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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.

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.

The primary objective of this study is to investigate whether the inclusion of convertible bond prices as important inputs into artificial neural networks can lead to improved accuracy in predicting Chinese stock prices. This novel approach aims to uncover the latent potential inherent in convertible bond dynamics, ultimately resulting in enhanced precision when forecasting stock prices.

The authors employed two machine learning models, namely the backpropagation neural network (BPNN) model and the extreme learning machine neural networks (ELMNN) model, on empirical Chinese financial time series data.

The results showed that the convertible bond price had a strong predictive power for low-market-value stocks but not for high-market-value stocks. The BPNN algorithm performed better than the ELMNN algorithm in predicting stock prices using the convertible bond price as an input indicator for low-market-value stocks. In contrast, ELMNN showed a significant decrease in prediction accuracy when the convertible bond price was added.

This study represents the initial endeavor to integrate convertible bond data into both the BPNN model and the ELMNN model for the purpose of predicting Chinese stock prices.

The purpose of this paper is to study large deviations for the solution processes of a stochastic equation incorporated with the effects of nonlocal condition.

A weak convergence approach is adopted to establish the Laplace principle, which is same as the large deviation principle in a Polish space. The sufficient condition for any family of solutions to satisfy the Laplace principle formulated by Budhiraja and Dupuis is used in this work.

Freidlin–Wentzell type large deviation principle holds good for the solution processes of the stochastic functional integral equation with nonlocal condition.

The asymptotic exponential decay rate of the solution processes of the considered equation towards its deterministic counterpart can be estimated using the established results.

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.

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.

The numerical methods are of great importance for approximating the solutions of a system of nonlinear singular ordinary differential equations. In this paper, the authors present the biorthogonal flatlet multiwavelet collocation method (BFMCM) as a numerical scheme for a class of system of Lane–Emden equations with initial or boundary or four-point boundary conditions.

The approach is involved in combining the biorthogonal flatlet multiwavelet (BFM) with the collocation method. The authors investigate the properties and procedure of the BFMCM for first time on this class of equations. By using the BFM and the collocation points, the method is constructed and it transforms the nonlinear differential equations problem into a system of nonlinear algebraic equations. The unknown coefficients of the assuming solution are determined by solving the obtained system. Additionally, convergence analysis and numerical stability of the suggested method are provided.

According to the attained results, the proposed BFMCM has more accurate results in comparison with results of other methods. The maximum absolute errors are calculated by using the BFMCM for comparison purposes provided.

The key desirable properties of BFMCM are its efficiency, simple applicability and minimizes errors. Therefore, the proposed method can be used to solve nonlinear problems or problems with singular points.

Present study aims to extend the lattice Boltzmann method (LBM) to simulate radiation in geometries with curved boundaries, as the first step to simulate radiation in complex porous media. In recent years, researchers have increasingly explored the use of porous media to improve the heat transfer processes. The lattice Boltzmann method (LBM) is one of the most effective techniques for simulating heat transfer in such media. However, the application of the LBM to study radiation in complex geometries that contain curved boundaries, as found in many porous media, has been limited.

The numerical evaluation of the effect of the radiation-conduction parameter and extinction coefficient on temperature and incident radiation distributions demonstrates that the proposed LBM algorithm provides highly accurate results across all cases, compared to those found in the literature or those obtained using the finite volume method (FVM) with the discrete ordinates method (DOM) for radiative information.

For the case with a conduction-radiation parameter equal to 0.01, the maximum relative error is 1.9% in predicting temperature along vertical central line. The accuracy improves with an increase in the conduction-radiation parameter. Furthermore, the comparison between computational performances of two approaches reveals that the LBM-LBM approach performs significantly faster than the FVM-DOM solver.

The difficulty of radiative modeling in combined problems involving irregular boundaries has led to alternative approaches that generally increase the computational expense to obtain necessary radiative details. To address the limitations of existing methods, this study presents a new approach involving a coupled lattice Boltzmann and first-order blocked-off technique to efficiently model conductive-radiative heat transfer in complex geometries with participating media. This algorithm has been developed using the parallel lattice Boltzmann solver.

This paper aims to develop a novel grey Bernoulli model with memory characteristics, which is designed to dynamically choose the optimal memory kernel function and the length of memory dependence period, ultimately enhancing the model's predictive accuracy.

This paper enhances the traditional grey Bernoulli model by introducing memory-dependent derivatives, resulting in a novel memory-dependent derivative grey model. Additionally, fractional-order accumulation is employed for preprocessing the original data. The length of the memory dependence period for memory-dependent derivatives is determined through grey correlation analysis. Furthermore, the whale optimization algorithm is utilized to optimize the cumulative order, power index and memory kernel function index of the model, enabling adaptability to diverse scenarios.

The selection of appropriate memory kernel functions and memory dependency lengths will improve model prediction performance. The model can adaptively select the memory kernel function and memory dependence length, and the performance of the model is better than other comparison models.

The model presented in this article has some limitations. The grey model is itself suitable for small sample data, and memory-dependent derivatives mainly consider the memory effect on a fixed length. Therefore, this model is mainly applicable to data prediction with short-term memory effect and has certain limitations on time series of long-term memory.

In practical systems, memory effects typically exhibit a decaying pattern, which is effectively characterized by the memory kernel function. The model in this study skillfully determines the appropriate kernel functions and memory dependency lengths to capture these memory effects, enhancing its alignment with real-world scenarios.

Based on the memory-dependent derivative method, a memory-dependent derivative grey Bernoulli model that more accurately reflects the actual memory effect is constructed and applied to power generation forecasting in China, South Korea and India.

This study aims to analyze the Prandtl fluid flow in the presence of better mass diffusion and heat conduction models. By taking into account a linearly bidirectional stretchable sheet, flow is produced. Heat generation effect, thermal radiation, variable thermal conductivity, variable diffusion coefficient and Cattaneo–Christov double diffusion models are used to evaluate thermal and concentration diffusions.

The governing partial differential equations (PDEs) have been made simpler using a boundary layer method. Strong nonlinear ordinary differential equations (ODEs) relate to appropriate non-dimensional similarity variables. The optimal homotopy analysis technique is used to develop solution.

Graphs analyze the impact of many relevant factors on temperature and concentration. The physical parameters, such as mass and heat transfer rates at the wall and surface drag coefficients, are also displayed and explained.

The reported work discusses the contribution of generalized flux models to note their impact on heat and mass transport.