Invariant characterization of third-order ordinary differential equations u''=f(x,u,u',u) with five-dimensional point symmetry group

The Cartan equivalence method is applied to provide an invariant characterization of the third-order ordinary differential equation u''=f(x,u,u',u') which admits a five-dimensional point symmetry Lie algebra. The invariant characterization is given in terms of the function f in a compact form. A simple procedure to construct the equivalent canonical form by use of an obtained constant invariant is also presented. We also show how one obtains the point transformation that does the reduction to linear form. Moreover, some applications are provided.Scopu

Invariant characterization of scalar third-order ODEs that admit the maximal point symmetry Lie algebra

The Cartan equivalence method is used to deduce an invariant characterization of the scalar third-order ordinary differential equation u"' = f(x, u, u', u"), which admits the maximal 7-dimensional point symmetry Lie algebra. The method provides auxiliary functions that can be used to efficiently obtain the point transformation that does the reduction to the simplest linear equation u"' = 0. Moreover, examples are given to illustrate the method. Copyright ? 2018 John Wiley & Sons, Ltd

Linearization of third-order ordinary differential equations u'''=f(x,u,u',u') via point transformations

The linearization problem for scalar third-order ordinary differential equations via point transformations was solved partially in the works of Al-Dweik et al by the use of the Cartan equivalence method. In order to solve this problem completely, the Cartan equivalence method is applied to provide an invariant characterization of the linearizable third-order ordinary differential equation (Formula presented.), which admits a four-dimensional point symmetry Lie algebra. The invariant characterization is given in terms of function f in a compact form. A simple procedure to construct the equivalent canonical form by use of an obtained invariant is also presented. The approach provides auxiliary functions, which can be effectively utilized to determine the point transformation that does the reduction to the equivalent canonical form. Furthermore, illustrations to the main theorem and applications are given.Scopu

Microscale Thermal Energy Transfer Over a Combined System of Thin Films: Analytical Approach

An analytical approach for the solution of the equation for phonon transport is presented for the combination thin films system. The transient phonon radiative transport model is considered and the combination of the silicon diamond silicon films is accommodated in the analysis. The multi-film system is thermally disturbed from the edges through introducing temperature difference across the combined films. Equivalent equilibrium temperature is considered quantifying the distribution of the phonon intensity in the films. Equivalent equilibrium temperature obtained from the analytical approach is compared with that predicted from the numerical solution. It is found that numerical predictions of equivalent equilibrium temperature agree well with those obtained from the analytical approach. The boundary scattering of phonons at the film edges causes equivalent equilibrium temperature jump at film edges, which becomes apparent in the early heating periods. Phonon scattering in the combined films causes sharp decay of temperature in the films, which is more pronounced in the silicon film than that of the diamond.Authors acknowledge the support of Deanship of Research, KFUPM, for the funded research project # RG181003 and King Abdullah City for Atamic and Renewable Energy (K.A. CARE) during the course of this work. In addition, Ahmad Y. Al-Dweik is thankful to Qatar University for its continuous support and excellent research facilities.Scopu