One-dimensional heat conduction equation with constant coefficients

1 ONE-DIMENSIONAL HEAT CONDUCTION EQUATION WITH CONSTANT COEFFICIENTS 

In this section we consider the one-dimensional heat conduction equation with constant coefficients and difference schemes in order to develop various methods for designing the appropriate difference schemes in the case of time-dependent problems. 

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The original problem. The heat diffusion process on a straight line is described by the heat conduction equation 

Indeed, by introducing x = x/a and denoting once again x by x we obtain (3). Where searching a solution to equation (2) on the segment 0 x I, it is sensible to pass to the dimensionless variables 

We concentrate primarily on the first boundary-value problem associated with equation- (3) in the rectangle D = 0 i l,0 T, in which it is required to find a continuous in D solution u = u(x,t) of the 

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One-dimensional heat conduction equation with constant coefficients... 

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Transient heat conduction or mass transfer in solids with constant physical properties (diffusion coefficient, thermal diffusivity, thermal conductivity, etc.) is usually represented by a linear parabolic partial differential equation. Steady state heat or mass transfer in solids, potential distribution in electrochemical cells is usually represented by elliptic partial differential equations. In this chapter, we describe how one can arrive at the analytical solutions for linear parabolic partial differential equations and elliptic partial differential equations in finite domains using a separation of variables method. The methodology is illustrated using a transient one dimensional heat conduction in a rectangle. 

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