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Analytical determination of harmonic components of current in electric circuits containing semiconductor converters with the use of a small parameter method (SPM) in frequency domain. The paper aims to discuss these issues.

A SPM realized in frequency domain was used in the analytical analysis of electric circuits with semiconductor converters. An automated method of formation of orthogonal harmonic components of electrical values on the basis of discrete convolution algorithm was used to provide the possibility of realization of calculation in frequency domain. A nonlinear characteristic of a semiconductor converter was presented by the method of numerical approximation. A numerical structured simulation method was applied to determination of the reference values of current in the analyzed circuit. Laws of theoretical electrical engineering were used for formation of the equations of voltage balance in the circuit with a nonlinear element.

It is shown that application of a SPM with its realization in frequency domain enables significant simplification of the process of the analysis of electric circuits with semiconductor converters in an analytical form and facilitation of calculation automation. Analytical and numerical calculation of a circuit with a diode under active-inductive load demonstrated efficiency and sufficient accuracy of the proposed method. It is shown that increase of the order of approximating polynomial and of the number of the analyzed harmonics provides the improvement of the accuracy of numerical calculations.

The results of the work can be used in calculation of electrotechnical devices containing semiconductor appliances and electric devices with nonlinear characteristics. Moreover, the obtained results enable studying the processes of compensation of current higher harmonics in electric networks with a nonlinear load containing semiconductor converters.

For the first time it was proposed to apply a SPM with its realization in frequency domain to the analysis of nonlinear electric circuits. The significance of the paper consists in the fact that the offered method makes it possible to carry out both circuit analytical and numerical analysis with the possibility of its automation.

This paper aims to present a simplified method to analyze the transient characteristics of a fractional-order very high frequency (VHF) resonant boost converter. The transient analytical solutions of state variables obtained by this method could be used as a guide for parameter design and circuit optimization.

The VHF converter is decoupled into a simplified equivalent circuit model and described by the differential equation. The solution of the simplified equivalent circuit model is taken as the main oscillation component of the transient state variable. And the equivalent small parameter method (ESPM) and Kalman filter technology are used to solve the differential equation of the converter to obtain the steady-state ripple component. Then, by superimposing the abovementioned two parts, the approximate transient analytical solution can be acquired. Finally, the influence of the fractional order of the energy storage elements on the transient process of the converter is discussed.

The results from the proposed method agree well with those from simulations, which indicates that the proposed method can effectively analyze the transient characteristic of the fractional-order VHF converter, and the analytical solution derived from the proposed mathematical model shows sufficient accuracy.

This paper proposes for the first time a method to analyze the transient characteristics of a fractional-order VHF resonant boost converter. By combining the main oscillated solution derived from the simplified equivalent circuit model with the steady-state solution based on ESPM, this method can greatly reduce the computation amount to estimate the transient solution. In addition, the discussion on the order of fractional calculus of energy storage components can provide an auxiliary guidance for the selection of circuit parameters and the study of stability.

This study aims to predict the unstable period-1 orbit (UPO-1) of DC–DC converters and find analytical expressions to describe it.

Nonlinear dynamical phenomena of a peak–current–mode controlled direct current–direct current (DC–DC) Boost converter are discussed briefly first. Then fast fourier transform (FFT) analysis of state variables under different dynamic states is provided, and the characteristic of the harmonic content in different states is summarized. Following these, a scientific hypothesis on the harmonic content of the UPO-1 is presented, and the Equivalent Small Parameter method is adopted then, thus analytic-form expressions of the UPO-1 can be derived.

According to results of theoretical analysis, numerical simulations and experiments, this paper illustrates that, like stable period-1 orbit, the UPO-1 is also made up of the DC component and harmonics with integer times of switching frequency.

This work provides an unreported approach for extracting the UPO-1 of DC–DC converters, which is mainly based on predicting the harmonic structure information of the orbit. According to experimental parts of the work, it shows that the stabilizer can be designed easier by using the proposed method. Additionally, from a broader perspective, the results could also have implications in a wide class of forced oscillation systems.

This paper aims to provide a symplectic conservation numerical analysis method for the study of nonlinear LC circuit.

The flux linkage control type nonlinear inductance model is adopted, and the LC circuit can be converted into the Hamiltonian system by introducing the electric charge as the state variable of the flux linkage. The nonlinear Hamiltonian matrix equation can be solved by perturbation method, which can be written as the sum of linear and nonlinear terms. Firstly, the linear part can be solved exactly. On this basis, the nonlinear part is analyzed by the canonical transformation. Then, the coefficient matrix of the obtained equation is still a Hamiltonian matrix, so symplectic conservation is achieved.

Numerical results reveal that the method proposed has strong stability, high precision and efficiency, and it has great advantages in long-term simulations.

This method provides a novel and effective way in studying the nonlinear LC circuit.

Multi‐point boundary value problems have important roles in the modelling of various problems in physics and engineering. This paper aims to present the solution of ordinary differential equations with multi‐point boundary value conditions by means of a semi‐numerical approach which is based on the homotopy analysis method.

The convergence of the obtained solution is expressed and some typical examples are employed to illustrate validity, effectiveness and flexibility of this procedure. This approach, in contrast to perturbation techniques, is valid even for systems without any small/large parameters and therefore it can be applied more widely than perturbation techniques, especially when there do not exist any small/large quantities.

Unlike other analytic techniques, this approach provides a convenient way to adjust and control the convergence of approximation series. Some applications will be briefly introduced.

The paper shows how an important boundary value problem is solved with a semi‐analytical method.

