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In this paper, we develop a Haar wavelet-based reconstruction method to recover missing data on an inaccessible part of the boundary from measured data on another accessible part. The technique is developed to solve inverse Cauchy problems governed by the Poisson equation which is severely ill-posed. The new method is mathematically simple to implement and can be easily applied to Cauchy problems governed by other partial differential equations appearing in various fields of natural science, engineering, and economics. To take into account the ill-conditioning of the obtained linear system due to the ill-posedness of the Cauchy problem, a preconditioning strategy combined with a regularization has been developed. Comparing the numerical results produced by a meshless method based on the polynomial expansion with those produced by the proposed technique illustrates the superiority of the latter. Other numerical results show its effectiveness. [ABSTRACT FROM AUTHOR]

In this study, we introduce an algorithm that utilizes the Haar wavelet collocation method to solve the time-dependent Emden–Fowler equation. This proposed method effectively addresses both linear and nonlinear partial differential equations. It is a numerical technique where the differential equation is discretized using Haar basis functions. A difference scheme is also applied to approximate the time derivative. By leveraging Haar functions and the difference scheme, we form a system of equations, which is then solved for Haar coefficients using MATLAB software. The effectiveness of this technique is demonstrated through various examples. Numerical simulations are performed, and the results are presented in graphical and tabular formats. We also provide a convergence analysis and an error analysis for this method. Furthermore, approximate solutions are compared with those obtained from other methods to highlight the accuracy, efficiency, and computational convenience of this technique. [ABSTRACT FROM AUTHOR]

Throughout this research paper, we represent a highly effective Haar wavelet technique to determine the solution of the complex nonlinear dynamical system with three variables chemical reaction model. The foremost objective of this study is to represent the dynamical behavior of chemical reaction model in the sense of Caputo derivative. The convergent analysis and stability analysis of the three variable chemical reaction model are discussed. The existence and uniqueness of the given model is also verified. Furthermore, the residual error analysis for this model is also presented. In addition, graphically the numerical solutions in a 2-dimensional and 3-dimensional manner are obtained by using MATLAB (2023a). [ABSTRACT FROM AUTHOR]

This research explores the dynamic behaviour of horn-shaped single-walled carbon nanotubes (HS-SWCNTs) conveying viscous nanofluid with pulsating the influence of a longitudinal magnetic field. The analysis utilizes Euler–Bernoulli beam model, considering the variable cross section, and incorporating Eringen's nonlocal theory to formulate the governing partial differential equation of motion. The instability domain of HS-SWCNTs is estimated using Galerkin's approach. Numerical analysis is performed using the Haar wavelet method. The critical buckling load obtained in this study is compared with previous research to validate the proposed model. The results highlight the effectiveness of the proposed model in assessing the vibrational characteristics of a complex multi-physics system involving HS-SWCNTs. Dispersion graphs and tables are presented to visualize the numerical findings pertaining to various system parameters, including the nonlocal parameter, magnetic flux, Knudsen number, and viscous factor. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Purpose: In this paper, the free vibration characteristics of a coupled functionally graded beam system(CFGBS) with arbitrary boundary conditions are presented based on the first-order shear deformation beam theory(FSDBT) and Haar wavelet method (HWM) for the first time. Method: A coupled beam system made of functionally graded material (FGM) consist of two straight beams and one curved beam, i.e., straight-curved-straight beam. For modeling the general boundary conditions, the artificial spring technique is employed. The individual beams of the coupled beam system are connected by continuous condition using the artificial spring technique. The displacement fields are derived based on the FSDBT and the governing equation of beam system is obtained by applying the Hamilton's principle. All displacement functions of the CFGBS are expanded by the Haar wavelet series and its integrals, and a discretized characteristic equation is obtained. Results: The accuracy and reliability of the proposed method for vibration characteristics analysis of the CFGBS considered in this paper are confirmed, and the new vibration characteristics results of the CFGBS with various geometric dimensions, boundary conditions, and material parameters are presented through numerical examples. These new numerical results can be used as benchmark data for research in this field. [ABSTRACT FROM AUTHOR]

