Purpose
This paper aims to review some effective methods for fully fourth-order nonlinear integral boundary value problems with fractal derivatives.
Design/methodology/approach
Boundary value ...problems arise everywhere in engineering, hence two-scale thermodynamics and fractal calculus have been introduced. Some analytical methods are reviewed, mainly including the variational iteration method, the Ritz method, the homotopy perturbation method, the variational principle and the Taylor series method. An example is given to show the simple solution process and the high accuracy of the solution.
Findings
An elemental and heuristic explanation of fractal calculus is given, and the main solution process and merits of each reviewed method are elucidated. The fractal boundary value problem in a fractal space can be approximately converted into a classical one by the two-scale transform.
Originality/value
This paper can be served as a paradigm for various practical applications.
This paper originally investigates the performance of a tri-stable piezoelectric energy harvester (TPEH) in rotational motion, aiming to solve the challenging issue of power supply for wireless ...sensors. Based on the Lagrange equation, the related theoretical model with the consideration of the effect of rotational motion is originally derived to describe its dynamic response and energy harvesting performance. In addition, the perturbation method is used to theoretically describe the TPEH for the oscillations around both non-zero and zero stable equilibrium positions. The numerical simulations and case studies are carried out to investigate the influence of the Kc coefficient on the dynamic response of the TPEH. More importantly, the corresponding experiments under different constant rotational speeds are performed to validate the energy harvesting enhancement of the presented TPEH and the accuracy of its theoretical model. It is experimentally verified that the TPEH can efficiently harvest energy in the wide rotational speed range (240–440 rpm), and the proposed theoretical model is suitable for the TPEH. Overall, the energy harvesting enhancement of the TPEH in rotational motion are verified, and the accuracy of the presented theoretical model is also experimentally validated.
•A tri-stable energy harvester in rotational motion is originally investigated.•The complete theoretical model is derived with considering the effect of rotational motion.•The perturbation method is used to obtain the theoretical solutions of the presented harvester.•Experiments validate the theoretical model and energy harvesting enhancement of the harvester.
Purpose
This study aims that very lately, Mohand transform is introduced to solve the ordinary and partial differential equations (PDEs). In this paper, the authors modify this transformation and ...associate it with a further analytical method called homotopy perturbation method (HPM) for the fractional view of Newell–Whitehead–Segel equation (NWSE). As Mohand transform is restricted to linear obstacles only, as a consequence, HPM is used to crack the nonlinear terms arising in the illustrated problems. The fractional derivatives are taken into the Caputo sense.
Design/methodology/approach
The specific objective of this study is to examine the problem which performs an efficient role in the form of stripe orders of two dimensional systems. The authors achieve the multiple behaviors and properties of fractional NWSE with different positive integers.
Findings
The main finding of this paper is to analyze the fractional view of NWSE. The obtain results perform very good in agreement with exact solution. The authors show that this strategy is absolutely very easy and smooth and have no assumption for the constriction of this approach.
Research limitations/implications
This paper invokes these two main inspirations: first, Mohand transform is associated with HPM, secondly, fractional view of NWSE with different positive integers.
Practical implications
In this paper, the graph of approximate solution has the excellent promise with the graphs of exact solutions.
Social implications
This paper presents valuable technique for handling the fractional PDEs without involving any restrictions or hypothesis.
Originality/value
The authors discuss the fractional view of NWSE by a Mohand transform. The work of the present paper is original and advanced. Significantly, to the best of the authors’ knowledge, no such work has yet been published in the literature.
Purpose
On a microgravity condition, a motion of soliton might be subject to a microgravity-induced motion. There is no theory so far to study the effect of air density and gravity on the motion ...property. Here, the author considers the air as discrete molecules and a motion of a soliton is modeled based on He’s fractal derivative in a microgravity space. The variational principle of the alternative model is constructed by semi-inverse method. The variational principle can be used to establish the conservation laws and reveal the structure of the solution. Finally, its approximate analytical solution is found by using two-scale method and homotopy perturbation method (HPM).
Design/methodology/approach
The author establishes a new fractal model based on He’s fractal derivative in a microgravity space and its variational principle is obtained via the semi-inverse method. The approximate analytical solution of the fractal model is obtained by using two-scale method and HPM.
Findings
He’s fractal derivative is a powerful tool to establish a mathematical model in microgravity space. The variational principle of the fractal model can be used to establish the conservation laws and reveal the structure of the solution.
Originality/value
The author proposes the first fractal model for the soliton motion in a microgravtity space and obtains its variational principle and approximate solution.
Here, the homotopy analysis method (HAM), which is a powerful and easy-to-use analytic tool for nonlinear problems and dose not need small parameters in the equations, is compared with the ...perturbation and numerical and homotopy perturbation method (HPM) in the heat transfer filed. The homotopy analysis method contains the auxiliary parameter
ℏ, which provides us with a simple way to adjust and control the convergence region of solution series.
•A new iteration regularization algorithm is proposed.•The proposed iteration regularization algorithm is strictly proved.•The present method is powerful for load identification for stochastic ...structure.•The newly developed method saves computational cost remarkably compared with MCS.•Numerical simulations validate the obtained theoretical results.
