Purpose

In the past few years, Haar wavelet-based numerical methods have been applied successfully to solve linear and nonlinear partial differential equations. This study aims to propose a wavelet collocation method based on Haar wavelets to identify a parameter in parabolic partial differential equations (PDEs). As Haar wavelet is defined in a very simple way, implementation of the Haar wavelet method becomes easier than the other numerical methods such as finite element method and spectral method. The computational time taken by this method is very less because Haar matrices and Haar integral matrices are stored once and used for each iteration. In the case of Haar wavelet method, Dirichlet boundary conditions are incorporated automatically. Apart from this property, Haar wavelets are compactly supported orthonormal functions. These properties lead to a huge reduction in the computational cost of the method.

Design/methodology/approach

The aim of this paper is to reconstruct the source control parameter arises in quasilinear parabolic partial differential equation using Haar wavelet-based numerical method. Haar wavelets possess various properties, for example, compact support, orthonormality and closed form expression. The main difficulty with the Haar wavelet is its discontinuity. Therefore, this paper cannot directly use the Haar wavelet to solve partial differential equations. To handle this difficulty, this paper represents the highest-order derivative in terms of Haar wavelet series and using successive integration this study obtains the required term appearing in the problem. Taylor series expansion is used to obtain the second-order partial derivatives at collocation points.

Findings

An efficient and accurate numerical method based on Haar wavelet has been proposed for parameter identification in quasilinear parabolic partial differential equations. Numerical results are obtained from the proposed method and compared with the existing results obtained from various finite difference methods including Saulyev method. It is shown that the proposed method is superior than the conventional finite difference methods including Saulyev method in terms of accuracy and CPU time. Convergence analysis is presented to show the accuracy of the proposed method. An efficient algorithm is proposed to find the wavelet coefficients at target time.

Originality/value

The outcome of the paper would have a valuable role in the scientific community for several reasons. In the current scenario, the parabolic inverse problem has emerged as very important problem because of its application in many diverse fields such as tomography, chemical diffusion, thermoelectricity and control theory. In this paper, higher-order derivative is represented in terms of Haar wavelet series. In other words, we represent the solution in multiscale framework. This would enable us to understand the solution at various resolution levels. In the case of Haar wavelet, this paper can achieve a very good accuracy at very less resolution levels, which ultimately leads to huge reduction in the computational cost.

中文翻译：

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使用Haar小波方法重构抛物线偏微分方程中的参数

目的

近年来，基于Haar小波的数值方法已成功应用于求解线性和非线性偏微分方程。本研究旨在提出一种基于 Haar 小波的小波搭配方法来识别抛物线偏微分方程 (PDE) 中的参数。由于Haar小波的定义非常简单，因此Haar小波方法的实现比其他数值方法如有限元方法和谱方法更容易。由于 Haar 矩阵和 Haar 积分矩阵存储一次并用于每次迭代，因此该方法占用的计算时间非常少。在 Haar 小波方法的情况下，Dirichlet 边界条件被自动合并。除了这个特性，Haar 小波是紧支持的正交函数。

设计/方法/方法

本文的目的是利用基于Haar小波的数值方法重构拟线性抛物线偏微分方程中出现的源控制参数。Haar 小波具有多种性质，例如紧支持、正交性和封闭形式表达。Haar 小波的主要困难在于它的不连续性。因此，本文不能直接使用 Haar 小波来求解偏微分方程。为了解决这个难题，本文用 Haar 小波级数表示最高阶导数，利用逐次积分得到问题中出现的所需项。泰勒级数展开用于获得配置点处的二阶偏导数。

发现

针对拟线性抛物偏微分方程中的参数辨识，提出了一种基于Haar小波的高效、准确的数值方法。从所提出的方法中获得了数值结果，并与从各种有限差分方法（包括Saulyev 方法）获得的现有结果进行了比较。结果表明，所提出的方法在精度和CPU时间方面优于包括Saulyev方法在内的传统有限差分方法。收敛性分析显示了所提出方法的准确性。提出了一种有效的算法来寻找目标时间的小波系数。

原创性/价值

出于多种原因，该论文的成果将在科学界发挥重要作用。在目前的情况下，抛物线反问题已成为非常重要的问题，因为它在许多不同的领域中都有应用，例如层析成像、化学扩散、热电和控制理论。在本文中，高阶导数用Haar小波级数表示。换句话说，我们代表了多尺度框架中的解决方案。这将使我们能够了解各种分辨率级别的解决方案。在 Haar 小波的情况下，本文可以在非常低的分辨率级别上获得非常好的精度，最终导致计算成本的大幅降低。