Bernoulli Wavelets Research Articles

Beyond conventional models: integer and fractional order analysis of nonlinear Michaelis-Menten kinetics in immobilised enzyme reactors

PurposeThis 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.Design/methodology/approachMathematical 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).FindingsThe 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.Research limitations/implicationsThe 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.Practical implicationsThe 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.Social implicationsThe 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.Originality/valueThis 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.

Numerical and theoretical analysis of the parabolic partial differential equation through the Bernoulli wavelet collocation scheme

Numerical and theoretical analysis of the parabolic partial differential equation through the Bernoulli wavelet collocation scheme

A numerical approach for solving nonlinear fractional Klein–Gordon equation with applications in quantum mechanics

Abstract In this paper, we propose a numerical approach for solving the nonlinear fractional Klein–Gordon equation (FKGE), a model of significant importance in simulating nonlinear waves in quantum mechanics. Our method combines the Bernoulli wavelet collocation scheme with a functional integration matrix to obtain approximate solutions for the proposed model. Initially, we transform the main problem into a system of algebraic equations, which we solve using the Newton–Raphson method to extract the unknown coefficients and achieve the desired approximate solution. To theoretically validate our method, we conduct a comprehensive convergence analysis, demonstrating its uniform convergence. We perform numerical experiments on various examples with different parameters, presenting the results through tables and figures. Our findings indicate that employing more terms in the utilized techniques enhances accuracy. Furthermore, we compare our approach with existing methods from the literature, showcasing its performance in terms of computational cost, convergence rate, and solution accuracy. These examples illustrate how our techniques yield better approximate solutions for the nonlinear model at a low computational cost, as evidenced by the calculated CPU time and absolute error. Additionally, our method consistently provides better accuracy than other methods from the literature, suggesting its potential for solving more complex problems in physics and other scientific disciplines.

Investigation of squeezing Cu–water Casson nanofluid flow with an internal heat source through the Bernoulli wavelet method

The present numerical Bernoulli wavelet technique aims at investigating the heat transfer of Casson nanofluid flow squeezed between parallel plates with the influence of an internal heat source. The fundamental governing equations describing the proposed flow problem is expressed in terms of highly nonlinear coupled ordinary differential equations via appropriate similarity transformations. The numerical Bernoulli wavelet method (BWM) is used to solve these nonlinear equations. Numerical outcomes of the BWM are compared with the different numerical results. Comparison between calculated results showed that the BWM is more suitable and correct than the any numerical method. The effects of each parameters, such as the Casson parameter, internal heat generation/absorption parameter, squeezing number, and nanoparticle volume fraction on the velocity and temperature are presented graphically and discussed. Overall, this research provides novel insights into the heat transfer characteristics of Casson nanofluids in squeezing flows with internal heating, which are highly relevant to several industrial processes and biomedical applications.

Heat and mass transfer on MHD squeezing flow through the porous media using the Bernoulli wavelet method

Heat and mass transfer on MHD squeezing flow through the porous media using the Bernoulli wavelet method

Bernstein polynomials method for solving multi-order fractional neutral pantograph equations with error and stability analysis

In this investigation, we present a new method for addressing fractional neutral pantograph problems, utilizing the Bernstein polynomials method. We obtain solutions for the fractional pantograph equations by employing operational matrices of differentiation, derived from fractional derivatives in the Caputo sense applied to Bernstein polynomials. Error analysis, along with Chebyshev algorithms and interpolation nodes, is employed for solution characterization. Both theoretical and practical stability analyses of the method are provided. Demonstrative examples indicate that our proposed techniques occasionally yield exact solutions. We compare the algorithms using several established analytical methods. Our results reveal that our algorithm, based on Bernstein series solution methods, outperforms others, exhibiting superior performance with higher accuracy orders compared to those obtained from Chebyshev spectral methods, Bernoulli wavelet method, and Spectral Tau method.

