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In this paper, the authors prove that the Douglas space of second kind with a generalised form of special (α, β)-metric F, is conformally invariant.

For, the authors have used the notion of conformal transformation and Douglas space.

The authors found some results to show that the Douglas space of second kind with certain (α, β)-metrics such as Randers metric, first approximate Matsumoto metric along with some special (α, β)-metrics, is invariant under a conformal change.

The authors introduced Douglas space of second kind and established conditions under which it can be transformed to a Douglas space of second kind.

Using a generalized translation operator, this study aims to obtain a generalization of Titchmarsh's theorem for the Laguerre–Bessel transform for functions satisfying the ψ-Laguerre–Bessel–Lipschitz condition in the space L²_α (K), where K=0,+∞×0,+∞[.

The author has employed the results developed by Titchmarsh, of reference number [1].

In this paper, an analogous of Titchmarsh's theorem is established for Laguerre–Bessel transform.

To the best of the authors’ findings, at the time of submission of this paper, the results reported are new and interesting.

This paper aims to present a novel adaptive sliding mode control (ASMC) method based on the predefined performance barrier function for reusable launch vehicle under attitude constraints and mismatched disturbances.

A novel ASMC based on barrier function is adopted to deal with matched and mismatched disturbances. The upper bounds of the disturbances are not required to be known in advance. Meanwhile, a predefined performance function (PPF) with prescribed convergence time is used to adjust the boundary of the barrier function. The transient performance, including the overshoot, convergence rate and settling time, as well as the steady-state performance of the attitude tracking error are retained in the predetermined region under the barrier function and PPF. The stability of the proposed control method is analyzed via Lyapunov method.

In contrast to conventional adaptive back-stepping methods, the proposed method is comparatively simple and effective which does not need to disassemble the control system into multiple first-order systems. The proposed barrier function based on PPF can adjust not only the switching gain in an adaptive way but also the convergence time and steady-state error. And the efficiency of the proposed method is illustrated by conducting numerical simulations.

A novel barrier function based ASMC method is proposed to fit in the amplitude of the mismatched and matched disturbances. The transient and steady-state performance of attitude tracking error can be selected as prior control parameters.

This study aims to investigate the most effective approach for governments and enterprises to combat desertification by considering the governance cycle. The focus is on understanding how the government can incentivize enterprises to actively engage in desertification combat efforts.

Both the government and the enterprise are treated as rational entities, making strategic choices for joint participation in combating desertification. Recognizing the dynamic nature of the desertification combat area, differential game models are employed to identify the optimal mode for combating desertification.

The findings underscore the significant influence of the governance cycle duration on the selection of desertification combat modes for government and enterprise. A cooperative mode is best suited to a short governance cycle, while an ecological subsidy mode is optimal for a longer cycle. Enhancing governance technology and shortening the governance cycle are conducive to combating desertification. Reducing taxes alone may not be an effective control strategy; rather, the government can better motivate enterprises by adopting tax rate policies aligned with the chosen governance mode.

This research contributes by elucidating the impact mechanism of the government cycle’s length on the desertification combat process. The results may offer valuable insights for governments in formulating strategies to encourage corporate participation in combating desertification and provide theoretical support for selecting optimal desertification combat modes.

We present an approach for pricing American put options with a regime-switching volatility. Our method reveals that the option price can be expressed as the sum of two components: the price of a European put option and the premium associated with the early exercise privilege. Our analysis demonstrates that, under these conditions, the perpetual put option consistently commands a higher price during periods of high volatility compared to those of low volatility. Moreover, we establish that the optimal exercise boundary is lower in high-volatility regimes than in low-volatility regimes. Additionally, we develop an analytical framework to describe American puts with an Erlang-distributed random-time horizon, which allows us to propose a numerical technique for approximating the value of American puts with finite expiry. We also show that a combined approach involving randomization and Richardson extrapolation can be a robust numerical algorithm for estimating American put prices with finite expiry.