The present study introduces the Haar wavelet method, which utilizes collocation points to approximate solutions to the Emden-Fowler Pantograph delay differential equations (PDDEs) of general order. This semi-analytic method requires the transformation of the original differential equation into a system of nonlinear differential equations, which is then solved to determine the Haar coefficients. The method’s application to fourth-, fifth-, and sixth-order PDDEs is discussed, along with an examination of convergence that involves the determination of an upper bound and the formulation of the rate of convergence for the method. Numerical simulations and error tables are presented to demonstrate the effectiveness and precision of this approach. The error tables clearly illustrate that the method’s accuracy improves progressively with increasing resolution.

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.

Longitudinal wave propagation in axially graded nanotubes was explored. The effect of shear deformation and lateral inertia on nanorods was considered using the nonlocal Raylegh–Bishop rod theory. As a novel approach, a nonlocal parameter was assumed in the graded formulation. The higher order Haar wavelet method was utilized for solving the governing equation of motion. The effects of material grading power-law index and nonlocal parameters on the longitudinal wave response of axially graded nanorods were investigated. Phase and group velocity variations of the axially graded nanorod were obtained. The present study may be useful in the modeling of advanced functional composite nanowires. [ABSTRACT FROM AUTHOR]

In this paper, firstly, the “ Haar wavelet method ” is used to give approximate solutions for coupled systems of linear fractional Fredholm integro-differential equations. Moreover, we consider the fractional derivative to be described in the Caputo sense. The general idea of this technique is simply based on reducing this kinds of coupled systems into systems of algebraic equations which are easily to deal with and solve. Also, Laplace transform operator is included to develop a sophisticated approach which we called “ Laplace Haar wavelet method ” as an adjustment to “ Haar wavelet method ” to reduce the error and computational time. We provide illustrative examples to confirm validity, efficiency, accuracy, and applicability of the proposed methods.

This study presents a robust and easy-to-implement stable numerical method based on Haar wavelets for the coupled degenerate reaction-diffusion partial differential equations having a nonlinear source with homogeneous Neumann boundary data and interacting with a system of nonlinear ODEs. The simple requirement of one-time computation of the associated Haar matrices and the Haar integral matrices during the entire calculation of the numerical solution makes the proposed method both cost-effective and simple. By projecting the higher-order derivatives into the Haar wavelet space the method effectively uses a discontinuous basis for solving the degenerate system. The Lax equivalence criterion is followed to prove the convergence of the proposed Haar wavelet method. The method is exponentially convergent and is directly extendable to the higher dimensions. All the quadratures in the proposed method are done symbolically without any approximation error. Multidimensional numerical examples, including a few models of medical significance dealing with cardiac electrical activity, are considered to illustrate the good performance of the proposed method. It is shown that solving the degenerate model equations using the proposed method is easy to implement and cost-effective while moving to higher dimensions in comparison to the finite element method. The problem with discontinuity in the parameter of the model, which is relevant to the case solving electrical activity problem in the presence of ischemia in cardiac tissue, is also solved using the proposed method. [ABSTRACT FROM AUTHOR]

Free magneto elastic waves in a pulsating viscous fluid conveying nanotube under the purview of Euler–Bernoulli beam model is addressed in this paper. The partial differential form of governing model is adopted via Eringen’s nonlocal theory. Galerkin’s method has been taken for instability and pulsation domains and Haar wavelet method (HWM) is considered for numerical evaluation. The dispersion graphs and tables are given for nonlocal value, magnetic field flux, Knudsen number and viscous parameter. The pulsation frequency and resolution output have a significant role over the physical variables. Critical buckling load is calculated and validated with the literature and received reasonable agreement.