Dufour and Soret effects on double diffusive Casson fluid flow with the influence of internal heat source

AbstractThis study focuses on the control of the cross-diffusion effects on the thermosolutal Casson fluid stream with an internal heat source. These effects have practical applications in geothermal energy extraction, cooling of electronic devices, petroleum engineering, and polymer processing. With the help of similarity transformations, the governing equations are transformed to nonlinear ordinary differential equations (ODEs). The highly nonlinear differential equations are solved with the help of Bernoulli wavelet numerical scheme, and the outputs are compared with previous literature to validate the findings. The study investigates the forces of various physical parameters on the velocity, temperature, and concentration of the fluid and presents the outcomes in graphical form. In addition, the study provides information on skin friction, heat and mass transfers in tabular format. Overall, the research contributes to a better understanding of the behaviour of non-Newtonian fluids under different thermal and concentration gradients and has practical implications in various industrial processes. Our findings demonstrate the remarkable effectiveness and accessibility of the Bernoulli wavelet method in solving coupled nonlinear ODEs of this nature. The results exhibit outstanding agreement, particularly in engineering applications involving coupled nonlinear ODEs.

Bernoulli wavelet application to the numerical solution of Jeffery–Hamel flow problem

In this study, we introduced the Bernoulli wavelet method (BWM) to address the nonlinear Jeffery–Hamel flow problem. Utilizing a newly devised operational matrix of integration with the Bernoulli wavelet, we apply the Bernoulli wavelet expansion in conjunction with the collocation method to convert the given differential equation into a series of nonlinear equations. These equations are subsequently solved using an appropriate iterative technique. The BWM demonstrates superior accuracy when compared to existing methods like differential transformation, homotopy perturbation, variational iteration, and Runge–Kutta (R–K). Our results indicate the effectiveness of BWM in providing more precise solutions for the Jeffery–Hamel flow, a significant phenomenon with wide applications in various engineering fields, including chemical, aerospace, civil, biomechanical, mechanical, and environmental engineering. The study also delves into the influence of Reynolds number and convergent/divergent channel configurations on Jeffery–Hamel flow applications.

A generalized Chebyshev operational method for Volterra integro-partial differential equations with weakly singular kernels

Volterra integro-partial differential equations with weakly singular kernels (VIPDEWSK) are utilized to model diverse physical phenomena. A matrix collocation method is proposed for determining the approximate solution of this functional equation category. The method employs shifted Chebyshev polynomials of the fifth kind (SCPFK) to construct two-dimensional pseudo-operational matrices of integration, avoiding the need for explicit integration and thereby speeding up computations. Error bounds are examined in a Chebyshev-weighted space, providing insights into approximation accuracy. The approach is applied to several experimental examples, and the results are compared with those obtained using the Bernoulli wavelets and Legendre wavelets methods.

Analysis of magnetohydrodynamic laminar boundary layer flow, heat, and mass transfer of nanofluids on a moving surface with thermal radiation via Bernoulli wavelets technique

This study investigates the nanofluid’s magnetohydrodynamic boundary layer flow (BLF) in the presence of thermal radiation using the Bernoulli wavelet collocation method (BWCM). The governing nonlinear partial differential equations (PDEs) are transformed into nonlinear coupled ordinary differential equations (ODEs) using the appropriate similarity transformations. The obtained ODEs are then transformed into a nonlinear system of algebraic equations using BWCM. By contrasting the BWCM with the NDSolve Mathematica command and the Haar wavelet method, the BWCM's validity is examined. The influence of the physical parameters on temperature, concentration, velocity, mass, and heat transfer rates are discussed through tables and graphs. It was found that the Prandtl number, thermophoresis parameter, Brownian motion parameter, radiation parameter, and Lewis number do not influence the skin friction coefficient.

