Haar Wavelet Collocation Method Research Articles

A New Solution for The Enzymatic Glucose Fuel Cell Model with Morrison Equation via Haar Wavelet Collocation Method

Integral equations are essential tools in various areas of applied mathematics. A computational approach to solving an integral equation is important in scientific research. The Haar wavelet collocation method (HWCM) with operational matrices of integration is one famous method which has been applied to solve systems of linear integral equations. In this paper, an approximated analytical method based on the Haar wavelet collocation method is applied to the system of diffusion convection partial differential equations with initial and boundary conditions. This system determines the enzymatic glucose fuel cell with the chemical reaction rate of the Morrison equation. The enzymatic glucose fuel cell model describes the concentration of glucose and hydrogen ion that can be converted into energy. During the process, the model reduces to the linear integral equation system including computational Haar matrices. The computational Haar matrices can be computed by HWCM coding in the Maple program. Illustrated examples are provided to demonstrate the preciseness and effectiveness of the proposed method. The results are shown as numerical solutions of glucose and hydrogen ion.

Treatment of Singularly Perturbed Differential Equations with Delay and Shift Using Haar Wavelet Collocation Method

An efficient Haar wavelet collocation method is proposed for the numerical solution of singularly perturbed differential equations with delay and shift. Taylor series (upto the first order) is used to convert the problem with delay and shift into a new problem without the delay and shift and then solved by Haar wavelet collocation method, which reduces the time and complexity of the system. Further, we apply the Haar wavelet collocation method directly to solve the problems. Also, we demonstrated several test examples to show the accuracy and efficiency of the Haar wavelet collocation method and compared our results with the finite difference and fitted operator finite difference method [11], [28].

Numerical investigation of dynamic Euler-Bernoulli equation via 3-Scale Haar wavelet collocation method

In this study, we analyze the performance of a numerical scheme based on 3-scale Haar wavelets for dynamic Euler-Bernoulli equation, which is a fourth order time dependent partial differential equation. This type of equations governs the behaviour of a vibrating beam and have many applications in elasticity. For its solution, we first rewrite the fourth order time dependent partial differential equation as a system of partial differential equations by introducing a new variable, and then use finite difference approximations to discretize in time, as well as 3-scale Haar wavelets to discretize in space. By doing so, we obtain a system of algebraic equations whose solution gives wavelet coefficients for constructing the numerical solution of the partial differential equation. To test the accuracy and reliability of the numerical scheme based on 3-scale Haar wavelets, we apply it to five test problems including variable and constant coefficient, as well as homogeneous and non-homogeneous partial differential equations. The obtained results are compared wherever possible with those from previous studies. Numerical results are tabulated and depicted graphically. In the applications of the proposed method, we achieve high accuracy even with small number of collocation points.

A reliable algorithm to compute the approximate solution of KdV-type partial differential equations of order seven.

The approximate solution of KdV-type partial differential equations of order seven is presented. The algorithm based on one-dimensional Haar wavelet collocation method is adapted for this purpose. One-dimensional Haar wavelet collocation method is verified on Lax equation, Sawada-Kotera-Ito equation and Kaup-Kuperschmidt equation of order seven. The approximated results are displayed by means of tables (consisting point wise errors and maximum absolute errors) to measure the accuracy and proficiency of the scheme in a few number of grid points. Moreover, the approximate solutions and exact solutions are compared graphically, that represent a close match between the two solutions and confirm the adequate behavior of the proposed method.

Haar wavelet collocation method for solving the system of linear Fredholm integral equations with constant coefficients

In this paper, we propose different procedure to solve the problem of linear system of Fredholm integral equations with constant coefficients. The proposed method gives the rapidly convergent successive approximation to the analytic solution. The reliability and efficiency of the method are illustrated by investigating the convergence results for some problems. In fact, comparisons are made between the contributed scheme and the generated results.

A new numerical scheme based on Haar wavelets for the numerical solution of the Chen–Lee–Liu equation

This work based on a new numerical scheme for the numerical solutions of the Chen–Lee–Liu equation using Haar wavelet collocation method. In this work, we combined the backward Euler difference (BED) formula with the Haar wavelet collocation method (HWCM). The BED formula approximates the time derivative of the Chen–Lee–Liu equation, while the HWCM approximates the space derivatives of the Chen–Lee–Liu equation, consequently converting it into a finite system of algebraic equations. Furthermore, we validate the accuracy of the proposed method numerically and graphically with the help of several examples. Moreover, we have also presented an error analysis to prove the convergence of the proposed method.