In the present work, a two‐dimensional Haar wavelet method is proposed to study the sensitivities of the price of an option. The method is appropriate to study these sensitivities as it explicitly gives the values of all the derivatives of the solution. A Black–Scholes model for European style options is considered to analyze the physical and numerical aspects of the put and the call option Greeks. We use the concept of coordinate transformation to make the Black–Scholes equation dimensionless and to resolve the obstacle in approximating the Greeks having non‐smoothness at the strike price. The infinite spatial domain is truncated into the finite domain to avoid large truncation errors. Through rigorous analysis, the method is shown first‐order accurate in the L2−norm. The numerical computations performed to approximate the option price and various Greeks, like delta, theta, gamma, and so forth, confirm the theoretical results in L2−norm. The relative errors and the maximum absolute errors are also presented. The motivational work of option Greeks analysis may leave a significant impact on financial institutes; it helps them to manage the risk by setting the portfolio's new value and to estimate the probability of losing money. [ABSTRACT FROM AUTHOR]

Recently, the two-dimensional elliptic singularly perturbed boundary value problems have received attention. These problems have not been much explored numerically. A highly accurate numerical scheme on different non-uniform meshes is suggested to solve such problems. In particular, the Haar wavelet method on a special type of non-uniform mesh and Shishkin mesh is proposed; because of the use of the block pulse function, it is easy to derive and handle the operator matrices. Also, it has been shown that Shiskin mesh provides better results. The use of the piecewise-uniform Shishkin mesh with the Haar wavelet scheme contains a novelty in itself. Through rigorous analysis, the method is shown as first-order convergent in L 2 -norm. The theoretical results are confirmed by computational results obtained in the maximum norm and L 2 -norm for two test problems. From the comparative results provided in tables, it is worth noting that the Shishkin mesh is an excellent choice to solve these problems instead of a particular type of non-uniform mesh for the proposed scheme. This can be justified because the grid points defined by the Shishkin mesh are adequately distributed in the layer and non-layer regions. [ABSTRACT FROM AUTHOR]

In this article, we consider an inverse problem related to the damped generalized regularized long wave equation with noisy data. We study numerically this problem by applying the quartic B-spline and Haar wavelet methods combined with the Tikhonov regularization method. We investigate the stability and the convergence analysis of this problem and show that O (k 2 + h 3) and O (1 M) are the convergence rates of the quartic B-spline method and the Haar wavelet method, respectively. The numerical results, by a comparative study, show that an excellent estimation of the unknown functions of the mentioned inverse problem. [ABSTRACT FROM AUTHOR]

In this paper, a Haar wavelet discretization method (HWDM) for analyzing the free vibration of the multi-stepped functionally graded (FG) curved beam with general boundary conditions is presented. It is assumed that the material properties of the beam change continuously in the thickness direction according to the distribution characteristics of the material determined by four parameters. The individual steps of the multi-stepped functionally graded curved beam are combined by a continuous condition. For generalization of boundary conditions, the artificial elastic spring technique is introduced. The displacement fields are determined by the first-order shear deformation theory (FSDT) and the motion equation is obtained by using Hamilton's principle. All displacement functions including boundary and continuity conditions are expressed in the form of a Haar wavelet series and its integral, and a characteristic equation discretized based on the Haar wavelet is obtained. The reliability and accuracy of the proposed method are confirmed through convergence and validation studies. The results indicate that this method has high accuracy and reliability, also a higher convergence rate in attaining the frequencies of the multi-stepped FG curved beam. Then, the frequency parameters and mode types of the multi-stepped FG curved beam obtained by the proposed method are given with numerical examples. These new numerical results can be used as benchmark data for research in this field. [ABSTRACT FROM AUTHOR]

In this paper, the free vibration characteristics of functionally graded double-beam system (FGDBS) elastically connected by an elastic layer with uniform elastic stiffness under arbitrary boundary conditions are investigated. It is assumption that the material distribution properties of the individual beams of FGDBS depends on different four-parameters. Timoshenko beam theory in which the effects of shear deformation and rotational inertia are considered is utilized to model the free vibration of FGDBS and displacement components can be obtained from the Haar wavelet series and their integral. Hamilton’s principle is applied to construct coupled governing equations and the virtual spring boundary technique is applied to generalize boundary conditions at four ends of FGDBS. The convergency and accuracy of the proposed method can be confirmed through the comparison results with previous literature and finite element method (FEM). A lot of new results have been proposed, such as the frequency characteristics of FGDBS under arbitrary boundary conditions which can be provided as reference data for future research.