The purpose of this study is to investigate the bending analysis of metal (Ti-6Al-4V)-ceramic (ZrO₂) functionally graded material (FGM) sandwich plate with material property gradation along length and thickness direction under thermo-mechanical loading using inverse trigonometric shear deformation theory (ITSDT). FGM sandwich plate with a ceramic core and continuous variation of material properties has been modelled using Voigt’s micro-mechanical model following the power law distribution method. The impact of bi-directional gradation of material properties over the bending response of FGM plate under thermo-mechanical loading has been investigated in this work.

In this study, gradation of material properties for FGM plates is considered along length and thickness directions using Voigt’s micromechanical model following the power law distribution method. This type of FGM is called bi-directional FGMs (BDFGM). Mechanical and thermal properties of BDFGM sandwich plates are considered temperature-dependent in the present study. ITSDT is a non-polynomial shear deformation theory which requires a smaller number of field variables for modelling of displacement function in comparison to poly-nominal shear deformation theories which lead to a reduction in the complexity of the problem. In the present study, ITSDT has been utilized to obtain the governing equations for thermo-mechanical bending of simply supported uni-directional FGM (UDFGM) and BDFGM sandwich plates. Analytical solution for bending analysis of rectangular UDFGM and BDFGM sandwich plates has been carried out using Hamilton’s principle.

The bending response of the BDFGM sandwich plate under thermo-mechanical loading has been analysed and discussed. The present study shows that centre deflection, normal stress and shear stress are significantly influenced by temperature-dependent material properties, bi-directional gradation exponents along length and thickness directions, geometrical parameters, sandwich plate layer thickness, etc. The present investigation also reveals that bi-directional FGM sandwich plates can be designed to obtain thermo-mechanical bending response with an appropriate selection of gradation exponents along length and thickness direction. Non-dimensional centre deflection of BDFGM sandwich plates decreases with increasing gradation exponents in length and thickness directions. However, the non-dimensional centre deflection of BDFGM sandwich plates increases with increasing temperature differences.

For the first time, the FGM sandwich plate with the bi-directional gradation of material properties has been considered to investigate the bending response under thermo-mechanical loading. In the literature, various polynomial shear deformation theories like first-order shear deformation theory (FSDT), third-order shear deformation theory (TSDT) and higher-order shear deformation theory (HSDT) have been utilized to obtain the governing equation for bending response under thermo-mechanical loading; however, non-polynomial shear deformation theory like ITSDT has been used for the first time to obtain the governing equation to investigate the bending response of BDFGM. The impact of bi-directional gradation and temperature-dependent material properties over centre deflection, normal stress and shear stress has been analysed and discussed.

Fractional Fokker-Planck equation (FFPE) and time fractional coupled Boussinesq-Burger equations (TFCBBEs) play important roles in the fields of solute transport, fluid dynamics, respectively. Although there are many methods for solving the approximate solution, simple and effective methods are more preferred. This paper aims to utilize Laplace Adomian decomposition method (LADM) to construct approximate solutions for these two types of equations and gives some examples of numerical calculations, which can prove the validity of LADM by comparing the error between the calculated results and the exact solution.

This paper analyzes and investigates the time-space fractional partial differential equations based on the LADM method in the sense of Caputo fractional derivative, which is a combination of the Laplace transform and the Adomian decomposition method. LADM method was first proposed by Khuri in 2001. Many partial differential equations which can describe the physical phenomena are solved by applying LADM and it has been used extensively to solve approximate solutions of partial differential and fractional partial differential equations.

This paper obtained an approximate solution to the FFPE and TFCBBEs by using the LADM. A number of numerical examples and graphs are used to compare the errors between the results and the exact solutions. The results show that LADM is a simple and effective mathematical technique to construct the approximate solutions of nonlinear time-space fractional equations in this work.

This paper verifies the effectiveness of this method by using the LADM to solve the FFPE and TFCBBEs. In addition, these two equations are very meaningful, and this paper will be helpful in the study of atmospheric diffusion, shallow water waves and other areas. And this paper also generalizes the drift and diffusion terms of the FFPE equation to the general form, which provides a great convenience for our future studies.

