Haar Wavelet Technique Research Articles

Numerical solution of generalized fractional Lane–Emden-Fowler equation using Haar wavelet method

This paper presents a numerical technique to solve the generalized fractional Lane–Emden–Fowler equation. We use the Haar wavelet technique to approximate the solution of the generalized fractional Lane–Emden–Fowler equation. In this method, a differential equation transforms into a system of nonlinear equations, and this system is further solved to obtain Haar coefficients. We also showed the convergence of this method by establishing an upper bound. The rate of convergence is also discussed in our algorithm. Many numerical examples have been taken, and the results of those numerical experiments have been written in tabular form. We also documented the experiments graphically to show the efficiency and accuracy of this method.

Application of generalized Haar wavelet technique on simultaneous delay differential equations

In this paper, the Haar wavelet technique is applied to a system of simultaneous linear and nonlinear proportional delay differential equations. By implementing this method, the delay differential equations are transformed into a series of linear and nonlinear algebraic equations, respectively. These equations are then solved using numerical code constructed in MATLAB. The obtained approximate solutions are compared with exact solutions, demonstrating the accuracy and efficacy of the Haar wavelet method. Using the fixed point theorem, the existence and uniqueness of the solutions is established. The reliability and efficiency of the proposed technique are illustrated through two numerical examples. Additionally, a comprehensive error analysis is presented and the convergence result is discussed, providing a thorough examination of the robustness of the method. This study not only shows that the Haar wavelet technique is a powerful tool for solving delay differential equations but also sets the groundwork for future research in this area.

Solving linear systems of fractional integro-differential equations by Haar and Legendre wavelets techniques

In this study, we present two highly effective approaches aimed at solving linear systems of equations, specifically focusing on the Fredholm and Volterra equations in fractional integro-differential formula. To tackle these challenging problems, we employ two distinct methodologies: the Haar wavelet technique (HWT) and the Legendre wavelet Technique (LWT). To accurately describe fractional derivatives with the equations, we adopt the Caputo sense, which provides a robust framework for handling fractional calculus. By applying HWT and LWT, we are able to transform the intricate equations of the fractional integro-differential system into algebraic equations. Fredholm and Volterra fractional integro-differential equations serve as benchmarks for evaluating the accuracy and efficiency of LWT and HWT. By comparing the approximate solutions obtained through these wavelet methods with the exact solutions, we are able to provide a comprehensive assessment of their performance. Through detailed numerical computations conducted using Mathematical 8,which demonstrate the effectiveness of both LWT and HWT in solving systems of Fredholm and Volterra equations in fractional integro-differential equations. The results obtained from these computations are presented through graphs, allowing for a clear and comparison between the approximate and exact solutions. The successful utilization of these wavelet methods provides researchers and practitioners with valuable tools for tackling complex mathematical problems in various fields of study. This study contributes to the progress of numerical techniques for solving of fractional integro-differential equations which aid in the development of more accurate and efficient computational approaches for analyzing and solving similar mathematical problems in the future.

A numerical study of the evolution of smoking habit model through Haar wavelet technique

A numerical study of the evolution of smoking habit model through Haar wavelet technique

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.

Two-dimensional Haar Wavelet Method for Numerical Solution of Delay Partial Differential Equations

In this paper, a two-dimensional Haar wavelet collocation method is applied to obtain the numerical solution of delay and neutral delay partial differential equations. Both linear and nonlinear problems can be solved using this method. Some benchmark test problems are given to verify the efficiency and accuracy of the aforesaid method. The results are compared with the exact solution and performance of the two-dimensional Haar collocation technique is measured by calculating the maximum absolute and root mean square errors for different numbers of grid points. The results are also compared with finite difference technique and one-dimensional Haar wavelet technique. The numerical results show that the two-dimensional Haar method is simply applicable, accurate and efficient.

