One-dimensional stationary problems

In order to develop a mathematical model, first, we answered the question “where” and “when” in Sec. 1.1.1. We described space and time for the considered process. It was easy to introduce time in- creasing form the initial time t ₀ to the end time t ₁. Frequently, it is convenient to take t ₀ = 0, sometimes t ₀ = −∞ and t ₁ = +∞. So, the time t ∈ ℝ is always one dimensional. The space variable x = (x, y, z) is a vector of the three-dimensional (3D) space ℝ³. However, as we saw before, sometimes the considered phenomenon is essentially 1D, i.e., depends only on one space variable, say x. The reduction of the 3D to the 1D geometry is displayed in Figs 6.1–6.2. Figure 6.1 1D perature distribution in the 3D wall. https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9781315277240/ef9fe5f1-ef7b-4c09-a56b-4913daf68eb6/content/fig6_1.tif"/> Figure 6.2 Let a function <italic>u</italic>(<italic>x, t</italic>) is defined in the half-plane {(<italic>x, y, z</italic>) ∈ ℝ³ : <italic>x ></italic> 0} and does not depend on <italic>y</italic> and <italic>z</italic>. Then, the function <italic>u</italic>(<italic>x, t</italic>) can be investigated as a function in the 1D space of variable <italic>x</italic> on the half-axis <italic>x</italic> > 0. https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9781315277240/ef9fe5f1-ef7b-4c09-a56b-4913daf68eb6/content/fig6_2.tif"/>