Bernoulli Wavelets Research Articles

Effects of heat transfer on MHD suction–injection model of viscous fluid flow through differential transformation and Bernoulli wavelet techniques

AbstractIn this study, a numerical solution of the velocity and heat transfer on the magnetohydrodynamic suction–injection model of viscous fluid flow has been studied. We use the differential transformation method and Bernoulli wavelet method to solve the highly nonlinear governing equations; applying appropriate similarity transformations and reducing governing equations to highly nonlinear coupled ordinary differential equations. The objective of this analysis is to determine how the suction parameter, Hartmann number, squeeze number, thermophoresis parameter, and Prandtl number affect the velocity and temperature profiles. When the current findings are compared with those that have already been published in the literature, confident suppositions are made, and it is discovered that there is considerable agreement. Graphs have been used to discuss the influence of nondimensional characteristics on velocity and temperature.

A Study on the Non-Linear Murray Equation Through the Bernoulli Wavelet Approach

A Study on the Non-Linear Murray Equation Through the Bernoulli Wavelet Approach

APPLICATION OF BERNOULLI WAVELET METHOD FOR NUMERICAL SOLUTION OF FUZZY LINEAR VOLTERRA-FREDHOLM INTEGRAL EQUATIONS

This work, Bernoulli wavelet method is formed to solve nonlinear fuzzy Volterra-Fredholm integral equations. Bernoulli wavelets have been created by dilation and translation of Bernoulli polynomials. First we introduce properties of Bernoulli wavelets and Bernoulli polynomials, and then we used it to transform the integral equations to the system of algebraic equations. We compared the result of the proposed method with the exact solution to show the convergence and advantages of the new method. The results got by present wavelet method are compared with that of by collocation method based on radial basis functions method. Finally, the numerical examples explain the accuracy of this method.

Modified Bernoulli wavelets functional matrix approach for the HIV infection of CD4+ T cells model

In this study, we generated a novel functional matrix using Bernoulli wavelets. Also, we developed a novel technique called the Bernoulli wavelets collocation method to obtain reasonably accurate solutions for the HIV-infection model of CD4+ T cells. This mathematical model is in the form of a system of a nonlinear ordinary differential equation (ODE). This approach obtains the solution for this model by transforming it into a system of nonlinear algebraic equations by expanding through Bernoulli wavelets with unknown coefficients. The collocation scheme is used to calculate these unknown coefficients. The consistency and proficiency of the developed approach are demonstrated through tables and graphs. Obtained results reveal that the current approach is more accurate than other methods in the literature. All computations have been made with the help of Mathematica software. Some properties of Bernoulli wavelets are discussed in terms of theorems.

Applications of the Bernoulli wavelet collocation method in the analysis of MHD boundary layer flow of a viscous fluid

This study focuses on the flow of viscous, electrically conducting incompressible fluid over a stretching plate. The Falkner–Skan equation is a nonlinear, third-order boundary value problem. No closed-form solutions are available for this two-point boundary value problem. Here, we developed a new functional matrix of integration using the Bernoulli wavelet and also generated a new technique called Bernoulli wavelet collocation method (BWCM) to solve the nonlinear differential equation that arises in the fluid flow over a stretching plate. The boundary layer model is transformed to a nonlinear ordinary differential equation called the Falkner-Skan type equation using suitable transformation. Using BWCM, we have solved the unbounded governing equations of different types that arise in the MHD boundary-layer flow of a viscous fluid over a stretching plate. Several aspects of this problem are justified using the Haar wavelet and the previously obtained theoretical results. It is observed that the boundary-layer thickness decreases as the pressure gradient and magnetic field parameters increase. The overshoots and undershoots are observed for some particular parameters using BWCM. Furthermore, our research yields dual solutions for some physical parameters, which are investigated for the first time in the literature using the Bernoulli wavelet approach. The nature of the flow problem is discussed through the graphs by varying the physical parameters.

Applications of Bernoulli wavelet collocation method in the analysis of Jeffery–Hamel flow and heat transfer in Eyring–Powell fluid

Applications of Bernoulli wavelet collocation method in the analysis of Jeffery–Hamel flow and heat transfer in Eyring–Powell fluid

Bernoulli wavelet method for non-linear fractional Glucose–Insulin regulatory dynamical system

In this paper, we propose a model that explains the glucose–insulin regulatory system (GIRS). The literature has demonstrated that fractional-order analysis can uncover new aspects of complex systems that were not previously explored. As a result of our studies, the GIRS is stable in its non-linear fractional-order form. Each numerical investigation is derived in order to demonstrate the effectiveness and applicability of fractional order analysis, including the Toufik–Atangana method and the Bernoulli wavelet operational matrix method (BWOMM). An unregulated glucose–insulin system causes a wide variety of biological disorders. These studies may permit a deeper understanding of the regulatory system.

