A Haar wavelets based approximation for nonlinear inverse problems influenced by unknown heat source. (20th September 2022)

• Record Type: Journal Article
• Authors: Ahsan, Muhammad; Shams‐ul Haq, Khawaja; Liu, Xuan; Ahmad Lone, Showkat; Nisar, Muhammad
• Is Part Of: Mathematical methods in the applied sciences. Volume 46:Number 2(2023)
• Page Start: 2475
• Page End: 2487
• Publication Date: 2022-09-20
• Subjects: Collocation methods -- Haar wavelets -- ill‐posed problems -- inverse problems -- well‐conditioned; Mathematics -- Periodicals; Technology -- Mathematics -- Periodicals; 519
• DOI: 10.1002/mma.8655
• Languages: English
• ISSNs: 0170-4214

Abstract

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.