A finite-difference and Haar wavelets hybrid collocation technique for non-linear inverse Cauchy problems. Issue 1 (2nd January 2022)

• Record Type: Journal Article
• Authors: Ahsan, Muhammad; Hussain, Iltaf; Ahmad, Masood
• Is Part Of: Applied mathematics in science and engineering. Volume 30:Issue 1(2022)
• Page Start: 121
• Page End: 140
• Publication Date: 2022-01-02
• Subjects: Inverse Cauchy problems -- Haar wavelets -- collocation method -- non-linear problems -- source term; 35-02 -- 35-04 -- 35-11; Science -- Mathematics -- Periodicals; Engineering mathematics -- Periodicals; Applied mathematics -- Periodicals; 501.51
• DOI: 10.1080/17415977.2022.2026350
• Languages: English
• ISSNs: 2769-0911

Abstract

Abstract : In this research work, a finite-difference and Haar wavelet hybrid collocation scheme is introduced for the ill-posed non-linear inverse Cauchy problem with a source depending on space variable along with an unknown solution and unknown right side boundary. The first-order finite-difference approach is adopted to approximate the ∂ u ∂ t part and two different Haar series are managed to approximate ∂ 2 u ∂ x 2 part and source term respectively. A simple linearization procedure is used to convert the non-linear problem into a linear form. In contradiction to various numerical schemes, the current introduced method generates a well-conditioned system of algebraic equations, therefore it is not required to apply a regularization approach. The results of the proposed method are stable and converge to the exact solution. Some numerical tests are also performed to confirm the accuracy, well-conditioning of the algebraic equations and easy applicability of the scheme on linear and non-linear cases.