKIE. The ADTM involves decomposing the fractional equation into a series of polynomials and solving each component iteratively. The HPTM is a modified perturbation method that uses a continuous deformation of a known solution to the desired solution. The results show that both methods can produce accurate and stable solutions for the time-fractional SKIE. In addition, we compare the numerical solutions obtained from both methods and demonstrate the superiority of the HPTM in terms of efficiency and accuracy. The study provides valuable insights into the wave solutions of shallow water dynamics and nonlinear waves in plasma, and has important implications for the study of fractional partial differential equations (FPDEs). In conclusion, the method offers effective and efficient solutions for the time-fractional SKIE and demonstrates their usefulness in solving nonlinear integrable PDEs.

A radiating straight fin with constant thermal properties by modified perturbation method

A radiating straight fin with constant thermal properties by modified perturbation method

Exploration of anisotropy on nonlinear stability of thermohaline viscoelastic porous convection

The nonlinear stability of thermohaline convection in a horizontal layer of an anisotropic porous medium saturated with an Oldroyd-B fluid is studied using a modified perturbation method. Anisotropy in permeability, thermal and solute diffusivities is modelled as second-rank tensors. The cubic Landau equations for steady and oscillatory bifurcating solutions are derived. It is apparent that subcritical instability is possible depending on the choices of governing parameters indicating the linear stability theory is not sufficient to capture the onset of convection. The effect of strain retardation parameter, solute Darcy-Rayleigh number, thermal anisotropy parameter as well as solute anisotropy parameter is to delay, while the effect of stress relaxation parameter, Lewis number and the mechanical anisotropy parameter is to hasten the onset of oscillatory convection. It is found that the stationary bifurcating solution is subcritical while that of oscillatory is supercritical at higher values of solute Darcy-Rayleigh number. The results of Maxwell fluid are obtained as a limiting case. Heat and mass transports are estimated using Nusselt number and Sherwood number, respectively and by tuning the anisotropy parameters it is possible to control the same.

A modified perturbation method in which the applicable range of a small parameter for the solution is extended much larger than that by the conventional perturbation method

A modified perturbation method in which the applicable range of a small parameter for the solution is extended much larger than that by the conventional perturbation method

Self-contact of a flexible loop under uniform hydrostatic pressure

Critical values of the pressure leading to self-contact in a flexible and inextensible loop subjected to uniform hydrostatic pressure have been predicted using variants and approximations of the harmonic balance method. Moreover, a modified perturbation method has been employed that has shown some advantages over conventional perturbation approach to the problem. A comprehensive comparison is made between the results we obtained using various techniques and the existing reported results for different pressure loadings and rotational symmetries.

A modified perturbation solution to the strongly nonlinear initial value problem of the KdV–KP equation

In this paper, a modified perturbation method to solve strongly nonlinear initial-value-problem of the KdV–KP equation is introduced. The method is a modification of the linearized perturbation expansion method which was developed to solve nonlinear initial value problems with no small parameters. The main idea is to embed an artificial parameter so that the dependent variable can later be expanded in series of this parameter. In addition, another arbitrary constant is included to take care of the secular terms that appear after integration. The current technique assumes the arbitrary constant can also be expanded in terms of the embedded parameter. This step gives an advantage of allowing for more iteration and more correction terms to be added to the perturbation solution. The obtained solution is compared with the exact solution obtained using the Lie group method and the results show good agreement.

A modified perturbation method for mathematical models with randomness: An analysis through the steady‐state solution to Burgers' partial differential equation

The variability of the data and the incomplete knowledge of the true physics require the incorporation of randomness into the formulation of mathematical models. In this setting, the deterministic numerical methods cannot capture the propagation of the uncertainty from the inputs to the model output. For some problems, such as the Burgers' equation (simplification to understand properties of the Navier–Stokes equations), a small variation in the parameters causes nonnegligible changes in the output. Thus, suitable techniques for uncertainty quantification must be used. The generalized polynomial chaos (gPC) method has been successfully applied to compute the location of the transition layer of the steady‐state solution, when a small uncertainty is incorporated into the boundary. On the contrary, the classical perturbation method does not give reliable results, due to the uncertainty magnitude of the output. We propose a modification of the perturbation method that converges and is comparable with the gPC approach in terms of efficiency and rate of convergence. The method is even applicable when the input random parameters are dependent random variables.

Truss layout design under nonprobabilistic reliability‐based topology optimization framework with interval uncertainties

SummaryA nonprobabilistic reliability‐based topology optimization (NRBTO) method for truss structures with interval uncertainties (or unknown‐but‐bounded uncertainties) is proposed in this paper. The cross‐sectional areas of levers are defined as design variables, while the material properties and external loads are regard as interval parameters. A modified perturbation method is applied to calculate structural response bounds, which are the prerequisite to obtain structural reliability. A deviation distance between the current limit state plane and the objective limit state plane, of which the expression is explicit, is defined as the nonprobabilistic reliability index, which serves as a constraint function in the optimization model. Compared with the deterministic topology optimization problem, the proposed NRBTO formulation is still a single‐loop optimization problem, as the reliability index is explicit. The sensitivity results are obtained from an analytical approach as well as a direct difference method. Eventually, the NRBTO problem is solved by a sequential quadratic programming method. Two numerical examples are used to testify the validity and effectiveness of the proposed method. The results show significant effects of uncertainties to the topology configuration of truss structures.

Axisymmetric Longitudinal-Bending Waves in a Cylindrical Shell Interacting with a Nonlinear Elastic Medium

A nonlinear differential equation is derived which describes the propagation of axisymmetric stationary longitudinal-bending waves in infinite cylindrical shell of Timoshenko type, interacting with the external nonlinear elastic medium. A modified perturbation method based on the use of diagonal Pade approximants was applied to build exact solitary-wave solutions of the derived equation in the form of traveling front and the traveling pulse. Numerical solutions of the equation, obtained by means of finite difference method, are in good agreement with the corresponding exact analytical ones.

