Haar Wavelet Collocation Method Research Articles

Haar wavelet collocation technique for numerical solution of porous media equations

This paper focuses on using efficient Haar wavelet collocation method to nonlinear parabolic differential equations in order to solve steady state pore-scale creaping fluid flow in porous medium, substituting finite Haar series for space derivatives, incontrast forward difference approach is used to estimate the time derivative for porous media equation. These equations has vast number of application in engineering and various other fields such as, in petroleum engineering, there are often situations where fluids are near critical conditions. As oil pressure reduces below the bubble point when gas forms liquid condensations, or when the surface tension become very low i.e, transition from miscibility to immiscibility. To model gas production by pressure decline requires an accurate description of physical mechanism involved in the appearance of the gas bubbles. The observed stability with theoretical convergence assertion, the conservation law of mass and energy are stated. This supports the convergence of the preferred paradigm. A convergent approximation to the porous media equation is obtained by suggested approach. By calculating the maximum error norm and the experimental rate of convergence for various problems, the method efficiency and dependability are demonstrated. The method is accurate, easily applicable, and efficient.

Numerical solution of Fisher’s equation through the application of Haar wavelet collocation method

This research article presents algorithm that employs the Haar wavelet collocation method (HWCM) to solve the convection diffusion reaction (CDR) model, specifically Fisher’s equation (F.E.). The proposed method is suitable for addressing both linear and nonlinear problems. The Haar wavelet method is a numerical technique that utilizes wavelet basis functions to provide an approximation of the solution for the Fisher’s equations. Additionally, a forward difference scheme is utilized to approximate the time derivative while the highest order space derivative is replaced by Haar function. The performance of the technique is demonstrated through some examples investigated in this study. The maximum absolute errors are computed for varying numbers of collocation points. The HWCM provides superior accuracy, flexibility, convenience, and computational efficiency compared to alternative methods.

THEORETICAL ANALYSIS AND NUMERICAL SOLUTION OF LINEAR AND NONLINEAR NEUTRAL DELAY DIFFERENTIAL EQUATIONS

In this paper, a collocation method based on the Haar wavelet is presented for the solution of both linear and nonlinear first-order neutral delay differential equations. The Haar functions are used to approximate the first-order derivative, and the approximate solution is obtained by using initial condition and integration. Some examples from the literature are used to test the suggested method efficiency and applicability. A comparison of exact and approximate solutions is given in figures for different numbers of collocation points. The root mean square and maximum absolute errors are calculated for different numbers of collocation points. The rate of convergence is calculated which is approximately equal to 2. The comparison of the present method with the other numerical methods is also given. The results demonstrate that the Haar wavelet collocation method is simple and effective for solving first-order linear and nonlinear neutral delay differential equations.

Numerical assessment of hyperbolic type double interface problems via Haar wavelets

In this manuscript, we have addressed wave propagation challenges within heterogeneous media and hyperbolic interface model. A hybrid numerical approach is introduced for solving these problems, combining the finite difference method and Haar wavelet collocation method. The method entails approximating the second-order space partial derivative through a truncated Haar wavelet series, while the temporal derivative is approximated using finite difference method. For linear hyperbolic interface model, the resulting algebraic systems are solved using the Gauss elimination technique. In the case of non-linear problems, the nonlinearity is addressed through the quasi-Newton linearization formula. To assess the accuracy of the proposed technique, we compute the computational rate of convergence, root mean square errors and maximum absolute errors employing various collocation points. The proposed method perform very well and produces a stable solution if sharp transitions exists in the solution space or if there is a discontinuity between initial and boundary conditions, whereas, the other existing method loses its accuracy in such cases. The numerical experiments, stability and rate of convergence both theoretical and computational, confirm the accuracy and diverse applicability of the method.

