Extension of Mpdx matrix for 4^th order partial differential equations with two independent variables

Numerical Modeling of a Propagation Process Using (Mpdx) Matrix Associated to Second Order Partial Diferential Equations with Two Independent Variables

“…Being used in various fields of engineering, a detailed presentation for Method of Matrix of Partial Derivatives of the State Vector associated with second order partial differential equations with two independent variables, time (t) and propagation space (s), is presented in literature [1][2][3][4]. It is considered a thermally insulated layer of thickness smax=200mm exposed on the upper part to a constant overtemperature u0=100 0 C uniform distributed over the entire heat transfer surface [5][6][7][8][9].…”

“…It is observed that temperatures presented in 𝑦 00 (𝑡, 𝑠 0 ) and 𝑦 00 (𝑡, 𝑠 𝑓 ) variants present aperiodic damped evolutions in relation to time 𝑡 with both possible inflexion points 𝑖 𝑇 and abscissa of inflection 𝑡 𝑖𝑛𝑓𝑙 [1][2][3][4][5].…”

“…In the Figure 2 is pesented an intuitive correlation variant between 𝑦 00 (𝑡 𝑓 , 𝑠) and 𝑦 00 (𝑠) between the two end points 𝑦 00 (𝑡 𝑓 , 𝑠 0 ) and 𝑦 00 (𝑡 𝑓 , 𝑠 𝑓 ). In a first stage, results that the same approximate function 𝑦 00 (𝑡 𝑓 , 𝑠) from figure 1 and 𝑦 00 (𝑡 𝑓 , 𝑠) from figure 2 becomes aperiodic decreasing with a possible inflection point 𝑖 𝑆 respectively abscissa of inflection 𝑠 𝑖𝑛𝑓𝑙 [1][2][3][4][5]. As a result, from 𝐹 0𝑇 (𝑡) = 𝑦 00 (𝑡, 𝑠 0 ) and 𝐹 0𝑆 (𝑡) = 𝑦 00 (𝑡 𝑓 , 𝑠) might be obtained, based on excess of terminology, an approximate solution expressed by the followig equation [1][2][3][4]:…”

“…Due to the fact that both time function 𝐹 0𝑇 (𝑡) and space function 𝐹 0𝑆 (𝑠) present aperiodic damped evolutions, with inflection abscesses [1][2][3][4]:…”

“…for 𝑇 2 > 𝑇 1 and 𝑆 2 > 𝑆 1 results that 𝐹 0𝑇 (𝑡) and 𝐹 0𝑆 (𝑠) might be approximated by two time constants 𝑇 1 , 𝑇 2 and two space constants 𝑆 1 , 𝑆 2 . Furthermore, the functions [1][2][3][4]:…”