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The purpose of the present work is to introduce a wavelet method for the solution of linear and nonlinear psi-Caputo fractional initial and boundary value problem.

The authors have introduced the new generalized operational matrices for the psi-CAS (Cosine and Sine) wavelets, and these matrices are successfully utilized for the solution of linear and nonlinear psi-Caputo fractional initial and boundary value problem. For the nonlinear problems, the authors merge the present method with the quasilinearization technique.

The authors have drived the orthogonality condition for the psi-CAS wavelets. The authors have derived and constructed the psi-CAS wavelets matrix, psi-CAS wavelets operational matrix of psi-fractional order integral and psi-CAS wavelets operational matrix of psi-fractional order integration for psi-fractional boundary value problem. These matrices are successfully utilized for the solutions of psi-Caputo fractional differential equations. The purpose of these operational matrices is to make the calculations faster. Furthermore, the authors have derived the convergence analysis of the method. The procedure of implementation for the proposed method is also given. For the accuracy and applicability of the method, the authors implemented the method on some linear and nonlinear psi-Caputo fractional initial and boundary value problems and compare the obtained results with exact solutions.

Since psi-Caputo fractional differential equation is a new and emerging field, many engineers can utilize the present technique for the numerical simulations of their linear/non-linear psi-Caputo fractional differential models. To the best of the authors’ knowledge, the present work has never been introduced and implemented for psi-Caputo fractional differential equations.

This paper aims to analyse the persistence of the S&P500 and DAX 30 stock indices as well as of the Fed’s Effective Federal Funds rate and of the European Central Bank’s Marginal Lending Facility rate, and the long-run linkages between stock prices and interest rates in the USA and Europe, respectively.

The methodology is based on the concepts of fractional integration and cointegration.

Using monthly data from January 1999 to December 2022, the results can be summarised as follows. All series examined are non-stationary: stock prices are found to be I(1) while interest rates display orders of integration substantially above 1, which implies a rejection of the hypothesis of mean reversion in all cases examined.

This paper uses an appropriate econometric framework to obtain new, reliable empirical evidence. All four series are highly persistent, and mean reversion does not occur in any single case. Moreover, the fractional cointegration analysis suggests that stock prices and interest rates are not linked in the long run.

This paper aims to model the relationship between oil price and stock returns for selected sectors in Nigeria using monthly data from January 2007 to December 2016.

The authors use both the linear (symmetric) autoregressive distributed lag (ARDL) by Pesaran et al. (2001) and non-linear (asymmetric) ARDL by Shin et al. (2014), and they also account for structural breaks using the Bai and Perron (2003) test that allows for multiple structural changes in regression models.

The results indicate that the strength of this relationship varies across sectors, albeit asymmetric and breaks. The authors identify two structural breaks that occur in 2008 and 2010/2011, which coincidentally correspond to the global financial crisis and the Arab spring (Libyan shutdowns), respectively. Moreover, the authors observe strong support for asymmetry and structural breaks for some sectors in the reaction of sector returns to movement in oil prices. These findings are robust and insensitive when considering different oil proxies. While further extensions can be pursued, the consideration of asymmetric effects as well as structural breaks should not be jettisoned when modelling this nexus.

This study is one of the very few studies that have investigated the sectoral behaviour of stocks to oil price shocks, particularly in Nigeria. This paper contributes to the oil stock literature using the recent technique of asymmetry and also considering the role structural breaks play in the relationship between oil price and stock returns.

With the increasing demands of industrial applications, it is imperative for robots to accomplish good contact-interaction with dynamic environments. Hence, the purpose of this research is to propose an adaptive fractional-order admittance control scheme to realize a robot–environment contact with high accuracy, small overshoot and fast response.

Fractional calculus is introduced to reconstruct the classical admittance model in this control scheme, which can more accurately describe the complex physical relationship between position and force in the interaction process of the robot–environment. In this control scheme, the pre-PID controller and fuzzy controller are adopted to improve the system force tracking performance in highly dynamic unknown environments, and the fuzzy controller is used to improve the trajectory, transient and steady-state response by adjusting the pre-PID integration gain online. Furthermore, the stability and robustness of this control algorithm are theoretically and experimentally demonstrated.

The excellent force tracking performance of the proposed control algorithm is verified by constructing highly dynamic unstructured environments through simulations and experiments. In simulations and experiments, the proposed control algorithm shows satisfactory force tracking performance with the advantages of fast response speed, little overshoot and strong robustness.

The control scheme is practical and simple in the actual industrial and medical scenarios, which requires accurate force control by the robot.

A new fractional-order admittance controller is proposed and verified by experiments in this research, which achieves excellent force tracking performance in dynamic unknown environments.

The discrete Fourier transform (dft) of a fractional process is studied. An exact representation of the dft is given in terms of the component data, leading to the frequency domain form of the model for a fractional process. This representation is particularly useful in analyzing the asymptotic behavior of the dft and periodogram in the nonstationary case when the memory parameter d≥12. Various asymptotic approximations are established including some new hypergeometric function representations that are of independent interest. It is shown that smoothed periodogram spectral estimates remain consistent for frequencies away from the origin in the nonstationary case provided the memory parameter d < 1. When d = 1, the spectral estimates are inconsistent and converge weakly to random variates. Applications of the theory to log periodogram regression and local Whittle estimation of the memory parameter are discussed and some modified versions of these procedures are suggested for nonstationary cases.

