Application of Haar wavelet method to eigenvalue problems of high order differential equations

In this work, we present a computational method for solving eigenvalue problems of high-order ordinary differential equations which based on the use of Haar wavelets. The variable and their derivatives in the governing equations are represented by Haar function and their integral. The first transform the spectral coefficients into the nodal variable values. The second, solve the obtained system of algebraic equation. The efficiency of the method is demonstrated by four numerical examples.     

Numerical solution of nonlinear Fredholm integral equations of the second kind using Haar wavelets

In this work, we present a computational method for solving nonlinear Fredholm integral equations of the second kind which is based on the use of Haar wavelets. Error analysis is worked out that shows efficiency of the method. Finally, we also give some numerical examples.     

Haar wavelet solutions of nonlinear oscillator equations

In this paper, we present a numerical scheme using uniform Haar wavelet approximation and quasilinearization process for solving some nonlinear oscillator equations. In our proposed work, quasilinearization technique is first applied through Haar wavelets to convert a nonlinear differential equation into a set of linear algebraic equations. Finally, to demonstrate the validity of the proposed method, it has been applied on three type of nonlinear oscillators namely Duffing, Van der Pol, and Duffing–van der Pol. The obtained responses are presented graphically and compared with available numerical and analytical solutions found in the literature. The main advantage of uniform Haar wavelet series with quasilinearization process is that it captures the behavior of the nonlinear oscillators without any iteration. The numerical problems are considered with force and without force to check the efficiency and simple applicability of method on nonlinear oscillator problems.     

Haar wavelet operational matrix of fractional order integration and its applications in solving the fractional order differential equations

Haar wavelet operational matrix has been widely applied in system analysis, system identification, optimal control and numerical solution of integral and differential equations. In the present paper we derive the Haar wavelet operational matrix of the fractional order integration, and use it to solve the fractional order differential equations including the Bagley-Torvik, Ricatti and composite fractional oscillation equations. The results obtained are in good agreement with the existing ones in open literatures and it is shown that the technique introduced here is robust and easy to apply.     

A numerical assessment of parabolic partial differential equations using Haar and Legendre wavelets

In this paper we present two new numerically stable methods based on Haar and Legendre wavelets for one- and two-dimensional parabolic partial differential equations (PPDEs). This work is the extension of the earlier work ,  and  from one- and two-dimensional boundary-value problems to one- and two- dimensional PPDEs. Two generic numerical algorithms are derived in two phases. In the first stage a numerical algorithm is derived by using Haar wavelets and then in the second stage Haar wavelets are replaced by Legendre wavelets in quest for better accuracy. In the proposed methods the time derivative is approximated by first order forward difference operator and space derivatives are approximated using Haar (Legendre) wavelets. Improved accuracy is obtained in the form of wavelets decomposition. The solution in this process is first obtained on a coarse grid and then refined towards higher accuracy in the high resolution space. Accuracy wise performance of the Legendre wavelets collocation method (LWCM) is better than the Haar wavelets collocation method (HWCM) for problems having smooth initial data or having no shock phenomena in the solution space. If sharp transitions exists in the solution space or if there is a discontinuity between initial and boundary conditions, LWCM loses its accuracy in such cases, whereas HWCM produces a stable solution in such cases as well. Contrary to the existing methods, the accuracy of both HWCM and LWCM do not degrade in case of Neumann’s boundary conditions. A distinctive feature of the proposed methods is its simple applicability for a variety of boundary conditions. Performances of both HWCM and LWCM are compared with the most recent methods reported in the literature. Numerical tests affirm better accuracy of the proposed methods for a range of benchmark problems.     

Haar wavelets method for solving Poisson equations with jump conditions in irregular domain

In this paper, Haar wavelets method is used to solve Poisson equations in the presence of interfaces where the solution itself may be discontinuous. The interfaces have jump conditions which need to be enforced. It is critical for the approximation of the boundaries of the irregular domain. An irregular domain can be treated by embedding the domain into a rectangular domain and Poisson equation is solved by using Haar wavelets method on the rectangle. Firstly, we demonstrate this method in the case of 1-D region, then we consider the solution of the Poisson equations in the case of 2-D region. The efficiency of the method is demonstrated by some numerical examples.     

A wavelet operational method for solving fractional partial differential equations numerically

Fractional calculus is an extension of derivatives and integrals to non-integer orders, and a partial differential equation involving the fractional calculus operators is called the fractional PDE. They have many applications in science and engineering. However not only the analytical solution existed for a limited number of cases, but also the numerical methods are very complicated and difficult. In this paper, we newly establish the simulation method based on the operational matrices of the orthogonal functions. We formulate the operational matrix of integration in a unified framework. By using the operational matrix of integration, we propose a new numerical method for linear fractional partial differential equation solving. In the method, we (1) use the Haar wavelet; (2) establish a Lyapunov-type matrix equation; and (3) obtain the algebraic equations suitable for computer programming. Two examples are given to demonstrate the simplicity, clarity and powerfulness of the new method.     

Rationalized Haar functions method for solving Fredholm and Volterra integral equations

This paper presents a computational technique for Fredholm integral equation of the second kind and Volterra integral equation of the second kind. The method is based upon Haar functions approximation. Properties of Rationalized Haar functions are first presented, the operational matrix of integration together with product operational matrix and Newton–Cotes nodes are utilized to reduce the computation of integral equations into some algebraic equations. The method is computationally attractive and applications are demonstrated through illustrative examples.     