The higher order Haar wavelet method (HOHWM) is used with a nonuniform grid to solve nonlinear partial differential equations numerically. The Burgers’ equation, the Korteweg–de Vries equation, the modified Korteweg–de Vries equation and the sine–Gordon equation are used as model equations. Adaptive as well as nonadaptive nonuniform grids are developed and used to solve the model equations numerically. The numerical results are compared to the known analytical solutions as well as to the numerical solutions obtained by application of the HOHWM on a uniform grid. The proposed methods of using nonuniform grid are shown to significantly increase the accuracy of the HOHWM at the same number of grid points.

In this article, we present a highly-accurate wavelet-based approximation to study and analyze the physical and numerical aspects of two-parameter singularly perturbed problems with Robin boundary conditions. To explore the swiftly changing behavior of such problems, we have used a special type of non-uniform mesh known as Shishkin mesh. Using Shishkin mesh with the Haar wavelet scheme contains a novelty in itself. We comprehensively explain an approach to solve the Robin boundary conditions involving the proposed Haar wavelet scheme. Through rigorous analysis, the order of convergence of the present scheme is shown quadratic and linear in the spatial and temporal directions, respectively. The robustness and proficiency of the contributed scheme are conclusively demonstrated with three test examples. Irrespective of the problem's geometry, the proposed method is highly accurate and very economical. [ABSTRACT FROM AUTHOR]

The essential objective of this work is to implement Adam Bashforth’s Moulton (ABM) and Haar wavelet method (HWM) to solve fractional chemical kinetics and another problem that relates the condensations of carbon dioxide (CO2) and phenyl glycidyl ether (PGE) with two variety of Drichlet and a mixed set of Neumann boundary and Drichlet type conditions respectively. We have been solved the above system of differential equations by Adam Bashforth’s Moulton and Haar wavelet operational method where this technique is to convert the system of differential equations into the system of algebraic equation which can be solved easily. This work is expects to contribute the vast advantage of Haar wavelets in chemical science. The Adam Bashforth’s Moulton and Haar wavelet method is impressive and convenient for obtaining numerical solutions of chemical engineering type problems. A complete agreement is acheived between Adam Bashforth’s Moulton solution and Haar wavelet solution. To manifest about the performance and applicability of the method, two test examples are deliberated.

This study uses the technique of incognito data and information into a container in the form of images combined with the addition of the password by using the method of Least Significant Bit (LSB) and the technique of Haar Wavelet. Testing the system by sending a message in the form of a text file and an image file with the process of the original image are transformed to wear haar wavelet divided into four zones of frequency, namely LL, LH, HL, and HH. Where Bitbit Readings are planted in a zone LL and attempted insertion of the bit of the file reading into the last bit in each byte of the image file and can show you back the results of the message. The research results, i.e., images processed by the LSB and Haar Wavelet method, do not change the file size, resolution, dpi, and physical form image. The advantage of this method is very simple computing, oriented computers, which need less space to store and are time-efficient. [ABSTRACT FROM AUTHOR]

The higher order Haar wavelet method (HOHWM) is used with a nonuniform grid to solve nonlinear partial differential equations numerically. The Burgers' equation, the Korteweg-de Vries equation, the modified Korteweg-de Vries equation and the sine-Gordon equation are used as model equations. Adaptive as well as nonadaptive nonuniform grids are developed and used to solve the model equations numerically. The numerical results are compared to the known analytical solutions as well as to the numerical solutions obtained by application of the HOHWM on a uniform grid. The proposed methods of using nonuniform grid are shown to significantly increase the accuracy of the HOHWM at the same number of grid points. [ABSTRACT FROM AUTHOR]