Wavelet-based techniques for Hygro-Magneto-Thermo vibration of nonlocal strain gradient nanobeam resting on Winkler-Pasternak elastic foundation

The primary objective of this article is to investigate the vibration characteristics of a nanobeam embedded in a Winkler-Pasternak elastic foundation adopting wavelet-based two relatively recent techniques, namely the Haar Wavelet Method (HWM) and the Higher Order Haar Wavelet Method (HOHWM). While subjected to a longitudinal magnetic field, the nanobeam is also exposed to hygroscopic and thermal environments. Hamilton's principle is being used to model the nanobeam within the frameworks of Euler-Bernoulli beam theory and nonlocal strain gradient theory. The effects of several parameters such as, the small scale, length scale, magnetic field intensity, hygroscopic, thermal, Winkler modulus and shear modulus on frequency parameters, are thoroughly investigated for PP (Pined-Pined) and CC (Clamped-Clamped) boundary conditions. Richardson's formula is used to determine the rate of convergence of both the Haar wavelet and higher order Haar wavelet methods, and the results obtained by the Haar wavelet technique are also extrapolated, revealing that the rate of convergence of extrapolated HWM is comparable to the higher order Haar wavelet method. The effectiveness of the higher order Haar wavelet method over the Haar wavelet method is demonstrated using PP (Pined-Pined) boundary condition as a test case. Furthermore, the results of this study have been cross-validated with previously published literature in specific circumstances, exhibiting exceptional agreement.

On solution of ordinary differential equations by using HWCM, ADM and RK4

The Haar wavelet collocation approach (HWCM) is an impressive numerical method for solving linear initial value problems when compared to the existing numerical methods (Adomian decomposition method (ADM) & Runge–Kutta method (RK4)). The objective of this study is to use the Haar-wavelet technique, Adomian decomposition technique (ADM) and Runge–Kutta (RK4) method to achieve the numerical solution of second-order ordinary differential equations. The proposed methods are applied to three different problems and the numerical results show that the HWCM has better agreement with analytic solutions than the other numerical methods.

Uniform Haar wavelet technique with Newton’s method for a kind of derivative dependent SBVPs

This work deals with the construction of a coupled technique, uniform Haar wavelet and Newton–Raphson method for a kind (Lane–Emden–Fowler type) of two-point derivative dependent singular differential equation $$\begin{aligned}&(g(x)y')'= h(x)f(x,y, g y'), \qquad x\in (0,1] \end{aligned}$$ with Dirichlet and Neumann–Robin boundary conditions (BCs) $$\begin{aligned}&y(0) =a,\qquad y(1)= b,\\&y'(0)= 0,\qquad \alpha _1 y(1)+\beta _1 y'(1)= \gamma _1. \end{aligned}$$ Here $$g(x)>0$$ on (0, 1) permit $$g(0)=0$$ . This collocation based numerical approach is employed for reducing the considered class of singular differential equations into algebraic one and hence, Newton–Raphson method is applied for numerical solution. The convergence study and its error analysis is developed to depict validity and the order of stability of the present method. From practical point of view, some numerical examples (both linear and nonlinear) have been illustrated for demonstrating the simplicity of compliance and outcomes based on the method. The $$L_{2}$$ norm and absolute error assist in demonstrating the enhancement of results with resolution J increases. In addition, the utilization of present method is found to be precise, straightforward, quick, adaptable, helpful, cost-friendly and computationally attractive.

Haar wavelet technique applied on the functionally graded carbon nanotube reinforced conical shells to study free vibration and buckling behaviors in thermal environments

This article is provided to investigate the free vibrational behaviors accompanied by buckling characteristics of carbon nanotube-reinforced conical shells via the Haar wavelet technique. For a better performance in the thermal environments, materials used in the structure obey the temperature-dependent relations. On the other hand, carbon nanotubes are embedded across the thickness uniformly or functionally. Using Hamilton’s principle through the first-order shear deformation theory yields governing equations. Both dynamic and stability states are evaluated in the vicinity of equilibrium state and are also derived with regard to initial thermal stresses that are obtained via linear membrane technique in static analysis. The obtained partial differential equations are converted into ordinary ones with the aid of separating variable technique. Subsequently, the Haar wavelet approach is used to discretize the equations meridionally and transform them into new algebraic ones. Capability of this approach in calculation of frequencies with only a few numbers of the collocation points is demonstrated. Finally, to verify the integrity and precision of the proposed approach, some comparison studies are made with those of relevant results in the literature primarily. Thereafter, some effective parameters such as the geometry of nanotube, different boundary conditions, temperatures, and material properties are studied.