A study on Chlamydia transmission in United States through the Haar wavelet technique

In this study, a novel method known as the Haar wavelet collocation method (HWCM) is used to analyze the mathematical model of Chlamydia transmission in the United States. We use dependent variables with various parameters, such as birth rate, mortality rate, and recovery rate, etc., to explore the nature of the Chlamydia disease in susceptible unvaccinated individuals, susceptible vaccinated individuals, infected, treated, and recovered individuals. The ordinary differential equations (ODEs) in this model are coupled nonlinear. In this method, the Chlamydia model is converted into a system of non-linear algebraic equations using properties of the operational matrix of Haar wavelets. Later, the Newton–Raphson approach is used to extract the unknown coefficients. Mathematica software has been used for all calculations. Tables and graphs contain tabulated results of calculations. As a result, it can be seen that the current approach is more precise than those used in the literature. Also, we discuss the effect of different parameters on susceptible unvaccinated individuals, susceptible vaccinated individuals, infected, treated, and recovered individuals through graphs.

Extension of Haar wavelet technique for numerical solution of three-dimensional linear and nonlinear telegraph equations

In this manuscript, a hybrid numerical technique is presented for solving three-dimensional hyperbolic telegraph equations. The proposed technique is based on the Haar wavelet collocation method and the finite difference method. In this technique, the space derivatives are estimated by truncated Haar wavelet series, while the time derivatives are approximated using finite difference method. The method is applied to some linear and non-linear three-dimensional hyperbolic telegraph equations. In the linear case, the resulting algebraic system is solved by the Gauss elimination technique. Whereas in the nonlinear case, the quasi-Newton linearization technique is utilized to remove the nonlinearity of the problem. The maximum absolute errors, root mean square errors and computational rate of convergence are calculated for a different number of collocation points at different time steps of the final time t=1. The stability analysis for the proposed method is also discussed. Overall, the recently devised numerical method boasts straightforward implementation, rapid convergence, and enhanced accuracy in addressing both linear and nonlinear problems.

Examining the role of activation energy and convective boundary conditions in nanofluid behavior of Couette-Poiseuille flow

Abstract This work investigates the behavior of a nanofluid in a horizontal channel under advection boundary conditions within the domain of magnetohydrodynamic radiative Couette-Poiseuille flow. We utilize the Haar wavelet collocation method (HWCM) to investigate the effects of energy activation. This research relies on the mathematical model introduced by Buongiorno, which effectively captures the flow dynamics and incorporates the influence of chemical processes. To streamline the governing flow equations, we employ boundary layer approximations. The HWCM is employed to numerically solve the non-linear coupled partial differential equations that regulate momentum, heat transport, and mass transfer processes. We examine the impact of several dimensionless convergence parameters on the velocity, temperature, and concentration profiles and give visual representations of these results. It is crucial to highlight that the activation energy of the specific chemical reaction is directly linked to the concentration of nanoparticles. The effect of Brownian motion on nanoparticle concentration varies from that of the thermophoresis parameter.

Consistency and Feasibility of Haar Wavelet Collocation Method for a Nonlinear Optimal Control Problem with Application

Haar wavelet-based numerical algorithms have recently been developed for various mathematical problems, including optimal control problems. However, no numerical algorithm is complete without its theoretical analysis. In this paper, we have shown the consistency and feasibility of the Haar wavelet-based collocation method for solving nonlinear optimal control problems that have a single state and a single control variable with constraints. The accuracy of the method has been shown through some application problems.

Haar wavelet approach to study the control of biological pest model in Tea plants

n this study, we consider a novel approach called the Haar wavelet collocation method (HWCM) toexamine the mathematical model of pest propagation in tea plants and how biological enemies might controlthem. This model is in the form of a system of coupled ordinary differential equations (ODEs). When studyingthe system, we consider tea plants, pests that harm the plants, biological enemies that are their reasonablecompetitors of pests, self-reproduction of the tea plants, natural death of pests and natural enemies, etc. Byturning the Mathematical model into a system of non-linear algebraic equations, we use the properties ofthe Haar wavelets. The opted method can solve the biological pest management problem in tea plants. Thevalues of the unknown coefficients are recovered using the collocation method and Newton Raphson method.The Mathematica program acquires the numerical results, nature, and uniformity. The acquired findings showthat the current method is more accurate than those indicated in tables and graphs.