In typical model-based calibration, linearization errors are derived inevitably, and non-negligible negative impact will be induced on the identification results if the rotational kinematic errors are not small enough or the lengths of links are too long, which is common in the industrial cases. Thus, an accurate two-step kinematic calibration method minimizing the linearization errors is presented for a six-DoF serial robot to improve the calibration accuracy.

The negative impact of linearization on identification accuracy is minimized by removing the responsible linearized kinematic errors from the complete kinematic error model. Accordingly, the identification results of the dimension-reduced new model are accurate but not complete, so the complete kinematic error model, which achieves high identification accuracy of the rest of the error parameters, is combined with this new model to create a two-step calibration procedure capable of highly accurate identification of all the kinematic errors.

The proportions of linearization errors in measured pose errors are quantified and found to be non-negligible with the increase of rotational kinematic errors. Thus, negative impacts of linearization errors are analyzed quantitatively in different cases, providing the basis for allowed kinematic errors in the new model. Much more accurate results were obtained by using the new two-step calibration method, according to a comparison with the typical methods.

This new method achieves high accuracy with no compromise on completeness, is easy to operate and is consistent with the typical method because the second step with the new model is conveniently combined without changing the sensors or measurement instrument setup.

A time-varying grey Fourier model (TVGFM(1,1,N)) is proposed for the simulation of variable amplitude seasonal fluctuation time series, as the performance of traditional grey models can't catch the time-varying trend well.

The proposed model couples Fourier series and linear time-varying terms as the grey action, to describe the characteristics of variable amplitude and seasonality. The truncated Fourier order N is preselected from the alternative order set by Nyquist-Shannon sampling theorem and the principle of simplicity, then the optimal Fourier order is determined by hold-out method to improve the robustness of the proposed model. Initial value correction and the multiple transformation are also studied to improve the precision.

The new model has a broader applicability range as a result of the new grey action, attaining higher fitting and forecasting accuracy. The numerical experiment of a generated monthly time series indicates the proposed model can accurately fit the variable amplitude seasonal sequence, in which the mean absolute percentage error (MAPE) is only 0.01%, and the complex simulations based on Monte-Carlo method testify the validity of the proposed model. The results of monthly electricity consumption in China's primary industry, demonstrate the proposed model catches the time-varying trend and has good performances, where MAPE_F and MAPE_T are below 5%. Moreover, the proposed TVGFM(1,1,N) model is superior to the benchmark models, grey polynomial model (GMP(1,1,N)), grey Fourier model (GFM(1,1,N)), seasonal grey model (SGM(1,1)), seasonal ARIMA model seasonal autoregressive integrated moving average model (SARIMA) and support vector regression (SVR).

The parameter estimates and forecasting of the new proposed TVGFM are studied, and the good fitting and forecasting accuracy of time-varying amplitude seasonal fluctuation series are testified by numerical simulations and a case study.

This study aims to investigate two newly developed (3 + 1)-dimensional Kairat-II and Kairat-X equations that illustrate relations with the differential geometry of curves and equivalence aspects.

The Painlevé analysis confirms the complete integrability of both Kairat-II and Kairat-X equations.

This study explores multiple soliton solutions for the two examined models. Moreover, the author showed that only Kairat-X give lump solutions and breather wave solutions.

The Hirota’s bilinear algorithm is used to furnish a variety of solitonic solutions with useful physical structures.

This study also furnishes a variety of numerous periodic solutions, kink solutions and singular solutions for Kairat-II equation. In addition, lump solutions and breather wave solutions were achieved from Kairat-X model.

The work formally furnishes algorithms for studying newly constructed systems that examine plasma physics, optical communications, oceans and seas and the differential geometry of curves, among others.

This paper presents an original work that presents two newly developed Painlev\'{e} integrable models with insightful findings.