A Mathematical Framework for Indeterminacy in Parabolic PDEs: The Neutrosophic Heat Equation

 

 

 

Ghassan AL-Thabhawee1, Hussein Alkattan2,3, El-Sayed M. El-kenawy4,5,*, Marwa M. Eid6,7

 

1Sciences of Mathematics, Computer Sciences, College of Health and Medical Techniques-Kufa Al-Furat Al-Awsat Technical University, Kufa, Iraq

 

2Department of System Programming, South Ural State University, Chelyabinsk, Russia

 

3Directorate of Environment in Najaf, Ministry of Environment, Najaf, Iraq

 

4Delta Higher Institute of Engineering and Technology Department for Communications and Electronics Mansoura 35511, Egypt

 

5Applied Science Research Center. Applied Science Private University, Amman, Jordan

 

6 Faculty of Artificial Intelligence, Delta University for Science and Technology, Mansoura, Egypt

 

7Jadara Research Center, Jadara University, Irbid 21110, Jordan

 

Emails: gmohammed@atu.edu.iq; alkattan.hussein92@gmail.com; skenawy@ieee.org; mmm@ieee.org

 

 

 

 

 

Abstract

 

We develop a neutrosophic framework for the 1-D transient heat equation that treats key thermal parameters as indeterminate rather than fixed or strictly probabilistic. Thermal diffusivity and source strength are represented by neutrosophic intervals; two extreme forward solves yield guaranteed envelopes  and , from which we compute a core field , an absolute width , and a relative indeterminacy index . Using an explicit FTCS discretization with stability enforced by , we report decision-oriented diagnostics: spatio-temporal maps of , and ; band plots along space/time sections; percentile trajectories of  over time; coverage curves quantifying the fraction of space-time with ; and response surfaces showing sensitivity of  to . Results demonstrate that, even when absolute spreads remain small, localized reliability losses can occur where  crosses zero, a regime routinely obscured by point-estimate modelling. The framework is transparent (envelopes + core), computationally light (two extreme runs), and compatible with neutrosophic statistics for data-driven interval setting. Beyond thermal diffusion, the method provides a conservative, explainable backbone for transport-driven decisions in materials, interfaces, and infrastructure subject to incomplete or evolving information.

 

Keywords: Neutrosophic modeling; Heat equation; Transient heat conduction; Interval uncertainty; Envelope propagation; Relative indeterminacy index (RII); Thermal diffusivity; Source amplitude; Uncertainty quantification