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This study explores the immobilisation of enzymes within porous catalysts of various geometries, including spheres, cylinders and flat pellets. The objective is to understand the irreversible Michaelis-Menten kinetic process within immobilised enzymes through advanced mathematical modelling.

Mathematical models were developed based on reaction-diffusion equations incorporating nonlinear variables associated with Michaelis-Menten kinetics. This research introduces fractional derivatives to investigate enzyme reaction kinetics, addressing a significant gap in the existing literature. A novel approximation method, based on the independent polynomials of the complete bipartite graph, is employed to explore solutions for substrate concentration and effectiveness factor across a spectrum of parameter values. The analytical solutions generated through the bipartite polynomial approximation method (BPAM) are rigorously tested against established methods, including the Bernoulli wavelet method (BWM), Taylor series method (TSM), Adomian decomposition method (ADM) and fourth-order Runge-Kutta method (RKM).

The study identifies two main findings. Firstly, the behaviour of dimensionless substrate concentration with distance is analysed for planar, cylindrical and spherical catalysts using both integer and fractional order Michaelis-Menten modelling. Secondly, the research investigates the variability of the dimensionless effectiveness factor with the Thiele modulus.

The study primarily focuses on mathematical modelling and theoretical analysis, with limited experimental validation. Future research should involve more extensive experimental verification to corroborate the findings. Additionally, the study assumes ideal conditions and uniform catalyst properties, which may not fully reflect real-world complexities. Incorporating factors such as mass transfer limitations, non-uniform catalyst structures and enzyme deactivation kinetics could enhance the model’s accuracy and broaden its applicability. Furthermore, extending the analysis to include multi-enzyme systems and complex reaction networks would provide a more comprehensive understanding of biocatalytic processes.

The validated bipartite polynomial approximation method presents a practical tool for optimizing enzyme reactor design and operation in industrial settings. By accurately predicting substrate concentration and effectiveness factor, this approach enables efficient utilization of immobilised enzymes within porous catalysts. Implementation of these findings can lead to enhanced process efficiency, reduced operating costs and improved product yields in various biocatalytic applications such as pharmaceuticals, food processing and biofuel production. Additionally, this research fosters innovation in enzyme immobilisation techniques, offering practical insights for engineers and researchers striving to develop sustainable and economically viable bioprocesses.

The advancement of enzyme immobilisation techniques holds promise for addressing societal challenges such as sustainable production, environmental protection and healthcare. By enabling more efficient biocatalytic processes, this research contributes to reducing industrial waste, minimizing energy consumption and enhancing access to pharmaceuticals and bio-based products. Moreover, the development of eco-friendly manufacturing practices through biocatalysis aligns with global efforts towards sustainability and mitigating climate change. The widespread adoption of these technologies can foster a more environmentally conscious society while stimulating economic growth and innovation in biotechnology and related industries.

This study offers a pioneering approximation method using the independent polynomials of the complete bipartite graph to investigate enzyme reaction kinetics. The comprehensive validation of this method through comparison with established solution techniques ensures its reliability and accuracy. The findings hold promise for advancing the field of biocatalysts and provide valuable insights for designing efficient enzyme reactors.

The objective of this work is to analyze the necessary conditions for chaotic behavior with fractional order and fractal dimension values of the fractal-fractional operator.

The numerical technique based on the fractal-fractional derivative is implemented over the fractional model and analyzes the condition at the distinct values of fractional order and fractal dimension.

The obtained numerical solution from the numerical technique is analyzed at distinct fractional order and fractal dimension values, and it has been figured out that the behavior of the solution either chaotic or non-chaotic agrees with the condition.

The necessary condition is associated with the fractional order only. So, our work not only studies the condition with fractional order but also examines the model by simultaneously adjusting fractal dimension values. It is found that the model still has chaotic or non-chaotic behavior at certain fractal dimension values and fractional order values corresponding to the condition.

The main aim of this paper is to investigate the fractional coupled nonlinear Helmholtz equation by two new analytical methods.

This article takes an inaugural look at the fractional coupled nonlinear Helmholtz equation by using the conformable derivative. It successfully finds new fractional periodic solutions and solitary wave solutions by employing methods such as the fractional method and the fractional simple equation method. The dynamics of these fractional periodic solutions and solitary wave solutions are then graphically represented in 3D with appropriate parameters and fractal dimensions. This research contributes to a deeper comprehension and detailed exploration of the dynamics involved in high dimensional solitary wave propagation.

The proposed two mathematical approaches are simple and efficient to solve fractional evolution equations.

The fractional coupled nonlinear Helmholtz equation is described by using the conformable derivative for the first time. The obtained fractional periodic solutions and solitary wave solutions are completely new.

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.

This study aims to treat a novel system of Volterra integro-differential equations with multiple delays and variable bounds, constituting a generic numerical method based on the matrix equation and a combinatoric-parametric Charlier polynomials. The proposed method utilizes these polynomials for the matrix relations at the collocation points.

Thanks to the combinatorial eligibility of the method, the functional terms can be transformed into the generic matrix relations with low dimensions, and their resulting matrix equation. The obtained solutions are tested with regard to the parametric behaviour of the polynomials with $\alpha$, taking into account the condition number of an outcome matrix of the method. Residual error estimation improves those solutions without using any external method. A calculation of the residual error bound is also fulfilled.

