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The current analysis produces the fractional sample of non-Newtonian Casson and Williamson boundary layer flow considering the heat flux and the slip velocity. An extended sheet with a nonuniform thickness causes the steady boundary layer flow’s temperature and velocity fields. Our purpose in this research is to use Akbari Ganji method (AGM) to solve equations and compare the accuracy of this method with the spectral collocation method.

The trial polynomials that will be utilized to carry out the AGM are then used to solve the nonlinear governing system of the PDEs, which has been transformed into a nonlinear collection of linked ODEs.

The profile of temperature and dimensionless velocity for different parameters were displayed graphically. Also, the effect of two different parameters simultaneously on the temperature is displayed in three dimensions. The results demonstrate that the skin-friction coefficient rises with growing magnetic numbers, whereas the Casson and the local Williamson parameters show reverse manners.

Moreover, the usefulness and precision of the presented approach are pleasing, as can be seen by comparing the results with previous research. Also, the calculated solutions utilizing the provided procedure were physically sufficient and precise.

This study aims to discuss the numerical solutions of weakly singular Volterra and Fredholm integral equations, which are used to model the problems like heat conduction in engineering and the electrostatic potential theory, using the modified Lagrange polynomial interpolation technique combined with the biconjugate gradient stabilized method (BiCGSTAB). The framework for the existence of the unique solutions of the integral equations is provided in the context of the Banach contraction principle and Bielecki norm.

The authors have applied the modified Lagrange polynomial method to approximate the numerical solutions of the second kind of weakly singular Volterra and Fredholm integral equations.

Approaching the interpolation of the unknown function using the aforementioned method generates an algebraic system of equations that is solved by an appropriate classical technique. Furthermore, some theorems concerning the convergence of the method and error estimation are proved. Some numerical examples are provided which attest to the application, effectiveness and reliability of the method. Compared to the Fredholm integral equations of weakly singular type, the current technique works better for the Volterra integral equations of weakly singular type. Furthermore, illustrative examples and comparisons are provided to show the approach’s validity and practicality, which demonstrates that the present method works well in contrast to the referenced method. The computations were performed by MATLAB software.

The convergence of these methods is dependent on the smoothness of the solution, it is challenging to find the solution and approximate it computationally in various applications modelled by integral equations of non-smooth kernels. Traditional analytical techniques, such as projection methods, do not work well in these cases since the produced linear system is unconditioned and hard to address. Also, proving the convergence and estimating error might be difficult. They are frequently also expensive to implement.

There is a great need for fast, user-friendly numerical techniques for these types of equations. In addition, polynomials are the most frequently used mathematical tools because of their ease of expression, quick computation on modern computers and simple to define. As a result, they made substantial contributions for many years to the theories and analysis like approximation and numerical, respectively.

This work presents a useful method for handling weakly singular integral equations without involving any process of change of variables to eliminate the singularity of the solution.

To the best of the authors’ knowledge, the authors claim the originality and effectiveness of their work, highlighting its successful application in addressing weakly singular Volterra and Fredholm integral equations for the first time. Importantly, the approach acknowledges and preserves the possible singularity of the solution, a novel aspect yet to be explored by researchers in the field.

This paper intends to construct a new adaptive grey seasonal model (AGSM) to promote the application of the grey forecasting model in quarterly GDP.

Firstly, this paper constructs a new accumulation operation that embodies the new information priority by using a hyperparameter. Then, a new AGSM is constructed by using a new grey action quantity, nonlinear Bernoulli operator, discretization operation, moving average trend elimination method and the proposed new accumulation operation. Subsequently, the marine predators algorithm is used to quickly obtain the hyperparameters used to build the AGSM. Finally, comparative analysis experiments and ablation experiments based on China's quarterly GDP confirm the validity of the proposed model.

AGSM can be degraded to some classical grey prediction models by replacing its own structural parameters. The proposed accumulation operation satisfies the new information priority rule. In the comparative analysis experiments, AGSM shows better prediction performance than other competitive algorithms, and the proposed accumulation operation is also better than the existing accumulation operations. Ablation experiments show that each component in the AGSM is effective in enhancing the predictive performance of the model.

A new AGSM with new information priority accumulation operation is proposed.

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.

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.

