Solving the Inverse Initial Value Problem for the Heat Conductivity Equation by Using the Picard Method

H. K. Al-Mahdawi¹, Mostafa Abotaleb^{*, 2}, Hussein Alkattan³, El-Sayed M. El-Kenawy⁴, E. M. Mohamed⁵

¹Electronic Computer Centre, University of Diyala

^{2, 3} Department of System Programming, South Ural State University, 454080 Chelyabinsk, Russia

⁴ Department of Communications and Electronics, Delta Higher Institute of Engineering and Technology, Mansoura, 35111, Egypt

⁵Basic science department, Delta higher institute for engineering and technology,
Mansoura, 35111, Egypt

Emails: hssnkd@gmail.com;  abotalebmostafa@bk.ru; alkattan.hussein92@gmail.com skenawy@ieee.org ; ehab_joo@yahoo.com

Abstract

In this work, the inverse initial value problem IVP for the heat equation is formulated and solved. Initial temperature (initial condition) distribution is unknown in this problem, and instead, the temperature spreading at period t= T> 0 is assumed. Among mathematical problems, a class of problems is singled out, the solutions of which are unstable to minor variations in the initial information. It is well identified that this problem is ill-posed. In order to solve the direct problem, we has used the separation of variables way. Note that the method of separation of variables is completely inapplicable for solving IVP, since it principals to rather errors, also divergent series. Ivanov V.K. noticed that if the inverse problem IP  is solved by the method separation of variables, and then the resulting series is changed by a incomplete sum of the series, in which the term number is depending on δ, N=N(δ), then as a result we obtain a stable approximate solution. The Picard method customs a regularizing family of operators  that map space  to same space.

Keywords: Inverse problem; Picard; Ill-posed; Initial value problem