A higher-order collocation method based on Haar wavelets for integro-differential equations with two-point integral condition

In this article, the higher-order Haar wavelet collocation method (HCMHW) is investigated to solve linear and nonlinear integro-differential equations (IDEs) with two types of conditions: simple initial condition and the point integral condition. We reproduce and compare the numerical results of the conventional Haar wavelet collocation method (CMHW) with those of HCMHW, demonstrating the superior performance of HCMHW across various conditions. Both methods effectively handle different types of given conditions. However, numerical results reveal that HCMHW exhibits a faster convergence rate than CMHW. To address nonlinear IDEs, we employ the quasi-linearization technique. The computational stability of both methods is evaluated through various experiments. Additionally, the article provides examples to illustrate the overall performance and accuracy of HCMHW compared to CMHW for both linear and nonlinear IDEs.

3D thin-film nanofluid flow with heat transfer on an inclined disc by using HWCM

Abstract Momentum and heat transmission influence the coated physical characteristics of wire product. As a result, understanding the polymeric movement and heat mass distribution is crucial. An increase in thermal efficiency is necessary for the wire covering technology. So, the aim of this work is to investigate the influence of nanomaterials on the heat and mass transport processes in wire coating analyses. A thin film nanofluid is used to investigate heat and mass transfer in three dimensions over a rotating inclined disc. Both the suction and injection effects of nanofluids and the thermal radiation of their fluxes are taken into account. By employing similarity variables, the set of governing equations can be transformed into a differential equation system. The necessary differential equation system is solved using the Haar wavelet collocation method. Plots and observations of the velocity distribution, concentration, and thermal fields within the boundary layer across an inclining, steadily rotating plane are made. Flow characteristics change as a result of varying embedded factors such as S , Sc , N b , Pr , S,{\rm{Sc}},{N}_{{\rm{b}}},\Pr , and thermophoretic parameters. Evidence suggests that as the number of rotation parameters grows, the thermal boundary layer weakens.

Haar wavelet for computing periodic responses of impact oscillators

Haar wavelets have an attractive property for non-smooth dynamical systems as they are capable of modelling sudden changes because of their local multi-resolution characteristics. In this paper, we applied the Haar wavelet collocation method embedding the segment technique to compute and detect periodic responses of an elastic impact oscillator. Comparisons between dynamical responses computed by a direct numerical simulation using a high accuracy Runge–Kutta algorithm and the proposed method are encouraging. Some key parameters of the impact oscillator were changed to demonstrate the effectiveness of the method. Moreover, the time–frequency index feature for the Haar wavelet coefficients describing impacts was proven to be an additional advantage of this method.

A numerical study of two-phase nanofluid MHD boundary layer flow with heat absorption or generation and chemical reaction over an exponentially stretching sheet by Haar wavelet method

This study examines the two-phase nanofluid magnetohydrodynamic (MHD) boundary layer flow model with heat absorption or generation and chemical reaction via porous media due to an exponential stretching sheet using the Haar wavelet collocation method (HWCM). First, the governing nonlinear partial differential equations (PDEs) are transformed into coupled, highly nonlinear, ordinary differential equations (ODEs) using similarity transformations. Then the coupled ODEs are translated from the infinite domain [0, ∞ ) to the finite domain [0, 1] via coordinate transformation because the wavelet plays a crucial role in [0, 1]. The obtained equations are solved using HWCM. The impacts of the physical parameters are discussed using tables and graphs. It was found that the Nusselt number R e x − 1 / 2 N u x is a decreasing function of the magnetic field, porous media parameter, heat generation or absorption parameter, and chemical reaction parameter.

A high-order multi-resolution wavelet method for nonlinear systems of differential equations

In this article, the applications of the new Haar wavelet collocation methods called as Haar wavelet collocation method (HWCM) and higher-order Haar wavelet collocation method (H-HWCM) are developed for the solution of linear and nonlinear systems of ordinary differential equations. The proposed H-HWCM is compared with a variety of other methods including the well-known HWCM. The quasi-linearization technique is introduced in the nonlinear cases. The stability and convergence of both techniques is studied in detail, which are the important parts to analyze the proposed methods. The efficiency of the methods is illustrated with certain numerical examples, but the H-HWCM is more accurate with faster convergence than the HWCM and other methods reported in the literature.

Solving General Fractional Lane-Emden-Fowler Differential Equations Using Haar Wavelet Collocation Method

This paper aims to solve general fractional Lane-Emden-Fowler differential equations using the Haar wavelet collocation method. This method transforms the fractional differential equation into a nonlinear system of equations, which is further solved for Haar coefficients using Newton’s method. We have constructed the higher-order Lane-Emden-Fowler equations. We have also discussed the convergence rate and stability analysis of our technique. We have explained the applications and numerically simulated the examples graphically and in tabular format to elaborate on the accuracy and efficiency of this approach.