Second‐Order Delay Differential Equations to Deal the Experimentation of Internet of Industrial Things via Haar Wavelet Approach

In this article, an efficient numerical approach for the solution of second‐order delay differential equations to deal with the experimentation of the Internet of Industrial Things (IIoT) is presented. With the help of the Haar wavelet technique, the considered problem is transformed into a system of algebraic equations which is then solved for the required results by using Gauss elimination algorithm. Some numerical examples for convergence of the proposed technique are taken from the literature. Maximum absolute and root mean square errors are calculated for various collocation points. The results show that the Haar wavelet method is an effective method for solving delay differential equations of second order. The convergence rate is also measured for various collocation points, which is almost equal to 2.

New Numerical Treatment for a Family of Two-Dimensional Fractional Fredholm Integro-Differential Equations

In this paper, we present a robust algorithm to solve numerically a family of two-dimensional fractional integro differential equations. The Haar wavelet method is upgraded to include in its construction the Laplace transform step. This modification has proven to reduce the accumulative errors that will be obtained in case of using the regular Haar wavelet technique. Different examples are discussed to serve two goals, the methodology and the accuracy of our new approach.

Haar Wavelet Technique for Solving Fractional Differential Equations with An Application

In this article, the approxmated solutions of ordinary differential equations of fractional order using Haar wavelet and B-spline bases are introduced. The algorithm of collection method is updated using two basis. Several initial value problems has been solved to show the applicability and efficacy of the Haar wavelet and B-spline basis. An application of Lane-Eman equation has been introduced and studied. The approximated results have clearly shown the advantage and the efficiency of the modified method in terms of accuracy and computational time.

A modified wavelet method for identifying transient features in time signals with applications to bean beetle maturation

We develop an averaging method, based on a modified Haar wavelet technique, for identifying when transient features occur in a time signal. We call this method the seaweed method and use it to identify different stages of bean beetle embryo maturation. We use randomized simulations to evaluate the seaweed method for accuracy and precision at different levels of noise. Our results support the efficacy of the seaweed method as a means for analyzing time-lapse photographs of bean beetle embryos and a wide variety of other time signals.

Planetary boundary layer height from CALIOP compared to radiosonde over China

Abstract. Accurate estimation of planetary boundary layer height (PBLH) is key to air quality prediction, weather forecast, and assessment of regional climate change. The PBLH retrieval from the Cloud-Aerosol Lidar with Orthogonal Polarization (CALIOP) is expected to complement ground-based measurements due to the broad spatial coverage of satellites. In this study, CALIOP PBLHs are derived from combination of Haar wavelet and maximum variance techniques, and are further validated against PBLHs estimated from ground-based lidar at Beijing and Jinhua. Correlation coefficients between PBLHs from ground- and satellite-based lidars are 0.59 at Beijing and 0.65 at Jinhua. Also, the PBLH climatology from CALIOP and radiosonde are compiled over China during the period from 2011 to 2014. Maximum CALIOP-derived PBLH can be seen in summer as compared to lower values in other seasons. Three matchup scenarios are proposed according to the position of each radiosonde site relative to its closest CALIPSO ground tracks. For each scenario, intercomparisons were performed between CALIOP- and radiosonde-derived PBLHs, and scenario 2 is found to be better than other scenarios using difference as the criteria. In early summer afternoon over 70 % of the total radiosonde sites have PBLH values ranging from 1.6 to 2.0 km. Overall, CALIOP-derived PBLHs are well consistent with radiosonde-derived PBLHs. To our knowledge, this study is the first intercomparison of PBLH on a large scale using the radiosonde network of China, shedding important light on the data quality of initial CALIOP-derived PBLH results.