Haar wavelet method for solving Fisher’s equation

In this paper, we develop an accurate and efficient Haar wavelet solution of Fisher’s equation, a prototypical reaction-diffusion equation. The solutions of Fisher’s equation are characterized by propagating fronts that can be very steep for large values of the reaction rate coefficient. There is an ongoing effort to better adapt Haar wavelet methods to the solution of differential equations with solutions that resemble shock waves or fronts typical of hyperbolic partial differential equations. Moreover the use of Haar wavelets is found to be accurate, simple, fast, flexible, convenient, small computation costs and computationally attractive.     

Solution of nonlinear Volterra–Fredholm–Hammerstein integral equations via rationalized Haar functions

Rationalized Haar functions are developed to approximate of the nonlinear Volterra–Fredholm–Hammerstein integral equations. The properties of rationalized Haar functions are first presented, and the operational matrix of integration together with the product operational matrix are utilized to reduce the computation of integral equations into some algebraic equations. The method is computationally attractive, and applications are demonstrated through illustrative examples.     

A Haar wavelet collocation approach for solving one and two-dimensional second-order linear and nonlinear hyperbolic telegraph equations

We have developed a new numerical method based on Haar wavelet (HW) in this article for the numerical solution (NS) of one- and two-dimensional hyperbolic Telegraph equations (HTEs). The proposed technique is utilized for one- and two-dimensional linear and nonlinear problems, which shows its advantage over other existing numerical methods. In this technique, we approximated both space and temporal derivatives by the truncated Haar series. The algorithm of the method is simple and we can implement easily in any other programming language. The technique is tested on some linear and nonlinear examples from literature. The maximum absolute errors (MAEs), root mean square errors (RMSEs), and computational convergence rate are calculated for different number of collocation points (CPs) and also some 3D graphs are also drawn. The results show that the proposed technique is simply applicable and accurate.     

Global exact boundary controllability for 1‐D quasilinear wave equations

Based on the local exact boundary controllability for 1‐D quasilinear wave equations, the global exact boundary controllability for 1‐D quasilinear wave equations in a neighborbood of any connected set of constant equilibria is obtained by an extension method. Similar results are also given for a kind of general 1‐D quasilinear hyperbolic equations. Copyright © 2010 John Wiley & Sons, Ltd.     

Solving the 2D and 3D nonlinear inverse source problems of elliptic type partial differential equations by a homogenization function method

In this article, the inverse source problems of 2D and 3D elliptic type nonlinear partial differential equations are resolved. For this purpose, a family of single-parameter homogenization functions that automatically meet the given boundary conditions are deduced and employed as the bases to expand the solution. We solve a linear algebraic equations system which satisfies the over-specified Neumann boundary condition to obtain the unspecified coefficients, and then the solution in the entire domain is permitted. Taking the solution into the governing equation, the unknown source function can be determined quickly. The present novel method is verified to be an accurate, effective, and robust scheme which is without solving nonlinear equations and iterations, and the additional data used are quite economical.     

Global well-posedness of the 3D magneto-micropolar equations with damping

The 3D magneto-micropolar equations with damping are considered in this paper. We prove the existence and uniqueness of strong solution for 3D magneto-micropolar equations with damping for $\beta \ge 4$ with any $\alpha >0$.     

Numerical solutions to initial and boundary value problems for linear fractional partial differential equations

In this article, Haar wavelets have been employed to obtain solutions of boundary value problems for linear fractional partial differential equations. The differential equations are reduced to Sylvester matrix equations. The algorithm is novel in the sense that it effectively incorporates the aperiodic boundary conditions. Several examples with numerical simulations are provided to illustrate the simplicity and effectiveness of the method.     

On a new 3D model for incompressible euler and navier-stokes equations

In this paper, we investigate some new properties of the incompressible Euler and Navier-Stokes equations by studying a 3D model for axisymmetric 3D incompressible Euler and Navier-Stokes equations with swirl. The 3D model is derived by reformulating the axisymmetric 3D incompressible Euler and Navier-Stokes equations and then neglecting the convection term of the resulting equations. Some properties of this 3D model are reviewed. Finally, some potential features of the incompressible Euler and Navier-Stokes equations such as the stabilizing effect of the convection are presented.     

Regularity of the 3D Navier–Stokes equations with viewpoint of 2D flow

The regularity of 2D Navier–Stokes flow is well known. In this article we study the relationship of 3D and 2D flow, and the regularity of the 3D Naiver–Stokes equations with viewpoint of 2D equations. We consider the problem in the Cartesian and in the cylindrical coordinates.     

Efficient hybrid method for solving special type of nonlinear partial differential equations

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.     

长方形区域上非分离非均匀Haar小波

定义了二维Haar尺度函数,构造了长方形区域上的二维非均匀Haar小波函数,给出了非均匀 Haar 小波的分解和重构公式,最后得到了单值重构算法.     

Well-posedness for the 2D dissipative quasigeostrophic equations in the Besov space

In this paper, we consider the initial value problem of the 2D dissipative quasi-geostrophic equations. Existence and uniqueness of the solution global in time are proved in the homogenous Besov space Bp,∞ s p with small data when 1 /2<α≤1,2/2α-1< p<∞,sp=2/p-(2α-1). Our proof is based on a new characterization of the homogenous Besov space and Kato's method.