Mathematics

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Numerical modeling of influence of source in heat transformation: An application in blacksmithing metal heating
(Department of mathematics, 2022) Kandel, Hari Prapanna
Partial differential equations (PDEs) are used to mimic a variety of real-world physical issues. A standard parabolic PDE of the form u ; ( > 0) is an 1D heat equation. In a regular form of domain, the heat equation has an analytical solution. Computing an analytical solution becomes challenging, if not impossible, any time the domain of such modeled issues has an uneven shape. In this case, numerical methods can be used to ﬁnd the numerical solution of these PDEs. Through the domain’s discretization into a limited number of areas. One of the numerical techniques used to determine the numerical solutions of PDEs is the ﬁnite difference method (FDM). Here, the FTCSSfor the one-dimensional heat equation and the numerical computation of its solution using FTCSS are discussed. Furthermore, numerical solution and analytic solution of heat equation has been compared and analyzed. Additionally, the 1D heat equation with variable starting conditions (ICs) and numerous initial conditions (ICs)has been solved numerically using FDMs. Blacksmiths heated the parts at various temperatures and locations to mold different metals into the necessary shapes. The numerical solution method for the 1D heat problem given here can be used to solve heat equations used in engineering and scientiﬁc disciplines. t = u xx