An MHD boundary layer flow of Casson fluid due to a moving wedge analyzed by the Bernoulli wavelet method

AbstractThe current study adopts a Bernoulli wavelet technique to explore the magnetohydrodynamic (MHD) boundary layer flow of a Casson fluid through a wedge. A nanofluid across a wedge has been investigated for its stable laminar MHD flow, heat, and mass transport characteristics. By choosing worthwhile non‐dimensional parameters, the governing boundary layer equation is reformed into a dimensionless Falkner–Skan equation. The consequent nonlinear equation is addressed via the Bernoulli wavelet method. For a range of different values of the physical parameters, the nature of the boundary layer flow is visually observed. Fluid velocity rises with rise in the values of Casson parameter, wedge parameter, and magnetic parameter. Local wall skin friction increases with increase in wedge parameter and magnetic parameter values. To ensure the accuracy of the findings, local wall skin friction is estimated and tested with other techniques that are already reported in the existing research.

A Comparative Study of Numerical Solution of Second-order Singular Differential Equations Using Bernoulli Wavelet Techniques

The main objective of this article is to discuss a numerical method for solving singular differential equations based on wavelets. Singular differential equations are first transformed into a system of linear algebraic equations, and then the linear system’s solution produces the unknown coefficients. Along with its estimated error, the convergence of the approximative solution is alsodetermined. Some numerical examples are thought to show that Bernoulli wavelet is better than Chebyshev and Legendre wavelet and other existing techniques.

Unsteady Casson fluid flow on MHD with an internal heat source

The present paper investigates the heat transfer features of a viscous fluid in an unsteady incompressible Casson fluid flow through a porous medium with an internal heat source under the influence of a magnetic field. A Casson fluid passes through the given porous medium using Darcy model. The study involves simulating the flow using mathematical models based on conservation laws and similarity transformations. The resulting highly nonlinear coupled ordinary differential equations are solved using the Bernoulli wavelet numerical approach. The study utilizes graphical tools to illustrate the flow behaviour under different physical factors, including the effect of the internal heat source. The normalization of the flow behaviour by the magnetic field suggests that it could potentially be used to control various flows. The study finds that various physical parameters which affects the velocity and temperature profiles of the fluid.

Wavelet methods for fractional electrical circuit equations

Classical electric circuits consists of resistors, inductors and capacitors which have irreversible and lossy properties that are not taken into account in classical analysis. FDEs can be interpreted as basic memory operators and are generally used to model the lossy properties or defects. Therefore, employing fractional differential terms in electric circuit equations provides accurate modelling of those circuit elements. In this paper, the numerical solutions of fractional LC, RC and RLC circuit equations are considered to better model those imperfections. To this end, the operational matrices for Bernoulli and Chebyshev wavelets are used to obtain the numerical solutions of those fractional circuit equations. Chebyshev wavelets are orthogonal, and under some circumstances, Bernoulli wavelets can be orthogonal. The wavelet methods’ quick convergence and minimal processing load depend on the orthogonality principle. In the proposed method, those FDEs are transformed into algebraic equation systems using operational matrices employing the discrete Wavelets. The performance of those two wavelet methods are compared and contrasted for computational load, speed, and absolute error values. The paper exploits discrete Bernoulli and Chebyshev wavelets for the numerical solution of fractional LC, RC and RLC circuit equations. The fast convergence, low processing burden, and compactness of the Bernoulli and Chebyshev wavelet methods for fractional circuit equation solutions represent the novel contributions of this paper. Numerical solutions and comparisons are also presented to validate the method.

Bernoulli polynomial based wavelets method for solving chaotic behaviour of financial model

This paper presents an algorithm for solving systems of non integer financial chaotic model. The Bernoulli wavelets function approximation applies to fractional order financial systems for the first time. The main objective of this process is to convert fractional differential equation systems into algebraic equation systems using wavelets approximation based on Bernoulli wavelets and their fractional integral operators. This fractional financial chaotic model that represents the effects of memory and chaos in a system. There are 3 compartments in this fractional order financial system: price indexes, interest rates, and investment demand. Analyses are depends on iteration method and Bernoulli wavelets method to determine the stability analysis, residual error and convergence analysis of the solutions. This study indicates that the presented approximate solutions fit exactly with the numerical solution and that the technique is accurate and effective. Our proposed model has been discussed for its uniqueness, boundedness, and non negativity. Its findings are extremely important for dealing with financial issues in the future.

