Document Type : Research Article

Authors

• Moslem Jamalpour Malekabadi ¹
• Ali Zakeri ²
• Amir Hossein Salehi Shayegan ³

¹ Faculty of Mathematics, K. N. Toosi University of Technology

² Faculty if marhematics, K. N. Toosi University of Technology

³ Department of Basic Sciences, Khatam-ol-Anbia (PBU) University

Abstract

In this paper, we propose an approximate solution to a one-dimensional inverse parabolic problem using radial basis functions (RBFs) and the Levenberg–Marquardt (LM) regularization method. This problem involves the backward heat equation. In particular, we first transform the well-known Black–Scholes equation into the heat equation through an appropriate change of variables. The resulting heat equation is then solved using the proposed numerical method.
To obtain the approximate solution for the unknown temperature values at the initial time, an optimization problem is formulated to minimize a cost functional. Since the system of equations is ill-conditioned, the LM regularization method is applied. We derive convergence rates for the LM iterates under a Holder stability estimate. Finally, a numerical example is presented to illustrate the method's accuracy and effectiveness.

Keywords

• Inverse parabolic problem
• Radial basis functions method
• Levenberg-Marquardt method
• Approximate solution