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Dec 14, 2020 · approach to the numerical solution of partial differential equations than other formulations. The main drawback of the ﬁnite difference methods is the ﬂexibility. Standard ﬁnite dif-ference methods requires more regularity of the solution (e.g. u2C2()) and the mesh (e.g. uniform grids). Difﬁculties also arise in imposing boundary ...

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Jan 27, 2017 · We have already seen the derivation of heat conduction equation for Cartesian coordinates. Now, consider a cylindrical differential element as shown in the figure. We can write down the equation in…

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fem2d_heat_rectangle, a program which applies the finite element method (FEM) to solve the time dependent heat equation on a square in 2D; fem2d_heat_square , a library which defines the geometry of a square region, as well as boundary and initial conditions for a given heat problem, and is called by fem2d_heat as part of a solution procedure.

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An alternative method is to use an alternate-direction-implicit (ADI) method [1]. The most common practice is to use TDMA to solve dependent variable along one direction of spatial coordinate implicitly while treating the dependent variable in the remaining spatial coordinate explicitly. Fig. 3 shows such an approach called "line-by line" method.

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With an ADI method the heat diffusion equation is first solved implicitly in the r-direction while leaving the other two directions explicit. But here all the possible six combinations, like r-[theta]-z, r-z-[theta], [theta]-r-z, [theta]-z-r, z-r-[theta], and z-[theta]-r, have been used to solve the heat diffusion equation and compared results.

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(2) and (3) we still pose the equation point-wise (almost everywhere) in time. In particular, one has to justify the point value u( 2;0) does make sense for an L type function which can be proved by the regularity theory of the heat equation. To easy the stability analysis, we treat tas a parameter and the function u= u(x;t) as a mapping u: [0 ...