Bernoulli wavelets functional matrix technique for a system of nonlinear singular Lane Emden equations

In the present paper, we developed the functional matrix of integration via Bernoulli wavelets and generated a competent numerical scheme to solve the nonlinear system of singular differential equations which is Lane Emden form by Bernoulli wavelets collocation technique (BWCT) with different physical conditions. The system of nonlinear singular models is not smooth to operate as they are singular and nonlinear. This approach obtains the solution for this system by transforming it into a-nonlinear system of algebraic equations by expanding through Bernoulli wavelets with unknown coefficients. These unknown coefficients are calculated using the collocation scheme. The consistency and proficiency of the developed approach are demonstrated via graphs and tables. Attained results confirm that the newly implemented approach is more effective and accurate than other techniques which are available in the literature. All computations have been made using Mathematica software. The convergence of this method is explained in terms of theorems.

An efficient hybrid numerical method for multi-term time fractional partial differential equations in fluid mechanics with convergence and error analysis

The fundamental purpose of this paper is to study the numerical solution of multi-term time fractional nonlinear Klein–Gordon equation, using regularized beta functions and fractional order Bernoulli wavelets. First, the exact formulas for the fractional integrals of the fractional order Bernoulli wavelets were obtained. Using properties of the regularized beta functions and their operational matrices the operational matrices of the fractional order Bernoulli wavelets were calculated. Through new operational matrices and appropriate collocation points, the time fractional nonlinear Klein–Gordon equation were transformed to a system of nonlinear algebraic equations. The convergence analysis and error bound of the proposed method were then performed. A sufficient number of numerical simulations were considered to show the effectiveness and validity of the presented numerical method and its theoretical analysis.

Approximation of functions by Bernoulli wavelet and its applications in solution of Volterra integral equation of second kind

In this paper, two new estimators E_{2^{k-1},0}^{(1)}(f) and E_{2^{k-1},M}^{(1)}(f) of characteristic function and an estimator E_{2^{k-1},M}^{(2)}(f) of function of Hddot{text {o}}lder’s class H^{alpha } [0,1) of order 0<alpha leqslant 1 have been established using Bernoulli wavelets. A new technique has been applied for solving Volterra integral equation of second kind using Bernoulli wavelet operational matrix of integration as well as product operational matrix. These matrices have been utilized to reduce the Volterra integral equation into a system of algebraic equations, which are easily solvable. Some examples are illustrated to show the validity and efficiency of proposed technique of this research paper.

Using Fractional Bernoulli Wavelets for Solving Fractional Diffusion Wave Equations with Initial and Boundary Conditions

In this paper, fractional-order Bernoulli wavelets based on the Bernoulli polynomials are constructed and applied to evaluate the numerical solution of the general form of Caputo fractional order diffusion wave equations. The operational matrices of ordinary and fractional derivatives for Bernoulli wavelets are set via fractional Riemann–Liouville integral operator. Then, these wavelets and their operational matrices are utilized to reduce the nonlinear fractional problem to a set of algebraic equations. For solving the obtained system of equations, Galerkin and collocation spectral methods are employed. To demonstrate the validity and applicability of the presented method, we offer five significant examples, including generalized Cattaneo diffusion wave and Klein–Gordon equations. The implementation of algorithms exposes high accuracy of the presented numerical method. The advantage of having compact support and orthogonality of these family of wavelets trigger having sparse operational matrices, which reduces the computational time and CPU requirements.

On the Numerical Solution of Partial Differential Equation with Convection Term by using Bernoulli Wavelets

On the Numerical Solution of Partial Differential Equation with Convection Term by using Bernoulli Wavelets

Bernoulli wavelet method for numerical solutions of system of fuzzy integral equations

In this paper, we have proposed an efficient numerical method to solve a system linear fuzzy Fredholm integral equations of the second kind based on Bernoulli wavelet method (BWM). Bernoulli wavelets have been generated by dilation and translation of Bernoulli polynomials. The aim of this paper is to apply Bernoulli wavelet method to obtain approximate solutions of a system of linear Fredholm fuzzy integral equations. First, we introduce properties of Bernoulli wavelets then we used it to transform the integral equations to the system of algebraic equations, the error estimates of the proposed method are given and compared by solving some numerical examples.

