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.     

Coupled Galerkin and parameterization methods for optimal control of discretely connected parallel beams

Galerkin and wavelet methods for optimal boundary control of a couple of discretely connected parallel beams are proposed. First, the problem with boundary controls is converted into a problem with distributed controls. The problem is, then, reduced by a Galerkin-based approach into determining the optimal control of a linear time-invariant lumped parameter system, which will be solved by a wavelet-based method using Legendre wavelets. The integration-operational matrix and Kronecker product are utilized to significantly simplify the optimization problem into a system of linear equations. A numerical example is presented to demonstrate the applicability and the efficiency of the proposed method.     

Numerical solutions of distributed order fractional differential equations in the time domain using the Müntz–Legendre wavelets approach

In this paper, a numerical method is presented to obtain and analyze the behavior of numerical solutions of distributed order fractional differential equations of the general form in the time domain with the Caputo fractional derivative. The suggested method is based on the Müntz–Legendre wavelet approximation. We derive a new operational vector for the Riemann–Liouville fractional integral of the Müntz–Legendre wavelets by using the Laplace transform method. Applying this operational vector and collocation method in our approach, the problem can be reduced to a system of linear and nonlinear algebraic equations. The arising system can be solved by the Newton method. Discussion on the error bound and convergence analysis for the proposed method is presented. Finally, seven test problems are considered to compare our results with other well‐known methods used for solving these problems. The results in the tabulated tables highlighted that the proposed method is an efficient mathematical tool for analyzing distributed order fractional differential equations of the general form.     

A Haar wavelets based approximation for nonlinear inverse problems influenced by unknown heat source

In this discussion, a new numerical algorithm focused on the Haar wavelet is used to solve linear and nonlinear inverse problems with unknown heat source. The heat source is dependent on time and space variables. These types of inverse problems are ill-posed and are challenging to solve accurately. The linearization technique converted the nonlinear problem into simple nonhomogeneous partial differential equation. In this Haar wavelet collocation method (HWCM), the time part is discretized by using finite difference approximation, and space variables are handled by Haar series approximation. The main contribution of the proposed method is transforming this ill-posed problem into well-conditioned algebraic equation with the help of Haar functions, and hence, there is no need to implement any sort of regularization technique. The results of numerical method are efficient and stable for this ill-posed problems containing different noisy levels. We have utilized the proposed method on several numerical examples and have valuable efficiency and accuracy.     

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.     

A meshless based numerical technique for traveling solitary wave solution of Boussinesq equation

In this paper, we employ the boundary-only meshfree method to find out numerical solution of the classical Boussinesq equation in one dimension. The proposed method in the current paper is a combination of boundary knot method and meshless analog equation method. The boundary knot technique is an integration free, boundary-only, meshless method which is used to avoid the known disadvantages of the method of fundamental solution. Also, we use the meshless analog equation method to replace the nonlinear governing equation with an equivalent nonhomogeneous linear equation. A predictor-corrector scheme is proposed to solve the resulted differential equation of the collocation. The numerical results and conclusions are obtained for both the ‘good’ and the ‘bad’ Boussinesq equations.     

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.     

Computation of eigenvalues and solutions of regular Sturm–Liouville problems using Haar wavelets

The paper presents a novel method for the computation of eigenvalues and solutions of Sturm–Liouville eigenvalue problems (SLEPs) using truncated Haar wavelet series. This is an extension of the technique proposed by Hsiao to solve discretized version of variational problems via Haar wavelets. The proposed method aims to cover a wider class of problems, by applying it to historically important and a very useful class of boundary value problems, thereby enhancing its applicability. To demonstrate the effectiveness and efficiency of the method various celebrated Sturm–Liouville problems are analyzed for their eigenvalues and solutions. Also, eigensystems are investigated for their asymptotic and oscillatory behavior. The proposed scheme, unlike the conventional numerical schemes, such as Rayleigh quotient and Rayleigh–Ritz approximation, gives eigenpairs simultaneously and provides upper and lower estimates of the smallest eigenvalue, and it is found to have quadratic convergence with increase in resolution.     

Legendre spectral method for the fractional Bratu problem

In this paper, the Legendre spectral collocation method (LSCM) is applied for the solution of the fractional Bratu's equation. It shows the high accuracy and low computational cost of the LSCM compared with some other numerical methods. The fractional Bratu differential equation is transformed into a nonlinear system of algebraic equations for the unknown Legendre coefficients and solved with some spectral collocation methods. Some illustrative examples are also given to show the validity and applicability of this method, and the obtained results are compared with the existing studies to highlight its high efficiency and neglectable error.     

State Analysis and Optimal Control of Time-Varying Discrete Systems via Haar Wavelets

The state analysis and optimal control of time-varying discrete systems via Haar wavelets are the main tasks of this paper. First, we introduce the definition of discrete Haar wavelets. Then, a comparison between Haar wavelets and other orthogonal functions is given. Based upon some useful properties of the Haar wavelets, a special product matrix and a related coefficient matrix are proposed; also, a shift matrix and a summation matrix are derived. These matrices are very effective in solving our problems. The local property of the Haar wavelets is applied to shorten the calculation procedures.     