Bernoulli Wavelets Numerical Approach for the Nonlinear Klein–Gordon and Benjamin–Bona–Mahony Equation

Bernoulli Wavelets Numerical Approach for the Nonlinear Klein–Gordon and Benjamin–Bona–Mahony Equation

A Numerical Technique Based on Bernoulli Wavelet Operational Matrices for Solving a Class of Fractional Order Differential Equations

In this paper, we present an efficient, new, and simple programmable method for finding approximate solutions to fractional differential equations based on Bernoulli wavelet approximations. Bernoulli Wavelet functions involve advantages such as orthogonality, simplicity, and ease of usage, in addition to the fact that fractional Bernoulli wavelets have exact operational matrices that improve the accuracy of the applied approach. A fractional differential equation was simplified to a system of algebraic equations using the fractional order integration operational matrices of Bernoulli wavelets. Examples are used to demonstrate the technique’s precision.

Application of Bernoulli wavelet method on triple‐diffusive convection in Jeffery–Hamel flow

AbstractIn this article, triple‐diffusive convection in the Jeffery–Hamel flow of viscous fluid is studied. The Jeffery–Hamel flow depends upon the radial component of velocity, while the peripheral velocity is zero. The problem has been articulated as nonlinear coupled partial differential equations (PDEs) together with the pertinent boundary conditions. The reduction of the nonlinear coupled PDEs into new nonlinear coupled ordinary differential equations is achieved via a collection of appropriate transformations, which is solved using the Bernoulli wavelet method. The obtained results were compared with the numerical results, confirming the good agreement between the present and previous numerical results. Dimensionless velocity, temperature, and concentration profiles are discussed for the relevant factors involved. Furthermore, skin friction, heat transfer, and solute diffusions are calculated.

Bernoulli wavelet least squares support vector regression: Robust numerical method for systems of fractional differential equations

In this study, a new hybrid method is developed to solve linear or nonlinear systems of fractional differential equations using Bernoulli wavelets (Bws) and the least squares support vector regression (LS‐SVR). The numerical methods based on operational matrices for solving various kinds of fractional equations have been widely studied in the last decade. In contrast to the existing methods, here we derive the operator of fractional integration, aiming to remove the approximation error. For this purpose, we present Bw operator of Riemann–Liouville fractional integration and use it in our scheme. In the proposed technique, we approximate the unknown functions via Bws, and then with the help of Bw operator of fractional integration and LS‐SVR, we reduce the problem to an algebraic system. In this way, we simplify the computation of the considered system. The error analysis of our method is proposed. Finally, we demonstrate the applicability of the present scheme by solving several numerical examples.

Bernoulli wavelet method for numerical solution of linear system of Fredholm integral equation of the second kind

One of the key tools for many fields of applied mathematics is the integral equations. Integral equations are widely utilized in many models, atmosphere–ocean dynamics, fluid mechanics, mathematical physics, and many other physical and engineering disciplines. A new numerical strategy based on the Bernoulli wavelet is introduced to solve system of Fredholm integral equations of second kind. In this paper, the Bernoulli wavelets are first built. Second, the system of Fredholm integral equations has been reduced into an algebraic system. In order to demonstrate the viability, and accuracy of the suggested Bernoulli wavelet approach, some numerical examples are offered at the end. The derived numerical results are examined with those from other numerical techniques and with exact solutions, demonstrating the superiority of the proposed method over those techniques. The novelty of proposed technique is that it can be extended for the numerical solution of two dimensional integral equations and differential equations appearing in engineering models, however some modifications will be required.