In this paper, we present a numerical method based on Haar wavelets for solving multidimensional parabolic inverse problem with a control parameter. In this method, the highest derivative appearing in the parabolic equation is expanded into Haar wavelet series. Taylor series expansion and cubic spline interpolation are used to approximate the control parameter. We have computed Haar matrices and Haar integral matrices only once and used the same for different time iterations which lead to a significant reduction in the computational cost. Numerical experiments are performed on some test problems. It is found that numerical results are in good agreement with the exact solution. [ABSTRACT FROM AUTHOR]

In this paper, two numerical techniques are presented to solve the nonlinear inverse generalized Benjamin–Bona–Mahony–Burgers equation using noisy data. These two methods are the quartic B-spline and Haar wavelet methods combined with the Tikhonov regularization method. We show that the convergence rate of the proposed methods is O (k 2 + h 3) and O 1 M for the quartic B-spline and Haar wavelet method, respectively. In fact, this work considers a comparative study between quartic B-spline and Haar wavelet methods to solve some nonlinear inverse problems. Results show that an excellent estimation of the unknown functions of the nonlinear inverse problem has been obtained. [ABSTRACT FROM AUTHOR]

In this research, we presented two classical numerical techniques to solve some oscillatory problems arised in various applications of sciences and engineering. Adomian decomposition method (ADM) and Haar wavelet method (HWM) are utilized for this purpose. Some numerical examples have been performed to illustrate the accuracy of the present methods. [ABSTRACT FROM AUTHOR]

The Lotka‐Volterra (LV) system is an interesting mathematical model because of its significant and wide applications in biological sciences and ecology. A fractional LV model in the Caputo sense is investigated in this paper. Namely, we provide a comparative study of the considered model using Haar wavelet and Adams‐Bashforth‐Moulton methods. For the first method, the Haar wavelet operational matrix of the fractional order integration is derived and used to solve the fractional LV model. The main characteristic of the operational method is to convert the considered model into an algebraic equation which is easy to solve. To demonstrate the efficiency and accuracy of the proposed methods, some numerical tests are provided. [ABSTRACT FROM AUTHOR]

The current study is focused on development and adaption of the higher order Haar wavelet method for solving nonlinear ordinary differential equations. The proposed approach is implemented on two sample problems—the Riccati and the Liénard equations. The convergence and accuracy of the proposed higher order Haar wavelet method are compared with the widely used Haar wavelet method. The comparison of numerical results with exact solutions is performed. The complexity issues of the higher order Haar wavelet method are discussed.

In this study, the thermal performance analysis of porous fin with temperature-dependent thermal conductivity and internal heat generation is carried out using Haar wavelet collocation method. The effects of various parameters on the thermal characteristics of the porous fin are investigated. It is found that as the porosity increases, the rate of heat transfer from the fin increases and the thermal performance of the porous fin increases. The numerical solutions by the Haar wavelet collocation method are in good agreement with the standard numerical solutions.

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.

We present here the numerical solution of damped forced oscillator problem using Haar wavelet and compare the numerical results obtained with some well-known numerical methods such as Runge-Kutta fourth order classical and Taylor Series methods. Numerical results show that the present Haar wavelet method gives more accurate approximations than above said numerical methods.