Spectral and Haar wavelet collocation method for the solution of heat generation and viscous dissipation in micro-polar nanofluid for MHD stagnation point flow

The aim and significance of paper presents, the semi-numerical investigation of magnetohydrodynamic flow of micropolar nanofluid with stagnation point is carried out under the influence of viscous dissipation and heat generation. The micropolar nanofluids are electrically conducting non-Newtonian fluids. The important applications of these fluids are observed in many research areas viz. bioengineering, biofuels and biomedical sectors etc. The appropriate similarity transformations are used to transform the governing equations into system of coupled nonlinear ordinary differential equations and are solved by using shifted Chebyshev collocation method and Haar wavelet collocation method. The variations in velocity, angular velocity, temperature and concentration profiles under the impact of various physical parameters, characterizing the flow field are discussed and are presented via graphs and tables. Temperature enhancement occurs with increment in each parameter except for Prandtl number. The concentration near the surface decreases with increment in the values of parameters and gradually it increases, except for Prandtl number and Schmidt number. The reverse trend of heat transfer occurs​ near a surface, when the dominance of stream velocity over stretching velocity is observed.

Numerical solution of general Emden–Fowler equation using Haar wavelet collocation method

This paper deals with the numerical solution of the general Emden–Fowler equation using the Haar wavelet collocation method. This method transforms the differential equation into a system of nonlinear equations. These equations are further solved by Newton's method to obtain the Haar coefficients, and finally the solution to the problem is acquired using these coefficients. We have taken many examples of fifth- and sixth-order equations and implemented our method on those examples. The graphs show the efficiency of the solution for resolution L = 3 and the maximum absolute error of our approach. The error tables give a good picture of the accuracy of this approach.

Theoretical and numerical study of a fluid flow of a slightly rarefied gas-free stream past a moving wall inside a porous medium

This study investigates the boundary layer flow problem arising in the viscous fluid flow of a slightly rarefied gas-free stream over a moving plate embedded in a porous medium. Previous works were mainly focused on numerically, and some properties of the solutions were discussed from numerical results. But, numerical solutions are only sometimes guaranteed the solution’s existence. In this work, we have adopted the famous topological shooting argument to prove the existence result of the solution. We have shown that the solution may be convex or concave depending on the parameter values. Also, for these reasons, there arise some difficulties. However, we have rectified all the challenges and proved that unique solutions exist for all the particular governing parameter values. Nevertheless, we found an exact solution for some specific parameter values. Furthermore, the Haar wavelet collocation method is used to address the numerical results of the nonlinear boundary value problem. We first validate the obtained results with the existing numerical outcomes. The impacts of permeability and slip parameters on the velocity profiles and shear stress are elucidated through tables and figures, and probable arguments are explained in detail.

Utilization of Haar wavelet collocation technique for fractal-fractional order problem

This work is devoted for establishing adequate results for the qualitative theory as well as approximate solution of “fractal-fractional order differential equations” (F-FDEs). For the required numerical results, we use Haar wavelet collocation (H-W-C) method which has very rarely utilized for F-FDEs. We establish the general algorithm for F-FDEs to compute numerical solution for the considered class. Also, we establish a result devoted to the qualitative theory via Banach fixed point result. A results devoted to Ulam-Hyers (U-H) stability are also included. Two pertinent examples are given along with the comparison and different norms of errors displayed in figures as well as tables.

A numerical Haar wavelet-finite difference hybrid method and its convergence for nonlinear hyperbolic partial differential equation

AbstractIn this research work, we proposed a Haar wavelet collocation method (HWCM) for the numerical solution of first- and second-order nonlinear hyperbolic equations. The time derivative in the governing equations is approximated by a finite difference. The nonlinear hyperbolic equation is converted into its full algebraic form once the space derivatives are replaced by the finite Haar series. Convergence analysis is performed both in space and time, where the computational results follow the theoretical statements of convergence. Many test problems with different nonlinear terms are presented to verify the accuracy, capability, and convergence of the proposed method for the first- and second-order nonlinear hyperbolic equations.