Remark on a constrained variational principle for heat conduction

Abstract

The heat conduction equation is re-studied by the semi-inverse method combined with separation of variables; a new variational principle for the heat conduction equation is obtained. Equivalence of the existed two in literature is shown. The significance of variable separation is confirmed once more

Lagrangians of the (2+1) -dimensional KP equation with variable coefficients and cross terms

“…(1.14) with respect to u and w are Similar results can be obtained for the Burgers equation [17] by the semi-inverse method [3]- [6], [8]. Some illustrating examples for construction of Lagrangian of a nonlinear equation are available in Refs [2,7,9,10,11,12,13,14,15,18].…”

Variational approach, soliton solutions and singular solitons for new coupled ZK system

“…10 describes the movement of the singular solitons obtained from Eqs. (26) for different times. Figs.…”

“…So we should search for a mathematical algorithm to discover the exact solutions of nonlinear partial differential equations. In recent years, powerful and efficient methods explored to find analytic solutions of nonlinear equations have drawn a lot of interest by a variety of scientists, such as Adomian decomposition method [2], the homotopy perturbation method [3,4], some new asymptotic methods searching for solitary solutions of nonlinear differential equations, nonlinear differential-difference equations and nonlinear fractional differential equations using the parameter-expansion method, the Yang-Laplace transform, the Yang-Fourier transform and ancient Chinese mathematics [4], the variational iteration method [5,6] which is used to introduce the definition of fractional derivatives [7,4], the He's variational approach [8], the extended homoclinic test approach [9,10], homogeneous balance method [11][12][13][14], Jacobi elliptic function method [15][16][17][18], Băclund transformation [19,20], G ′ /G expansion method for nonlinear partial differential equation [21,22], and fractional differential-difference equations of rational type [23][24][25] It is important to point out that a new constrained variational principle for heat conduction is obtained recently by the semi-inverse method combined with separation of variables [26], which is exactly the same with He-Lee's variational principle [27]. A short remark on the history of the semi-inverse method for establishment of a generalized variational principle is given in [28].…”

A short remark on Chien’s variational principle of maximum power losses for viscous fluids

“…The variational theory for viscous flow has been caught much attention, and various variational formulations were established for some special cases (Bogner, 2008;Chen et al, 2010;Ecer, 1980;He, 1998;Petrov, 2015;Fei et al, 2013;Jia et al, 2014;Li and Liu, 2013;Tao and Chen, 2013). As early as 1868, Helmholtz proposed a minimum principle for viscous fluids neglecting kinetic effect (Finlayson, 1972), and it was once proved that there exists no variational representation for Navier-Stokes equations (Finlayson, 1972).…”