Efficient Numerical Scheme for the Solution of Tenth Order Boundary Value Problems by the Haar Wavelet Method

In this paper, an accurate and fast algorithm is developed for the solution of tenth order boundary value problems. The Haar wavelet collocation method is applied to both linear and nonlinear boundary value problems. In this technqiue, the tenth order derivative in boundary value problem is approximated using Haar functions and the process of integration is used to obtain the expression of lower order derivatives and approximate solution for the unknown function. Three linear and two nonlinear examples are taken from literature for checking validation and the convergence of the proposed technique. The maximum absolute and root mean square errors are compared with the exact solution at different collocation and Gauss points. The experimental rate of convergence using different number of collocation points is also calculated, which is nearly equal to 2.

Numerical Solution of Nonlinear Fifth-Order KdV-Type Partial Differential Equations via Haar Wavelet

In this research article, the numerical solution of different forms of widely used one-dimensional Korteweg–de Vries equation is discussed. For this purpose, Haar wavelet collocation method is implemented. A simple algorithm is constructed which is based on the proposed method. The presented method is tested on fifth-order Lax equation, Sawada–Kotera equation, Caudrey–Dodd–Gibbon equation, Kaup–Kuperschmidt equation and Ito equation. The obtained approximate results are displayed using tables and figures. The numerical results show good accuracy of the proposed method.

A new approach for solving one-dimensional fractional boundary value problems via Haar wavelet collocation method

In this paper, we propose the numerical method for positive solutions to the m-point boundary value problems of fractional differential equations with p-Laplacian operator and prove the convergence for our scheme. The operational matrix of fractional integration, based on the accurate calculation of fractional integrals of Haar wavelet functions, is combined with the collocation method and the fixed point iterative method to introduce the function approximation and obtain the approximate solution to the given problem. Also, we present some numerical examples to illustrate our main results.

Solution of singularly perturbed differential difference equations and convection delayed dominated diffusion equations using Haar wavelet

In this paper, we apply Haar wavelet collocation method to solve the linear and nonlinear second-order singularly perturbed differential difference equations and singularly perturbed convection delayed dominated diffusion equations, arising in various modeling of chemical processes. First, we transform delay term by using Taylor expansion and then apply Haar wavelet method. To show the robustness, accuracy and efficiency of the method, three problems of second-order singularly perturbed differential difference equations and three problems of convection delayed dominated diffusion equations have been solved. Also, results are compared with the exact solution of the problems and methods existing in the literature, which confirms the superiority of the Haar wavelet collocation method. We obtained accurate numerical solution of problems by increasing the level of resolutions.

Collocation method for solving nonlinear fractional optimal control problems by using Hermite scaling function with error estimates

SummaryThis article presents an efficient numerical method for solving fractional optimal control problems (FOCPs) by utilizing the Hermite scaling function operational matrix of fractional‐order integration. The proposed technique is applied to transform the state and control variables into nonlinear programming (NLP) parameters at collocation points. The NLP solver is then used to solve FOCP. Furthermore, the L2‐error estimates in the approximation of unknown variables and the approximation of block pulse operational matrix of fractional‐order integration are derived and illustrative examples are included to demonstrate the applicability of the proposed method. Moreover, the results are compared with the Haar wavelet collocation method, hybrid of block‐pulse and Taylor polynomials method, Bernstein polynomials method, and the Boubaker hybrid function method to show the superiority of the proposed method.

Efficient sustainable algorithm for numerical solutions of systems of fractional order differential equations by Haar wavelet collocation method

This manuscript deals a numerical technique based on Haar wavelet collocation which is developed for the approximate solution of some systems of linear and nonlinear fractional order differential equations (FODEs). Based on these techniques, we find the numerical solution to various systems of FODEs. We compare the obtain solution with the exact solution of the considered problems at integer orders. Also, we compute the maximum absolute error to demonstrate the efficiency and accuracy of the proposed method. For the illustration of our results we provide four test examples. The experimental rates of convergence for different number of collocation point is calculated which is approximately equal to 2. Fractional derivative is defined in the Caputo sense.

COMPARATIVE STUDY OF WAVELET METHODS FOR SOLVING BERNOULLI’S EQUATIONS

A comparative study of two numerical techniques is presented for solving nonlinear differential equations of the Bernoulli’s type. Proposed techniques are based on the conversion of nonlinear differential equations into linear differential equations by substituting particular factor and utilization of Haar wavelet collocation method (HWCM) and Hermite wavelet collocation method (HeWCM) to these linear equations. Searching for numerical solutions of such equations has attracted a considerable amount of research work where computer symbolic systems facilitate the computational work.