The purpose of this study is to provide a robust numerical method for a two parameter singularly perturbed delay parabolic initial boundary value problem (IBVP).

To solve the problem, the authors have used a hybrid scheme combining the midpoint scheme, the upwind scheme and the second-order central difference scheme for the spatial derivatives. The backward Euler scheme on a uniform mesh is used to approximate the time derivative. Here, the authors have used Shishkin type meshes for spatial discretization.

It is observed that the proposed method converges uniformly with almost second-order spatial accuracy with respect to the discrete maximum norm.

This paper deals with the numerical study of a two parameter singularly perturbed delay parabolic IBVP. To solve the problem, the authors have used a hybrid scheme combining the midpoint scheme, the upwind scheme and the second-order central difference scheme for the spatial derivatives. The backward Euler scheme on a uniform mesh is used to approximate the time derivative. The convergence analysis is carried out. It is observed that the proposed method converges uniformly with almost second-order spatial accuracy with respect to the discrete maximum norm. Numerical experiments illustrate the efficiency of the proposed scheme.

The purpose of this paper is to obtain analytic solutions of the (1+1) and (2+1)‐dimensional dispersive long wave equations by the homotopy analysis and the homotopy Padé methods.

The obtained approximation by using homotopy method contains an auxiliary parameter which is a simple way to control and adjust the convergence region and rate of solution series.

The approximation solutions by [m,m] homotopy Padé technique is often independent of auxiliary parameter ℏ and this technique accelerates the convergence of the related series.

In this paper, analytic solutions of the (1+1) and (2+1)‐dimensional dispersive long wave equations are obtained by the homotopy analysis and the homotopy Padé methods. The obtained approximation by using homotopy method contains an auxiliary parameter which is a simple way to control and adjust the convergence region and rate of solution series. The approximation solutions by [m,m] homotopy Padé technique are often independent of auxiliary parameter ℏ and this technique accelerates the convergence of the related series.

The purpose of this study is to propose a generalized integral transform technique (GITT) to investigate the bending behavior of rectangular thin plates with linearly varying thickness resting on a double-parameter foundation.

The bending of plates with linearly varying thickness resting on a double-parameter foundation is analyzed by using the GITT for six combinations of clamped, simply-supported and free boundary conditions under linearly varying loads. The governing equation of plate bending is integral transformed in the uniform-thickness direction, resulting in a linear system of ordinary differential equations in the varying thickness direction that is solved by a fourth-order finite difference method. Parametric studies are performed to investigate the effects of boundary conditions, foundation coefficients and geometric parameters of variable thickness plates on the bending behavior.

The proposed hybrid analytical-numerical solution is validated against a fourth-order finite difference solution of the original partial differential equation, as well as available results in the literature for some particular cases. The results show that the foundation coefficients and the aspect ratio b/a (width in the y direction to height of plate in the x direction) have significant effects on the deflection of rectangular plates.

The present GITT method can be applied for bending problems of rectangular thin plates with arbitrary thickness variation along one direction under different combinations of loading and boundary conditions.

The purpose of this paper was to study laminar fluid flow and convective heat transfer in a conical gap at small conicity angles up to 4° for the case of disk rotation with a fixed cone.

In this paper, the improved asymptotic expansion method developed by the author was applied to the self-similar Navier–Stokes equations. The characteristic Reynolds number ranged from 0.001 to 2.0, and the Prandtl numbers ranged from 0.71 to 10.

Compared to previous approaches, the improved asymptotic expansion method has an accuracy like the self-similar solution in a significantly wider range of Reynolds and Prandtl numbers. Including radial thermal conductivity in the energy equation at small conicity angle leads to insignificant deviations of the Nusselt number (maximum 1.23%).

This problem has applications in rheometry to experimentally determine viscosity of liquids, as well as in bioengineering and medicine, where cone-and-disk devices serve as an incubator for nurturing endothelial cells.

The study can help design more effective devices to nurture endothelial cells, which regulate exchanges between the bloodstream and the surrounding tissues.

To the best of the authors’ knowledge, for the first time, novel approximate analytical solutions were obtained for the radial, tangential and axial velocity components, flow swirl angle on the disk, tangential stresses on both surfaces, as well as static pressure, which varies not only with the Reynolds number but also across the gap. These solutions are in excellent agreement with the self-similar solution.

This paper aims to develop resonant vibratory gyroscopes for high sensitive detection. The dynamic characteristics of resonant vibratory gyroscopes are investigated.

Firstly, the working principle and the dynamic output characteristics of the resonant vibratory gyroscope could be described by the damped Mathieu equation. Moreover, an approximate analytical method based on the small parameter perturbation has been used for the purpose of investigating the approximate solution of the damped Mathieu equation. Finally, to verify the feasibility of the approximate analytical method of the damped Mathieu equation, dynamic output characteristics’ experiments of the resonant vibratory gyroscope are built.

The theoretical analysis and numerical simulations show that the approximate solution of the damped Mathieu equation is close to the dynamic output characteristics of the resonant vibratory gyroscope. On the other hand, it is concluded from the tested result that there exists a correlation between the theoretical curve and the experimental data processing result, meaning the damped dynamics analytical method is effective in building resonant vibratory gyroscopes.

This paper seeks to establish a foundation for optimizing and testing the performance of the resonant vibratory gyroscope. To this end, the approximate analytical method of the damped Mathieu equation was discussed. The result of this research has proved that the dynamic characteristics based on the damped Mathieu equation is an effective approach and is instructional in the practical resonant sensor design.