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

Muhammad Ahsan*, Khwaja SHams-ul Haq, Xuan Liu, Showkat Ahmad Lone, Muhammad Nisar

*Corresponding author for this work

    Research output: Contribution to journalArticlepeer-review

    7 Citations (Scopus)

    Abstract

    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.
    Original languageEnglish
    Pages (from-to)2475-2487
    Number of pages13
    JournalMathematical Methods in the Applied Sciences
    Volume46
    Issue number2
    Early online date20 Sept 2022
    DOIs
    Publication statusPublished - 30 Jan 2023

    Keywords

    • Collocation methods
    • Haar wavelets
    • ill-posed problems
    • inverse problems
    • well-conditioned

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