Wavelet Collocation Method for Optimal Control Problems

A Haar wavelet technique is discussed as a method for discretizing the nonlinear system equations for optimal control problems. The technique is used to transform the state and control variables into nonlinear programming (NLP) parameters at collocation points. A nonlinear programming solver can then be used to solve optimal control problems that are rather general in form. Here, general Bolza optimal control problems with state and control constraints are considered. Examples of two kinds of optimal control problems, continuous and discrete, are solved. The results are compared to those obtained by using other collocation methods.

웨이블릿 및 시스템 분할을 이용한 특이섭동 선형 시스템 해석

A Haar wavelet based numerical method for solving singularly perturbed linear time invariant system is presented in this paper. The reduced pure slow and pure fast subsystems are obtained by decoupling the singularly perturbed system and differential matrix equations are converted into algebraic Sylvester matrix equations via Haar wavelet technique. The operational matrix of integration and its inverse matrix are utilized to reduce the computational time to the solution of algebraic matrix equations. Finally a numerical example is given to demonstrate the validity and applicability of the proposed method.

Hierarchical synopses with optimal error guarantees

Hierarchical synopsis structures offer a viable alternative in terms of efficiency and flexibility in relation to traditional summarization techniques such as histograms. Previous research on such structures has mostly focused on a single model, based on the Haar wavelet decomposition. In previous work, we have introduced a more refined, wavelet-inspired hierarchical index structure for synopsis construction: the Haar + tree. The chief advantages of this structure are twofold. First, it achieves higher synopsis quality at the task of summarizing data sets with sharp discontinuities than state-of-the-art histogram and Haar wavelet techniques. Second, thanks to its search space delimitation capacity, Haar + synopsis construction operates in time linear in the size of the data set for any monotonic distributive error metric. Contemporaneous research has introduced another hierarchical synopsis structure, the compact hierarchical histogram (CHH). In this article, we elaborate on both these structures. First, we formally prove that the CHH, in its default binary-hierarchy form, is a simplified variant of a Haar + tree. We then focus on the summarization problem, with both these hierarchical synopsis structures, in which an error guarantee expressed by a maximum-error metric is required. We show that this problem is most efficiently solved through its dual, space-minimization counterpart, which can also achieve optimal quality . In this case, there is a benefit to be gained by specializing the algorithm for each structure; hence, our algorithm for optimal-quality maximum-error CHH requires low polynomial time; on the other hand, optimal-quality Haar + synopses for maximum-error metrics are constructed in exponential time; hence, we also develop a low-polynomial-time approximation scheme for the maximum-error Haar + case. Furthermore, we extend our approach for both general-error and maximum-error Haar + synopses to arbitrary dimensionality. In our experimental study, (i) we confirm the theoretically expected superiority of Haar + synopses over Haar wavelet methods in both construction time and achieved quality for representative error metrics; (ii) we demonstrate that Haar + synopses are also constructed faster than optimal plain histograms, and, moreover, achieve higher synopsis quality with highly discontinuous data sets; such an advantage of a hierarchical synopsis structure over a histogram had been intuitively expressed, but never experimentally verified; and (iii) we show that Haar + synopsis quality supersedes that of a CHH.

Numerical solution of differential equations using Haar wavelets

Haar wavelet techniques for the solution of ODE and PDE is discussed. Based on the Chen–Hsiao method [C.F. Chen, C.H. Hsiao, Haar wavelet method for solving lumped and distributed-parameter systems, IEE Proc.—Control Theory Appl. 144 (1997) 87–94; C.F. Chen, C.H. Hsiao, Wavelet approach to optimising dynamic systems, IEE Proc. Control Theory Appl. 146 (1997) 213–219] a new approach—the segmentation method—is developed. Five test problems are solved. The results are compared with the result obtained by the Chen–Hsiao method and with the method of piecewise constant approximation [C.H. Hsiao, W.J. Wang, Haar wavelet approach to nonlinear stiff systems, Math. Comput. Simulat. 57 (2001) 347–353; S. Goedecker, O. Ivanov, Solution of multiscale partial differential equations using wavelets, Comput. Phys. 12 (1998) 548–555].