Optimal Control of Linear Time-Varying Systems via Haar Wavelets

This paper introduces the application of Haar wavelets to the optimal control synthesis for linear time-varying systems. Based upon some useful properties of Haar wavelets, a special product matrix, a related coefficient matrix, and an operational matrix of backward integration are proposed to solve the adjoint equation of optimization. The results obtained by the proposed Haar approach are almost the same as those obtained by the conventional Riccati method.     

Vieta–Fibonacci wavelets: Application in solving fractional pantograph equations

In this paper, the Vieta–Fibonacci wavelets as a new family of orthonormal wavelets are generated. An operational matrix concerning fractional integration of these wavelets is extracted. A numerical scheme is established based on these wavelets and their fractional integral matrix together with the collocation technique to solve fractional pantograph equations. The presented method reduces solving the problem under study into solving a system of algebraic equations. Several examples are provided to show the accuracy of the method.     

Solving fractional integral equations by the Haar wavelet method

Haar wavelets for the solution of fractional integral equations are applied. Fractional Volterra and Fredholm integral equations are considered. The proposed method also is used for analysing fractional harmonic vibrations. The efficiency of the method is demonstrated by three numerical examples.     

一类非线性微分方程组的有理化Haar小波解法

利用有理化Haar小波性质和方法,建立了一类非线性微分方程组在任意区间[a,b）的求解算法．基于该算法,运用计算机代数系统Maple,给出了求解非线性微分方程组的程序．并运用此程序给出了一类微分方程组的计算实例,从数值模拟来看可以达到较高的精度,并对方程组的动力学行为给出较好的描述．     

三阶微分方程的多区域Legendre-Petrov-Galerkin谱方法

提出三阶微分方程初边值问题的多区域Legendre-Petrov-Galerkin谱方法.对于三阶线性微分方程,证明该方法全离散格式的稳定性,并给出L~2-误差估计.进而将该方法和Legendre配置方法相结合,应用于某些非线性问题.数值算例对单区域和多区域方法的结果进行比较.     

Solving 2D and 3D Poisson equations and biharmonic equations by the Haar wavelet method

In this paper, we present a computational method for solving 2D and 3D Poisson equations and biharmonic equations which based on the use of Haar wavelets. The highest derivative appearing in the differential equation is expanded into the Haar series, this approximation is integrated while the boundary conditions are incorporated by using integration constants. In 2D the first transform the spectral coefficients into the nodal variable values and then use Kronecker products to construct the approximations for derivatives over a tensor product grid of the horizontal and vertical blocks. Finally, solutions to four test problems are investigated.     

Discontinuous Legendre wavelet element method for elliptic partial differential equations

By incorporating the Legendre multiwavelet into the discontinuous Galerkin (DG) method, this paper presents a novel approach for solving Poisson’s equation with Dirichlet boundary, which is known as the discontinuous Legendre multiwavelet element (DLWE) method, derive an adaptive algorithm for the method, and estimate the approximating error of its numerical fluxes. One striking advantage of our method is that the differential operator, boundary conditions and numerical fluxes involved in the elementwise computation can be done with lower time cost. Numerical experiments demonstrate the validity of this method. Furthermore, this paper generalizes the DLWE method to the general elliptic equations defined on a bounded domain and describes the possibilities of constructing optimal adaptive algorithm. The proposed method and its generalizations are also applicable to some other kinds of partial differential equations.     

Klein-Gordon-Zakharov方程初边值问题的Legendre谱方法

研究Klein-Gordon-Zakharov方程初边值问题的Legendre谱方法.在先验估计的基础上,证明了该格式的稳定性和收敛性,并得到最优阶误差估计.另外,还设计了一个半隐格式,并给出数值例子.在文章的后面给出了多区域谱格式,数值结果表明精度要高于单区域.     

A new approach by two‐dimensional wavelets operational matrix method for solving variable‐order fractional partial integro‐differential equations

In this paper, a new computational scheme based on operational matrices (OMs) of two‐dimensional wavelets is proposed for the solution of variable‐order (VO) fractional partial integro‐differential equations (PIDEs). To accomplish this method, first OMs of integration and VO fractional derivative (FD) have been derived using two‐dimensional Legendre wavelets. By implementing two‐dimensional wavelets approximations and the OMs of integration and variable‐order fractional derivative (VO‐FD) along with collocation points, the VO fractional partial PIDEs are reduced into the system of algebraic equations. In addition to this, some useful theorems are discussed to establish the convergence analysis and error estimate of the proposed numerical technique. Furthermore, computational efficiency and applicability are examined through some illustrative examples.     

Cahn-Hilliard方程的拟谱逼近

该文讨论用Ｌｅｇｅｎｄｒｅ拟谱方法数值求解非线性Ｃａｈｎ Ｈｉｌｌｉａｒｄ方程的Ｄｉｒｉｃｈｌｅｔ问题．建立了其半离散和全离散逼近格式，它们保持原问题能量耗散的性质．证明了离散解的存在唯一性，并给出了最佳误差估计．数值实验也证实了我们的结果．