aspg-28-1816-1792155152 20261016125232 American Scientific Publishing Group admin@americaspg.com American Scientific Publishing Group Journal of Artificial Intelligence and Metaheuristics JAIM 2833-5597 2022 2 2 H. K. Al-Mahdawi Electronic Computer Centre, University of Diyala Mostafa Abotaleb Department of System Programming, South Ural State University, 454080 Chelyabinsk, Russia Hussein Alkattan Department of System Programming, South Ural State University, 454080 Chelyabinsk, Russia El-Sayed M. El-Kenawy Department of Communications and Electronics, Delta Higher Institute of Engineering and Technology, Mansoura, 35111, Egypt E. M. Mohamed Basic science department, Delta higher institute for engineering and technology, Mansoura, 35111, Egypt 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 style='font-size:10.0pt;font-family:"Cambria Math",serif;mso-bidi-font-family: "Times New Roman";mso-bidi-theme-font:major-bidi'> m:val="p"/>{ 10.0pt;mso-ansi-font-size:10.0pt;mso-bidi-font-size:10.0pt;font-family:"Cambria Math",serif; mso-ascii-font-family:"Cambria Math";mso-hansi-font-family:"Cambria Math"; mso-bidi-font-family:"Times New Roman";mso-bidi-theme-font:major-bidi'> style='font-size:10.0pt;font-family:"Cambria Math",serif;mso-bidi-font-family: "Times New Roman";mso-bidi-theme-font:major-bidi'> m:val="p"/>R 10.0pt;font-family:"Cambria Math",serif;mso-bidi-font-family:"Times New Roman"; mso-bidi-theme-font:major-bidi'> m:val="p"/>N 10.0pt;font-family:"Cambria Math",serif;mso-bidi-font-family:"Times New Roman"; mso-bidi-theme-font:major-bidi'>} that map space style='font-size:10.0pt;mso-ansi-font-size:10.0pt;mso-bidi-font-size:10.0pt; font-family:"Cambria Math",serif;mso-ascii-font-family:"Cambria Math"; mso-hansi-font-family:"Cambria Math";mso-bidi-font-family:"Times New Roman"; mso-bidi-theme-font:major-bidi'> style='font-size:10.0pt;font-family:"Cambria Math",serif;mso-bidi-font-family: "Times New Roman";mso-bidi-theme-font:major-bidi'> m:val="p"/>L 10.0pt;font-family:"Cambria Math",serif;mso-bidi-font-family:"Times New Roman"; mso-bidi-theme-font:major-bidi'> m:val="p"/>2 10.0pt;font-family:"Cambria Math",serif;mso-bidi-font-family:"Times New Roman"; mso-bidi-theme-font:major-bidi'>[0,1] to same space. 2022 46 55 1816 https://creativecommons.org/licenses/by/4.0/ 10.54216/JAIM.020205 https://www.americaspg.com/journal/28/article/1816 https://www.americaspg.com/storage/61684735975.pdf