Haar Wavelet Collocation Method for Solving the System of Linear Fredholm Integral Equations with Constant Coefficients

In this paper, we propose different procedure to solve the problem of linear system of Fredholm integral equations with constant coefficients. The proposed method gives the rapidly convergent successive approximation to the analytic solution. The reliability and efficiency of the method are illustrated by investigating the convergence results for some problems. In fact, comparisons are made between the contributed scheme and the generated results.

Haar wavelet collocation method for linear first order stiff differential equations

In general, there are countless types of problems encountered from different disciplines that can be represented by differential equations. These problems can be solved analytically in simpler cases; however, computational procedures are required for more complicated cases. Right at this point, the wavelet-based methods have been using to compute these kinds of equations in a more effective way. The Haar Wavelet is one of the appropriate methods that belongs to the wavelet family using to solve stiff ordinary differential equations (ODEs). In this study, The Haar Wavelet method is applied to stiff differential problems in order to demonstrate the accuracy and efficacy of this method by comparing the exact solutions. In comparison, similar to the exact solutions, the Haar wavelet method gives adequate results to stiff differential problems.

Haar wavelet approximation for the solution of cubic nonlinear Schrodinger equations

In this study, Haar wavelet collocation method is used for the numerical solution of 1D and 2D cubic nonlinear Schrodinger equations with initial and Dirichlet boundary conditions. The space derivatives are estimated through Haar wavelet collocation method whereas for time derivative we have used Crank–Nicolson scheme. The proposed method is implemented upon several test problems and the numerical results of these test problems establish that the proposed method is accurate.

A novel model for the contamination of a system of three artificial lakes

<p style='text-indent:20px;'>In this study, a new model has been developed to monitor the contamination in connected three lakes. The model has been motivated by two biological models, i.e. cell compartment model and lake pollution model. Haar wavelet collocation method has been proposed for the numerical solutions of the model containing a system of three linear differential equations. In addition to the solutions of the system, convergence analysis has been briefly given for the proposed method. The contamination in each lake has been investigated by considering three different pollutant input cases, namely impulse imposed pollutant source, exponentially decaying imposed pollutant source, and periodic imposed pollutant source. Each case has been illustrated with a numerical example and results are compared with the exact ones. Regarding the results in each case it has been seen that, Haar wavelet collocation method is an efficient algorithm to monitor the contamination of a system of lakes problem.

Haar wavelet collocation approach for Lane-Emden equations arising in mathematical physics and astrophysics

In this paper, we propose the Haar wavelet collocation method for the numerical solution of the Lane-Emden equation with boundary conditions. To overcome the singular behavior at the origin, we first transform the Lane-Emden equation into an integral equation. The Haar wavelet collocation method is utilized to reduce the integral equation into a system of algebraic equations in the unknown expansion coefficients, then Newton’s iterative process is employed for numerical solutions. We also discuss the convergence and error analysis of the technique. Six singular problems are presented to demonstrate the accuracy and applicability of the method, including; thermal explosion, oxygen-diffusion in a spherical cell, heat conduction through a solid with heat generation and shallow membrane caps problems. The numerical results are compared with existing numerical and exact solutions. The use of the Haar wavelet is found to be accurate, fast, flexible and convenient.

Haar wavelet collocation method for solving singular and nonlinear fractional time-dependent Emden\u2013Fowler equations with initial and boundary conditions

In this paper, we have applied an iterative method to the singular and nonlinear fractional partial differential of Emden–Fowler equations types. Haar wavelets operational matrix of fractional integration will be used to solve the problem with the Picard technique. The singular equations turn to Sylvester equations that will be solved so that numerically solvable is very cost- effective. Moreover, the proposed technique is reliable enough to overcome the difficulty of the singular point at x = 0. Numerical examples are providing to illustrate the efficiency and accuracy of the technique.

A simple algorithm for numerical solution of nonlinear parabolic partial differential equations

In this paper, numerical solution of nonlinear two-dimensional parabolic partial differential equations with initial and Dirichlet boundary conditions is considered. The time derivative is approximated using finite difference scheme whereas space derivatives are approximated using Haar wavelet collocation method. The proposed method is developed for semilinear and quasilinear cases, however, it can easily be extended to other types of nonlinearities as well. The proposed method is also illustrated for nonlinear heat equation and Burgers’ equation. The proposed method is implemented upon five test problems and the numerical results are shown using tables and figures. The numerical results validate the accuracy and efficiency of the proposed method.