Theoretical Analysis and Numerical Simulation of Resonances and Stability of a Piecewise Linear-Nonlinear Vibration Isolation System

A methodology is presented to study the resonance and stability for a single-degree-of-freedom (SDOF) system with a piecewise linear-nonlinear stiffness term (i.e., one piece is linear and the other is weakly nonlinear). Firstly, the exact response of the linear governing equation is obtained, and a modified perturbation method is applied to finding the approximate solution of the weakly nonlinear equation. Then, the primary and 1/2 subharmonic resonances are obtained by imposing continuity conditions and periodicity conditions. Furthermore, Jacobian matrix is derived to investigate the stability of resonance responses. Finally, the results of theoretical study are compared with numerical results, and a good agreement is observed.

Solving a class of burning disturbed problem with shock layers

A class of combustion problem with shock layers is considered. A modified perturbation method is presented. Using this simple and valid technique, we construct the boundary and the shock layers solution to the problem, and the asymptotic behavior of the solution is discussed. The modifying perturbation method is shown to be a valid method.

A Three Dimensional Microwave Tomography Employing an Adjoint Network Theorem Based Sensitivity Matrix

A reconstruction algorithm for three dimensional microwave imaging based on the modified perturbation method is proposed. Both the object conductivity (sigma) and permittivity (epsiv <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">r</sub> ) are reconstructed. The original Perturbation method developed for static problems was modified in order to apply in time harmonic problem at microwave frequencies. The Jacobian matrix is calculated in closed form employing an adjoint network theorem in conjunction with the electromagnetics reciprocity theorem. A number of successful reconstructions were carried out for different complex permittivity profiles, but all of them based on a computer phantom approach.

A higher order asymptotic approximation for the fundamental frequency of a multiply connected membrane

The fundamental frequency of a fixed membrane is the square root of the lowest eigenvalue of negative Laplace operator with Dirichlet boundary conditions. A multiply connected membrane with inner cores of vanishing maximal dimensions 2 c j is considered in the present article. The modified perturbation method developed for a doubly connected membrane is extended to provide a general formula for the fundamental frequency of the multiply connected membrane. A higher order asymptotic approximation (as c j → 0 ) for the fundamental frequency of a membrane with inner circular cores of radius c j is specified. It is an excellent extension of the results in the literature. Moreover, a second-order asymptotic approximation (as c → 0 ) for the fundamental frequency of a circular membrane of radius 1 with finitely many inner circular cores of small radius c is found and computed explicitly. The effects of the positions of the inner cores on the second-order asymptotic approximation are investigated. The accuracy of the second-order asymptotic approximation is also shown by the comparisons among the asymptotic approximations and the numerical values computed by other investigators.

Solution for Aircraft Anti-Icing System Simulation by a Modified Perturbation Method

A solution of the surface temperature and the runback water mass flow rate distributions along the body contour for an electrothermal or a hot-air anti-icing system of an aircraft, when water droplets impinge on the surface, is obtained by a modified perturbation method. In the method, a small dimensionless parameter a that represents the ratio of the external convective heat transfer rate from the airfoil surface to the internal heating rate is used. Because of the high nonlinearity in the energy equation, obtaining a convergent solution by normal use of the perturbation method is virtually impossible. To improve the convergence properties of the solution, the calculation procedure is modified. The main idea behind the modification is splitting of the nonlinear term into linear and nonlinear terms and introducing a new variable e, by which a in each linear term is replaced. These modifications of the perturbation method help reduce the contribution of the nonlinear term, which drastically improves the convergence characteristics of the solution. The solutions agree well with the results obtained by the Runge-Kutta method and experimental and numerical results found in literature.

A Modified Perturbation Method for Three-Dimensional Time Harmonic Impedance Tomography

A reconstruction algorithm for three dimensional time harmonic impedance imaging based on the Modified Perturbation Method (MPM), [1], is proposed. Both the object conductivity (σ) and permittivity (e) are reconstructed. The original Perturbation method developed for static problems was modified in order to apply in time harmonic problem in higher frequency. In this case complex permittivity and complex voltages are involved. So, major modifications have been made in order to achieve accepted results. The jacobian matrix is expressed in complex form and the modified perturbation reconstruction algorithm formulated accordingly. A number of successful reconstructions were carried out for different complex permittivity profiles, but all of them based on a computer phantom approach.

Free-parameter perturbation-method solutions of the nonlinear stability of shallow spherical shells

The free-parameter perturbation method is applied to solve the problems of nonlinear stability of spehrical shallow shells under uniform load. As a modified perturbation method, the free-parameter perturbation method enables researchers to obtain all characteristic relations without choosing the certain perturbation parameter. Some examples were discussed to study the variety regulations of deflections and stress of shells in the process of buckling, and the results were compared with those of other researchers.

Fundamental frequency of a doubly connected membrane: a modified perturbation method

The fundamental frequency of a membrane is the square root of the lowest eigenvalue of the negative Laplace operator with Dirichlet boundary conditions. A doubly connected membrane with the inner region of vanishing maximal dimension 2c is considered in this paper. A modified perturbation method is developed to provide an asymptotic expansion (c → 0) for the fundamental frequency of the membrane. The first three order terms of the asymptotic expansion for the fundamental frequency of a doubly connected membrane with the circular inner region are derived explicitly. The results are compared with the exact solutions and the approximations determined by other investigators. The error of the perturbation calculations compared with the exact values is less than 1% as c is less than or equal to 0.25 and is less than 4% as c is less than or equal to 0.35.