A new higher order Haar wavelet method (HOHWM) has been developed for solving differential and integro-differential equations. Generalized approach has been proposed for wavelet expansion allowing improvement of the accuracy and the rate of convergence of the solution. The sample problem considered shows, that applying the approach proposed allows to improve the order of convergence of the HWM from two to four and to reduce the absolute error by several orders of magnitude (depending on mesh). Furthermore, in the case of sample problem considered, the computational and implementation complexities are kept in the same range with widely used HWM. [ABSTRACT FROM AUTHOR]

In this paper, the Haar wavelet method is employed to analyze the parametric instability of truncated conical shells under static and time dependent periodic axial loads. The present work is based on the Love first-approximation theory for classical thin shells. The displacement field is expressed as the Haar wavelet series in the axial direction and trigonometric functions in the circumferential direction. Then the partial differential equations are reduced into a system of coupled Mathieu-type ordinary differential equations describing dynamic instability behavior of the shell. Using Bolotin’s method, the first-order and second-order approximations of principal instability regions are determined. The correctness of present method is examined by comparing the results with those in the literature and very good agreement is observed. The difference between the first-order and second-order approximations of principal instability regions for tensile and compressive loads is also investigated. Finally, numerical results are presented to bring out the influences of various parameters like static load factors, boundary conditions and shell geometrical characteristics on the domains of parametric instability of conical shells. [ABSTRACT FROM AUTHOR]

In this article, an efficient hybrid method has been developed for solving some special type of nonlinear partial differential equations. Hybrid method is based on tanh-coth method, quasilinearization technique and Haar wavelet method. Nonlinear partial differential equations have been converted into a nonlinear ordinary differential equation by choosing some suitable variable transformations. Quasilinearization technique is used to linearize the nonlinear ordinary differential equation and then the Haar wavelet method is applied to linearized ordinary differential equation. A tanh-coth method has been used to obtain the exact solutions of nonlinear ordinary differential equations. It is easier to handle nonlinear ordinary differential equations in comparison to nonlinear partial differential equations. A distinct feature of the proposed method is their simple applicability in a variety of two and three-dimensional nonlinear partial differential equations. Numerical examples show better accuracy of the proposed method as compared with the methods described in past. Error analysis and stability of the proposed method have been discussed. [ABSTRACT FROM AUTHOR]

In this paper, we apply Haar wavelet method for solving seconed order initial value problems of ordinary differential equation. The fundamental idea of Haar wavelet method is to convert the differential equations into a group of algebraic equations, which involves a finite number or variables. We compared this numerical results with the exact solution and other known methods. [ABSTRACT FROM AUTHOR]

The current study focuses on the evaluation of the Haar wavelet method, i.e. its comparison with widely used strong formulation based methods (FDM-finite difference method and DQM-differential quadrature method). A solid element 3D finite element model is developed and the numerical results obtained by using simplified approaches are confirmed. [ABSTRACT FROM AUTHOR]

In this paper, numerical solutions of one dimensional coupled KdV equation has been investigated by Haar Wavelet method. Time derivatives given in this equation are discretized by finite differences and nonlinear terms appearing in the equations are linearized by some linearization techniques and space derivatives are discretized by Haar wavelets. For examining performance of the proposed method, single soliton solution and conserved quantities of some test problems are used. Also error analysis of numerical scheme is investigated and numerical results are compared with some results already existing in the literature. [ABSTRACT FROM AUTHOR]

We present, a collocation method based on Haar wavelet and Kronecker tensor product for solving three-dimensional partial differential equations. The method is based on approximating a sixth-order mixed derivative by a series of Haar wavelet basis functions. The present method is suitable for numerical solution of all kinds of three-dimensional Poisson and Helmholtz equations. Numerical examples are solving to establish the efficiency and accuracy of the present method. Numerical results obtained are better as compared to numerical results obtained in past. [ABSTRACT FROM AUTHOR]

A mathematical model of the transient free-surface model of unsteady ship motions is discussed. In this paper, the two reliable wavelet methods are proposed for the numerical solutions of the transient free-surface model. Mathematical derivations show that in fact the horizontal and vertical derivatives can be expressed by TFSGF. Additionally, the wavelet-based computational methods have been applied to solve the radiation problem of a floating hemisphere at zero speed. The approximation results of hydrodynamic coefficients show good agreement with the analytical solutions. Some numerical examples are presented to demonstrate the validity and applicability of the proposed method. [ABSTRACT FROM AUTHOR]