The purpose of this study is to solve two unique but difficult partial differential equations: the foam drainage equation and the nonlinear time-fractional fisher’s equation. Through our methods, we aim to provide accurate solutions and gain a deeper understanding of the intricate behaviors exhibited by these systems.

In this study, we use a dual technique that combines the Aboodh residual power series method and the Aboodh transform iteration method, both of which are combined with the Caputo operator.

We develop exact and efficient solutions by merging these unique methodologies. Our results, presented through illustrative figures and data, demonstrate the efficacy and versatility of the Aboodh methods in tackling such complex mathematical models.

Owing to their fractional derivatives and nonlinear behavior, these equations are crucial in modeling complex processes and confront analytical complications in various scientific and engineering contexts.

This study aims to propose and numerically assess different ways of discretising a very weak formulation of the Poisson problem.

We use integration by parts twice to shift smoothness requirements to the test functions, thereby allowing low-regularity data and solutions.

Various conforming discretisations are presented and tested, with numerical results indicating good accuracy and stability in different types of problems.

This is one of the first articles to propose and test concrete discretisations for very weak variational formulations in primal form. The numerical results, which include a problem based on real MRI data, indicate the potential of very weak finite element methods for tackling problems with low regularity.

Robotic arms’ interactions with the external environment are growing more intricate, demanding higher control precision. This study aims to enhance control precision by establishing a dynamic model through the identification of the dynamic parameters of a self-designed robotic arm.

This study proposes an improved particle swarm optimization (IPSO) method for parameter identification, which comprehensively improves particle initialization diversity, dynamic adjustment of inertia weight, dynamic adjustment of local and global learning factors and global search capabilities. To reduce the number of particles and improve identification accuracy, a step-by-step dynamic parameter identification method was also proposed. Simultaneously, to fully unleash the dynamic characteristics of a robotic arm, and satisfy boundary conditions, a combination of high-order differentiable natural exponential functions and traditional Fourier series is used to develop an excitation trajectory. Finally, an arbitrary verification trajectory was planned using the IPSO to verify the accuracy of the dynamical parameter identification.

Experiments conducted on a self-designed robotic arm validate the proposed parameter identification method. By comparing it with IPSO1, IPSO2, IPSOd and least-square algorithms using the criteria of torque error and root mean square for each joint, the superiority of the IPSO algorithm in parameter identification becomes evident. In this case, the dynamic parameter results of each link are significantly improved.

A new parameter identification model was proposed and validated. Based on the experimental results, the stability of the identification results was improved, providing more accurate parameter identification for further applications.

Accurate prediction of seasonal power consumption trends with impact disturbances provides a scientific basis for the flexible balance of the long timescale power system. Consequently, it fosters reasonable scheduling plans, ensuring the safety of the system and improving the economic dispatching efficiency of the power system.

First, a new seasonal grey buffer operator in the longitudinal and transverse dimensional perspectives is designed. Then, a new seasonal grey modeling approach that integrates the new operator, full real domain fractional order accumulation generation technique, grey prediction modeling tool and fruit fly optimization algorithm is proposed. Moreover, the rationality, scientificity and superiority of the new approach are verified by designing 24 seasonal electricity consumption forecasting approaches, incorporating case study and amalgamating qualitative and quantitative research.

Compared with other comparative models, the new approach has superior mean absolute percentage error and mean absolute error. Furthermore, the research results show that the new method provides a scientific and effective mathematical method for solving the seasonal trend power consumption forecasting modeling with impact disturbance.

Considering the development trend of longitudinal and transverse dimensions of seasonal data with impact disturbance and the differences in each stage, a new grey buffer operator is constructed, and a new seasonal grey modeling approach with multi-method fusion is proposed to solve the seasonal power consumption forecasting problem.

The highlights of the paper are as follows:

1. A new seasonal grey buffer operator is constructed.

2. The impact of shock perturbations on seasonal data trends is effectively mitigated.

3. A novel seasonal grey forecasting approach with multi-method fusion is proposed.

4. Seasonal electricity consumption is successfully predicted by the novel approach.

5. The way to adjust China's power system flexibility in the future is analyzed.

A new seasonal grey buffer operator is constructed.

The impact of shock perturbations on seasonal data trends is effectively mitigated.

A novel seasonal grey forecasting approach with multi-method fusion is proposed.

Seasonal electricity consumption is successfully predicted by the novel approach.

The way to adjust China's power system flexibility in the future is analyzed.

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.