%0 Journal Article
%T Thermomechanical Dynamics (TMD) and Bifurcation-Integration Solutions in Nonlinear Differential Equations with Time-Dependent Coefficients
%A Hiroshi Uechi
%A Lisa Uechi
%A Schun T. Uechi
%J Journal of Applied Mathematics and Physics
%P 1733-1743
%@ 2327-4379
%D 2024
%I Scientific Research Publishing
%R 10.4236/jamp.2024.125108
%X The new independent solutions of the nonlinear differential equation with time-dependent coefficients (NDE-TC) are discussed, for the first time, by employing experimental device called a drinking bird whose simple back-and-forth motion develops into water drinking motion. The solution to a drinking bird equation of motion manifests itself the transition from thermodynamic equilibrium to nonequilibrium irreversible states. The independent solution signifying a nonequilibrium thermal state seems to be constructed as if two independent bifurcation solutions are synthesized, and so, the solution is tentatively termed as the <i>bifurcation-integration</i> <i>solution</i>. The bifurcation-integration solution expresses the transition from mechanical and thermodynamic equilibrium to a nonequilibrium irreversible state, which is explicitly shown by the nonlinear differential equation with time-dependent coefficients (NDE-TC). The analysis established a new theoretical approach to nonequilibrium irreversible states, <i>thermomechanical</i> <i>dynamics</i> (TMD). The TMD method enables one to obtain thermodynamically consistent and time-dependent progresses of thermodynamic quantities, by employing the bifurcation-integration solutions of NDE-TC. We hope that the basic properties of bifurcation-integration solutions will be studied and investigated further in mathematics, physics, chemistry and nonlinear sciences in general.
%K The Nonlinear Differential Equation with Time-Dependent Coefficients
%K The Bifurcation-Integration Solution
%K Nonequilibrium Irreversible States
%K Thermomechanical Dynamics (TMD)
%U http://www.scirp.org/journal/PaperInformation.aspx?PaperID=133333