This paper investigates the Korteweg-de Vries equation within the scope of the local fractional derivative formulation. The exact traveling wave solutions of non-differentiable type with the generalized functions defined on Cantor sets are analyzed. The results for the non-differentiable solutions when fractal dimension is 1 are also discussed. It is shown that the exact solutions for the local fractional Korteweg-de Vries equation characterize the fractal wave on shallow water surfaces.

On exact traveling-wave solutions for local fractional Korteweg-de Vries equation.

There are various types of fractional derivatives in literature. One of the most natural and well-behaved fractional derivatives is recently introduced by the authors Khalil et al. [34], namely the conformable fractional derivative. In this paper, some more results about conformable fractional Laplace transform introduced by Abdeljawad [43] are investigated. The invariant subspace method is developed to get the exact solutions of various conformable time fractional differential equations. Finally, this theory is extended for the coupled system of conformable fractional differential equations, as well.

Invariant subspaces admitted by fractional differential equations with conformable derivatives

In this study, the generalized Kudryashov method (GKM) is handled to find exact solutions of time-fractional Burgers equation, time-fractional Cahn-Hilliard equation, and time-fractional generalized third-order KdV equation. These time-fractional equations can be turned into another nonlinear ordinary differantial equation by travelling wave transformation. Then, GKM has been implemented to attain exact solutions of time-fractional Burgers equation, time-fractional Cahn-Hilliard equation, and time-fractional generalized third-order KdV equation. Also, some new hyperbolic function solutions have been obtained by using this method. It can be said that this method is a generalized form of the classical Kudryashov method.

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Generalized Kudryashov Method for Time-Fractional Differential Equations

In this paper, we derive the complete algebra of Lie point symmetries for the class of time-fractional nonlinear dispersive equation. By means of the classical Lie symmetry method, the associated vector fields are obtained which in turn are utilized for the reduction of the equation. In particular, the conservation laws of the equation are obtained.

http://kora.kpu.ca/islandora/object/kora%3A47/datastream/PDF/view

Symmetry analysis and conservation laws for the class of time-fractional nonlinear dispersive equation

Exact solutions for some time‐fractional evolution equations using Lie group theory

Lie symmetry analysis of the time fractional KdV-type equation

Strain coupling and relaxation dynamics associated with the ferrimagnetic/ferroelastic phase transition at ${T}_{c}\ensuremath{\approx}310\phantom{\rule{0.28em}{0ex}}\mathrm{K}$ in double perovskite ${\mathrm{Ba}}_{2}{\mathrm{FeReO}}_{6}$ with a high degree of Fe/Re order have been investigated with resonant ultrasound spectroscopy through the temperature interval $\ensuremath{\sim}5--600\phantom{\rule{0.28em}{0ex}}\mathrm{K}$ and with applied magnetic field of up to $\ifmmode\pm\else\textpm\fi{}2\phantom{\rule{0.28em}{0ex}}\mathrm{T}$. Strain analysis using diffraction data from the literature is consistent with a Landau model of the transition as $Fm\overline{3}m{1}^{\ensuremath{'}}\ensuremath{\rightarrow}I4/m{m}^{\ensuremath{'}}{m}^{\ensuremath{'}}$, improperly ferroelastic, driven by a magnetic order parameter with symmetry ${\mathrm{\ensuremath{\Gamma}}}_{4}^{+}$. Ferroelastic shear strain of up to $\ensuremath{\sim}0.0015$ arises from spin/orbit coupling and is smaller than is typical of coupling with octahedral tilting. It provides the underlying cause of softening of the shear modulus observed over an interval of $\ensuremath{\sim}100\phantom{\rule{0.28em}{0ex}}\mathrm{K}$ below ${T}_{\mathrm{c}}$, though with order/disorder rather than displacive character for the transition. Hysteretic effects suggest that precursor microstructures and mixed magnetic/ferroelastic domains below ${T}_{\mathrm{c}}$ depend on the thermal history of the sample and can evolve on a timescale of hours and days at room temperature. Elasticity data collected as a function of external magnetic field reveal that poled samples are slightly softer than those with multiple magnetic domains at 4 K. At 300 K there is a time-dependent viscous component of the response to the field that relates to the bulk modulus and, hence, to volume changes associated with magnetic ordering. A loss peak seen $\ensuremath{\sim}20--50\phantom{\rule{0.28em}{0ex}}\mathrm{K}$ below ${T}_{\mathrm{c}}$ in AC magnetic measurements made at frequencies of $0.2--1\phantom{\rule{0.28em}{0ex}}\mathrm{kHz}$ yielded an activation energy $\ensuremath{\sim}0.4\phantom{\rule{0.28em}{0ex}}\mathrm{eV}$ and has been attributed to freezing of magnetic/ferroelastic domain walls. No equivalent loss peak was seen in the acoustic data measured at $\ensuremath{\sim}100--500\phantom{\rule{0.28em}{0ex}}\mathrm{kHz}$, however, implying that these domain walls are mobile in response to a dynamic magnetic field on a timescale of $\ensuremath{\sim}{10}^{\ensuremath{-}2}\text{--}{10}^{\ensuremath{-}3}\phantom{\rule{0.28em}{0ex}}\mathrm{s}$ but immobile in response to a dynamic stress field applied on a timescale of $\ensuremath{\sim}{10}^{\ensuremath{-}5}\text{--}{10}^{\ensuremath{-}6}\phantom{\rule{0.28em}{0ex}}\mathrm{s}$. Debye-like acoustic loss peaks at temperatures below $\ensuremath{\sim}100\phantom{\rule{0.28em}{0ex}}\mathrm{K}$ yielded activation energies of $\ensuremath{\sim}0.02--0.1\phantom{\rule{0.28em}{0ex}}\mathrm{eV}$ and are discussed in terms of pinning/freezing of polarons. ${\mathrm{Ba}}_{2}{\mathrm{FeReO}}_{6}$ is a material with magnetoelastic and magnetoelectric heterogeneities that might be tuned by choice of thermal history and cation order. 

Magnetoelastic coupling and microstructure dynamics associated with spin-orbit coupling in the ferrimagnetic/ferroelastic ordered double perovskite 
Ba2FeReO6

Juan Hu

Papers

Lie symmetry analysis of the time fractional KdV-type equation

Magnetoelastic coupling and microstructure dynamics associated with spin-orbit coupling in the ferrimagnetic/ferroelastic ordered double perovskite Ba2FeReO6