Investigation of the smoking model is important as it has a direct effect on human health. This paper focuses on the numerical analysis of the fractional order giving up smoking model. Nonetheless, due to observational or experimental errors, or any other circumstance, it may contain some incomplete information. Fuzzy sets can be used to deal with uncertainty. Yet, there may be some inconsistency in the membership as well. As a result, the primary goal of this proposed work is to numerically solve the model in a type-2 fuzzy environment.

Triangular perfect quasi type-2 fuzzy numbers (TPQT2FNs) are used to deal with the uncertainty in the model. In this work, concepts of r2-cut at r1-plane are used to model the problem's uncertain parameter. The Legendre wavelet method (LWM) is then utilised to solve the giving up smoking model in a type-2 fuzzy environment.

LWM has been effectively employed in conjunction with the r2-cut at r1-plane notion of type-2 fuzzy sets to solve the model. The LWM has the advantage of converting the non-linear fractional order model into a set of non-linear algebraic equations. LWM scheme solutions are found to be well agreed with RK4 scheme solutions. The existence and uniqueness of the model's solution have also been demonstrated.

To deal with the uncertainty, type-2 fuzzy numbers are used. The use of LWM in a type-2 fuzzy uncertain environment to achieve the model's required solutions is quite fascinating, and this is the key focus of this work.

The purpose of this paper is to examine the degree of persistence in four precious metal prices (i.e. gold, palladium, platinum and silver) during the last four US recessions.

Using daily price data for gold, palladium, platinum and silver running from July 2, 1990, to March 21, 2022, and dating of business cycles in the USA provided by NBER (2022), the paper uses fractional integration to test the degree of persistence of precious metal prices.

The empirical analysis shows the unrelenting prominence of gold in relation to other precious metals (palladium, platinum and silver) as a hedge against market uncertainty in the post-pandemic new era.

Two are the main contributions of the paper. Firstly, the authors contribute to the commodity markets and finance literature on precious metal price modelling. Secondly, the authors also contribute to the literature on commodity markets and business cycles with a special focus on recessionary periods.

In particular, in this article, the authors investigate the degree of persistence in the credit-to-gross domestic product (GDP) ratio in 44 Organisation for Economic Co-operation and Development (OECD) economies in the context of nonlinear deterministic trends.

The authors use Chebyshev's polynomials in time, which allow us to model changes in the data in a smoother way than by structural breaks.

This study’s results indicate that approximately one-quarter of the series display non-linear structures, and only Argentina displays a mean reverting pattern.

Policy implications of the results obtained are discussed at the end of the manuscript.

The authors use an approach developed that allows for non-linear trends based on Chebyshev polynomials in time, with the residuals being fractionally integrated or integrated of order d, where d can be any real value.

The purpose of this paper is to originally present the generic analytical models of memelement and inverse memelement with time-dependent memory effect.

The variable order forward Grünwald–Letnikov fractional derivative and the memristor and inverse memristor models proposed by Fouda et al. have been adopted as the basis. Both analytical and numerical studies have been conducted. The applications to the candidate practical memristor and inverse memelements have also been presented.

The generic analytical models of memelement and inverse memelement with time-dependent memory effect, the simplified ones for DC and AC signal-based analyses and the equations of crucial parameters have been derived. Besides the well-known opposite relationships with frequency, the Lissajous patterns of memelement and inverse memelement also use the opposite relationships with the time. The proposed models can be well applied to the practical elements.

To the best of the authors’ knowledge, for the first time, the models’ memelement and inverse memelement with time-dependent memory effect have been presented. A new contrast between these elements has been discovered. The resulting models are applicable to the practical elements.

The fractional order HIV model has an important role in biological science. To study the HIV model in a better way, the model is presented with the help of Atangana- Baleanu operator which is in Caputo sense. Also, the characteristics of the solutions are described briefly with the help of the advance numerical techniques for the different values of fractional order derivatives. This paper aims to discuss the aforementioned objectives.

In this work, Adams-Bashforth method and Euler method are used to get the solution of the HIV model. These are the important numerical methods. The comparison results also are described with the physical meaning of the solutions of the model.

HIV model is analyzed under the view of fractional and AB derivative in Atangana-Baleanu-Caputo sense. The uniqueness of the solution is proved by using Banach Fixed point. The solution is derived with the help of Sumudu transform. Further, the authors employed fractional Adam-Bashforth method and Euler method to enumerate numerical results. The authors have used several values of fractional orders to present the outcomes graphically. The above calculations have been done with the help of MATLAB (R2016a). The numerical scheme used in the proposed study is valid and fruitful, and the same can be used to explore other real issues.

This investigation can be done for the real data sets.

This paper aims to express the solution of the HIV model in a better way with the effect of non-locality, this work is very useful.

In this work, HIV model is developed with the help of Atangana- Baleanu operator in Caputo sense. By using Banach Fixed point, the authors proved that the solution is unique. Also, the solution is presented with the help of Sumudu transform. The behaviors of the solutions are checked for different values of fractional order derivatives with the physical meaning with help of the Adam-Bashforth method and the Euler method.