1-D And 2-D Heat Equation - Engineering Assignment Help

Assignment Task

In mathematics, if given an open subset U of ?ⁿ and a subinterval I of ?, one says that a function u : U × I → ? is a solution of the heat equation if

where (x₁, …, x_n, t) denotes a general point of the domain. It is typical to refer to t as "time" and x₁, …, x_n as "spatial variables," even in abstract contexts where these phrases fail to have their intuitive meaning. The collection of spatial variables is often referred to simply as x. For any given value of t, the right-hand side of the equation is the Laplacian of the function u(⋅, t) : U → ?. As such, the heat equation is often written more compactly as

In physics and engineering contexts, especially in the context of diffusion through a medium, it is more common to fix a Cartesian coordinate system and then to consider the specific case of a function u(x, y, z, t) of three spatial variables (x, y, z) and time variable t. One then says that u is a solution of the heat equation if

in which α is a positive coefficient called the diffusivity of the medium. In addition to other physical phenomena, this equation describes the flow of heat in a homogeneous and isotropic medium, with u(x, y, z, t) being the temperature at the point (x, y, z) and time t. If the medium is not homogeneous and isotropic, then α would not be a fixed coefficient, and would instead depend on (x, y, z); the equation would also have a slightly different form. In the physics and engineering literature, it is common to use ∇² to denote the Laplacian, rather than ?.

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