New Procedures of a Fractional Order Model of Novel Coronavirus (COVID-19) Outbreak via Wavelets Method

Coronaviruses are a group of RNA (ribonucleic acid) viruses with the capacity for rapid mutation and recombination. Coronaviruses are known to cause respiratory or intestinal infections in humans and animals. In this paper, a biologically compatible set of nonlinear fractional differential equations governing the outbreak of the novel coronavirus is suggested based on a model previously proposed in the literature. Then, this set is numerically solved utilizing two new methods employing sine–cosine and Bernoulli wavelets and their operational matrices. Moreover, the convergence of the solution is experimentally studied. Furthermore, the accuracy of the solution is proved via comparing the results with those obtained in previous research for the primary model. Furthermore, the computational costs are compared by measuring the CPU running time. Finally, the effects of the fractional orders on the outbreak of the COVID-19 are investigated.

Application and comparison of the two efficient algorithms to solve the pantograph Volterra fuzzy integro-differential equation

In this paper, collocation method based on sinc function and Bernoulli wavelet is proposed to find numerical solution of pantograph Volterra fuzzy integro-differential equation. In the first step, the fuzzy integro-differential equation is converted to a system of integro-differential equations in crisp case. Then by using the proposed algorithms, this system is reduced to a system of algebraic equations which is solved by Newton’s method. Also convergence analysis of the methods is given. We illustrate the accuracy of these algorithms by two examples and compare the obtaining results and the computational implementations.

Solving Fractional Two-Dimensional Nonlinear Partial Volterra Integral Equation by Using Bernoulli Wavelet

Solving Fractional Two-Dimensional Nonlinear Partial Volterra Integral Equation by Using Bernoulli Wavelet

Orthonormal Bernoulli wavelets neural network method and its application in astrophysics

The purpose of this paper is to present two efficient computational schemes for solving Lane–Emden type equations based on an artificial neural network. The mentioned neural network consists of three layers (input layer, hidden layer, output layer). We use orthonormal Bernoulli wavelets and arctanh(x) functions as activation functions of the hidden layer and output layer, respectively. Finally, the classical optimization and collocation methods are applied to train this neural network in first and second methods, respectively. In addition, to show the accuracy and efficiency of the current techniques, we test those on various types of Lane–Emden equations and our schemes are compared with some other numerical methods.

Bernoulli wavelet method for numerical solution of anomalous infiltration and diffusion modeling by nonlinear fractional differential equations of variable order

In this paper, generalized fractional-order Bernoulli wavelet functions based on the Bernoulli wavelets are constructed to obtain the numerical solution of problems of anomalous infiltration and diffusion modeling by a class of nonlinear fractional differential equations with variable order. The idea is to use Bernoulli wavelet functions and operational matrices of integration. Firstly, the generalized fractional-order Bernoulli wavelets are constructed. Secondly, operational matrices of integration are derived and utilize to convert the fractional differential equations (FDE) into a system of algebraic equations. Finally, some numerical examples are presented to demonstrate the validity, applicability and accuracy of the proposed Bernoulli wavelet method.

Solution of third-order Emden–Fowler-type equations using wavelet methods

PurposeThe numerical solution of third-order boundary value problems (BVPs) has a great importance because of their applications in fluid dynamics, aerodynamics, astrophysics, nuclear reactions, rocket science etc. The purpose of this paper is to develop two computational methods based on Hermite wavelet and Bernoulli wavelet for the solution of third-order initial/BVPs.Design/methodology/approachBecause of the presence of singularity and the strong nonlinear nature, most of third-order BVPs do not occupy exact solution. Therefore, numerical techniques play an important role for the solution of such type of third-order BVPs. The proposed methods convert third-order BVPs into a system of algebraic equations, and on solving them, approximate solution is obtained. Finally, the numerical simulation has been done to validate the reliability and accuracy of developed methods.FindingsThis paper discussed the solution of linear, nonlinear, nonlinear singular (Emden–Fowler type) and self-adjoint singularly perturbed singular (generalized Emden–Fowler type) third-order BVPs using wavelets. A comparison of the results of proposed methods with the results of existing methods has been given. The proposed methods give the accuracy up to 19 decimal places as the resolution level is increased.Originality/valueThis paper is one of the first in the literature that investigates the solution of third-order Emden–Fowler-type equations using Bernoulli and Hermite wavelets. This paper also discusses the error bounds of the proposed methods for the stability of approximate solutions.

Bernoulli Wavelets Operational Matrices Method for the Solution of Nonlinear Stochastic Itô-Volterra Integral Equations

This article gives an effective strategy to solve nonlinear stochastic Itô-Volterra integral equations (NSIVIE). These equations can be reduced to a system of nonlinear algebraic equations with unknown coefficients, using Bernoulli wavelets, their operational matrix of integration (OMI), stochastic operational matrix of integration (SOMI) and these equations can be solved numerically. Error analysis of the proposed method is given. Moreover, the results obtained are compared to exact solutions with numerical examples to show that the method described is accurate and precise.