In this paper, we propose a novel fast convergence iterative regularization method on the basis of a new iterative regularization operator to solve the dynamic load identification of stochastic structures. Firstly, the validity and convergence of the proposed algorithm is proved by strict mathematical theory and numerical results of engineering example. Secondly, the dynamic load identification of uncertain structures can be transformed into a series of deterministic inverse problems by the matrix perturbation method. Finally, we also assess the statistical characteristics of identified loads using the statistical theory. Particularly, the proposed method successfully solves the dynamic load identification of stochastic structure. Numerical simulations are given to validate the feasibility of the presented method in this paper.
Employing variable horizontal and vertical shifts for the unperturbed solution, a new perturbation method is proposed. The new method is named as the shift perturbation method. The existing ...perturbation methods are interpreted with respect to the horizontal and vertical shifts for the unperturbed solution. The new method is applied to some well-known vibrational problems. The method is capable of producing admissible approximate solutions for a wide range of problems such as the free and forced vibrations of the Duffing equation, equations with quadratic and cubic nonlinearities, variable amplitude nonlinear problems, and boundary layer problems. A variant of the method also produced valid solutions for large perturbation parameters.
The full-polarimetric ground-penetrating radar (FP-GPR) can obtain the polarimetric attributes of targets and achieve more accurate identification compared with traditional GPRs. However, in most ...cases, the polarimetric signals collected by GPRs are not only from the targets but also from the ground surface. According to the classical Fresnel formulas, the polarized directions of the waves will change after transmissions due to the difference in the transmission coefficients between horizontally and vertically polarized waves. This effect, called induced field rotation (IFR), will interfere with the acquisition of polarimetric attributes and also exists in the field measurements on a rough surface. In this study, we have derived the association between the measured FP-GPR data and transmission coefficients of the rough surface provided by the small perturbation method (SPM). The effects of IFRs from the rough surface on H-Alpha decomposition are analyzed later. The results show that the parameters of the rough surface will affect the values of components in the scattering matrix, but will not change the matrix and H-Alpha decomposition result; the incident angle and relative permittivity play main roles; for the H-Alpha decomposition results of three typical targets, the flat and dihedral are little influenced by IFRs, but the cylinder is seriously affected; simultaneously, a template for discriminating whether the calibration for H-Alpha decomposition is necessary is established. Both numerical and experimental tests validate the conclusions. Finally, a novel application strategy of H-Alpha decomposition is proposed, which has taken IFR from the rough surface into considerations.
A modified perturbation method (Kato and Matsuda, 2021) is used to obtain the solution of the heat transfer problem of a radiating straight fin with constant thermal properties. The main procedure of ...the modified perturbation method (MPM) is: (1) A perturbation parameter ε is assumed to be included in the nonlinear term of the differential equation. The solution θ is expressed by θ = φ + θf, where θf is an initial approximation of the solution. (In this paper, θf is assumed to be a constant) (2) θ = φ + θf is substituted into the differential equation and the nonlinear term is split into linear and nonlinear terms. (3) ε which is not in the nonlinear term is replaced by a newly introduced parameter ε´. (4) An asymptotic expansion of φ in powers of ε is assumed for the solution of the differential equation, from which we obtain the perturbation solution of φ including ε and ε´. (5) ε´ in the perturbation solution of φ is replaced by ε. Then we obtain the perturbation solution of θ. The obtained solutions by MPM are found to be in good agreement with the numerical results by the finite difference method. The solutions are also compared with those by the conventional perturbation method (CPM). It is found that MPM can extend the applicable range of the small parameter ε (radiation-conduction parameter) drastically compared with that by CPM. The modifications of the perturbation method by splitting the nonlinear term help reduce the contribution of the nonlinear term, which drastically improve the convergence characteristics of the solution.
A modified perturbation method (Kato and Matsuda, 2021) is used to obtain the solution of the heat transfer problem of a radiating straight fin with constant thermal properties. The main procedure of ...the modified perturbation method (MPM) is: (1) A perturbation parameter ε is assumed to be included in the nonlinear term of the differential equation. The solution θ is expressed by θ = φ + θf, where θf is an initial approximation of the solution. (In this paper, θf is assumed to be a constant) (2) θ = φ + θf is substituted into the differential equation and the nonlinear term is split into linear and nonlinear terms. (3) ε which is not in the nonlinear term is replaced by a newly introduced parameter ε´. (4) An asymptotic expansion of φ in powers of ε is assumed for the solution of the differential equation, from which we obtain the perturbation solution of φ including ε and ε´. (5) ε´ in the perturbation solution of φ is replaced by ε. Then we obtain the perturbation solution of θ. The obtained solutions by MPM are found to be in good agreement with the numerical results by the finite difference method. The solutions are also compared with those by the conventional perturbation method (CPM). It is found that MPM can extend the applicable range of the small parameter ε (radiation-conduction parameter) drastically compared with that by CPM. The modifications of the perturbation method by splitting the nonlinear term help reduce the contribution of the nonlinear term, which drastically improve the convergence characteristics of the solution.