All computations are carried out by a special programming module. The accuracy and productivity of the method are scrutinized via numerical and graphical results. Based on the discussions, one can point out that the method is very proper to solve a system in question.

This paper introduces a generic computational numerical method containing the matrix expansions of the combinatoric Charlier polynomials, in order to treat the system of Volterra integro-differential equations with multiple delays and variable bounds. Thus, the method enables to evaluate stiff differential and integral parts of the system in question. That is, these parts generates two novel components in terms of unknown terms with both differentiated and delay arguments. A rigorous error analysis is deployed via the residual function. Four benchmark problems are solved and interpreted. Their graphical and numerical results validate accuracy and efficiency of the proposed method. In fact, a generic method is, thereby, provided into the literature.

Understanding the mechanical and thermal behavior of materials is the goal of the branch of study known as fractional thermoelasticity, which blends fractional calculus with thermoelasticity. It accounts for the fact that heat transfer and deformation are non-local processes that depend on long-term memory. The sphere is free of external stresses and rotates around one of its radial axes at a constant rate. The coupled system equations are solved using the Laplace transform. The outcomes showed that the viscoelastic deformation and thermal stresses increased with the value of the fractional order coefficients.

The results obtained are considered good because they indicate that the approach or model under examination shows robust performance and produces accurate or reliable results that are consistent with the corresponding literature.

This study introduces a proposed viscoelastic photoelastic heat transfer model based on the Moore-Gibson-Thompson framework, accompanied by the incorporation of a new fractional derivative operator. In deriving this model, the recently proposed Caputo proportional fractional derivative was considered. This work also sheds light on how thermoelastic materials transfer light energy and how plasmas interact with viscoelasticity. The derived model was used to consider the behavior of a solid semiconductor sphere immersed in a magnetic field and subjected to a sudden change in temperature.

This study introduces a proposed viscoelastic photoelastic heat transfer model based on the Moore-Gibson-Thompson framework, accompanied by the incorporation of a new fractional derivative operator. In deriving this model, the recently proposed Caputo proportional fractional derivative was considered. This work also sheds light on how thermoelastic materials transfer light energy and how plasmas interact with viscoelasticity. The derived model was used to consider the behavior of a solid semiconductor sphere immersed in a magnetic field and subjected to a sudden change in temperature.

This paper aims to study qualitative properties and approximate solutions of a thermostat dynamics system with three-point boundary value conditions involving a nonsingular kernel operator which is called Atangana-Baleanu-Caputo (ABC) derivative for the first time. The results of the existence and uniqueness of the solution for such a system are investigated with minimum hypotheses by employing Banach and Schauder's fixed point theorems. Furthermore, Ulam-Hyers (UH) stability, Ulam-Hyers-Rassias UHR stability and their generalizations are discussed by using some topics concerning the nonlinear functional analysis. An efficiency of Adomian decomposition method (ADM) is established in order to estimate approximate solutions of our problem and convergence theorem is proved. Finally, four examples are exhibited to illustrate the validity of the theoretical and numerical results.

This paper considered theoretical and numerical methodologies.

This paper contains the following findings: (1) Thermostat fractional dynamics system is studied under ABC operator. (2) Qualitative properties such as existence, uniqueness and Ulam–Hyers–Rassias stability are established by fixed point theorems and nonlinear analysis topics. (3) Approximate solution of the problem is investigated by Adomain decomposition method. (4) Convergence analysis of ADM is proved. (5) Examples are provided to illustrate theoretical and numerical results. (6) Numerical results are compared with exact solution in tables and figures.

The novelty and contributions of this paper is to use a nonsingular kernel operator for the first time in order to study the qualitative properties and approximate solution of a thermostat dynamics system.

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.

To achieve sustainable development in shipping, accurately identifying the impact of artificial intelligence on shipping carbon emissions and predicting these emissions is of utmost importance.

A multivariable discrete grey prediction model (WFTDGM) based on weakening buffering operator is established. Furthermore, the optimal nonlinear parameters are determined by Grey Wolf optimization algorithm to improve the prediction performance, enhancing the model’s predictive performance. Subsequently, global data on artificial intelligence and shipping carbon emissions are employed to validate the effectiveness of our new model and chosen algorithm.

To demonstrate the applicability and robustness of the new model in predicting marine shipping carbon emissions, the new model is used to forecast global marine shipping carbon emissions. Additionally, a comparative analysis is conducted with five other models. The empirical findings indicate that the WFTDGM (1, N) model outperforms other comparative models in overall efficacy, with MAPE for both the training and test sets being less than 4%, specifically at 0.299% and 3.489% respectively. Furthermore, the out-of-sample forecasting results suggest an upward trajectory in global shipping carbon emissions over the subsequent four years. Currently, the application of artificial intelligence in mitigating shipping-related carbon emissions has not achieved the desired inhibitory impact.

This research not only deepens understanding of the mechanisms through which artificial intelligence influences shipping carbon emissions but also provides a scientific basis for developing effective emission reduction strategies in the shipping industry, thereby contributing significantly to green shipping and global carbon reduction efforts.

The multi-variable discrete grey prediction model developed in this paper effectively mitigates abnormal fluctuations in time series, serving as a valuable reference for promoting global green and low-carbon transitions and sustainable economic development. Furthermore, based on the findings of this paper, a grey prediction model with even higher predictive performance can be constructed by integrating it with other algorithms.

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.