In this paper, we develop and analyze a nodal discontinuous Galerkin method for the linearized fractional Cahn–Hilliard equation containing derivatives of fractional order in space. The Caputo derivative is chosen as the representation of spatial derivative. The linearized fractional Cahn–Hilliard problem has been expressed as a system of low order differential/integral equations. We adopt the nodal discontinuous Galerkin methods for the full spatial discretization using a high-order nodal basis set of orthonormal Lagrange–Legendre polynomials of arbitrary order in space on each element of computational domain. Moreover, we prove the stability and optimal order of convergence N+1 for the linearized fractional Cahn–Hilliard problem when polynomials of degree N are used. Numerical experiments are displayed to verify the theoretical results.

In this paper, we develop and analyze a nodal discontinuous Galerkin method for the linearized fractional Cahn–Hilliard equation containing derivatives of fractional order in space. The Caputo derivative is chosen as the representation of spatial derivative. The linearized fractional Cahn–Hilliard problem has been expressed as a system of low order differential/integral equations. We adopt the nodal discontinuous Galerkin methods for the full spatial discretization using a high-order nodal basis set of orthonormal Lagrange–Legendre polynomials of arbitrary order in space on each element of computational domain. Moreover, we prove the stability and optimal order of convergence N+1 for the linearized fractional Cahn–Hilliard problem when polynomials of degree N are used. Numerical experiments are displayed to verify the theoretical results.

In this article, we first introduce a singular fractional Sturm–Liouville problem (SFSLP) on unbounded domain. The associated fractional differential operator is both Weyl and Caputo type. The properties of spectral data for fractional operator on unbounded domain have been investigated. Moreover, it has been shown that the eigenvalues of the singular problem are real-valued and the corresponding eigenfunctions are orthogonal. The analytical eigensolutions of SFSLP are obtained and defined as generalized Laguerre fractional-polynomials. The optimal approximation of such generalized Laguerre fractional-polynomials in suitably weighted Sobolev spaces involving fractional derivatives has been derived. We construct an efficient generalized Laguerre fractional-polynomials-Petrov–Galerkin methods for a class of fractional initial value problems and fractional boundary value problems. As a numerical example, we examine space fractional advection–diffusion equation. Our theoretical results are confirmed by associated numerical results.

In this article, we first introduce a singular fractional Sturm–Liouville problem (SFSLP) on unbounded domain. The associated fractional differential operator is both Weyl and Caputo type. The properties of spectral data for fractional operator on unbounded domain have been investigated. Moreover, it has been shown that the eigenvalues of the singular problem are real-valued and the corresponding eigenfunctions are orthogonal. The analytical eigensolutions of SFSLP are obtained and defined as generalized Laguerre fractional-polynomials. The optimal approximation of such generalized Laguerre fractional-polynomials in suitably weighted Sobolev spaces involving fractional derivatives has been derived. We construct an efficient generalized Laguerre fractional-polynomials-Petrov–Galerkin methods for a class of fractional initial value problems and fractional boundary value problems. As a numerical example, we examine space fractional advection–diffusion equation. Our theoretical results are confirmed by associated numerical results.

In this article, we first introduce a singular fractional Sturm–Liouville problem (SFSLP) on unbounded domain. The associated fractional differential operator is both Weyl and Caputo type. The properties of spectral data for fractional operator on unbounded domain have been investigated. Moreover, it has been shown that the eigenvalues of the singular problem are real-valued and the corresponding eigenfunctions are orthogonal. The analytical eigensolutions of SFSLP are obtained and defined as generalized Laguerre fractional-polynomials. The optimal approximation of such generalized Laguerre fractional-polynomials in suitably weighted Sobolev spaces involving fractional derivatives has been derived. We construct an efficient generalized Laguerre fractional-polynomials-Petrov–Galerkin methods for a class of fractional initial value problems and fractional boundary value problems. As a numerical example, we examine space fractional advection–diffusion equation. Our theoretical results are confirmed by associated numerical results.

The Kinetics and mechanism of morphology and oxygen electro reduction reaction at PdCo catalysts synthesized on carbon black XC72 have been investigated. The effect of annealing temperature and aging time on the catalytic activity and stability were studied in more details. It is found that, the average particle size increases with increasing the aging time and leds to decrease the catalytic activity. The optimal heat-treatment temperature is found to be 300 ◦C for 3 h in the case of higher percent of Co and in the case of lower percentage of Co it is found to be 700˚C for 4 h. The highest oxygen reduction reaction (ORR) catalytic activity, are obtained. A typical transmission electron microscopy (TEM) micrograph of the (Pd32Co68/C) catalyst, heat-treated at 300 ◦C aged at different times reveals that the average particle size is 25 nm with a relatively narrow size distribution.

The synthesized carbon-supported Pd-Fe alloy electrocatalysts were characterized for the purpose of the fuel cell cathode oxygen reduction reaction (ORR). The synthesized catalysts were characterized in terms of structural morphology and catalytic activity by XRD and electrochemical measurements. Surface cyclic voltammetry was used to confirm the formation of the Pd–Fe alloy. The catalysts were heat-treated at temperatures ranging from 300 ◦C to 700 ◦C for different aging times, in order to improve activity and stability. The average particle size of 10.16 nm, and the highest ORR catalytic activity were obtained at the optimal heat-treatment temperature 300 ◦C for 3h.

Characterization of the different precipitates developed in supersaturated Al–1.12Mg2Si–0.35Si (mass%) alloy by thermoelectric power (TEP) and electrical resistivity (ER) measurements was considered. The effect of precipitation of coherent, semi-coherent and noncoherent phases on the TEP was found impressive in describing different precipitates. Upon growing and coherency loss of b0 particles, the TEP increases to reach a maximum value. TEP begins to approach a stable value by the formation of the equilibrium b (Mg2Si) precipitates and then stabilizes with the complete growth of more stable precipitates b phase and Si particles. The first-order coefficient a of the ER–temperature dependence was found dominating in the temperature range 300–635 K. Above this temperature range, the second-order coefficient b starts sharing effectively the resistivity–temperature dependence. Furthermore, it has been shown that the range of temperature in which the first-order coefficient a is dominating slightly increases after slow cooling twice to the room temperature in two successive runs. Correlation of a and b with the lattice rigidity of the alloy under investigation was established. Quenching and slow cooling affect strongly the observed correlation. All measurements in the present investigation were taken under non-isothermal conditions.

Characterization of the different precipitates developed in supersaturated Al–1.12Mg2Si–0.35Si (mass%) alloy by thermoelectric power (TEP) and electrical resistivity (ER) measurements was considered. The effect of precipitation of coherent, semi-coherent and noncoherent phases on the TEP was found impressive in describing different precipitates. Upon growing and coherency loss of b0 particles, the TEP increases to reach a maximum value. TEP begins to approach a stable value by the formation of the equilibrium b (Mg2Si) precipitates and then stabilizes with the complete growth of more stable precipitates b phase and Si particles. The first-order coefficient a of the ER–temperature dependence was found dominating in the temperature range 300–635 K. Above this temperature range, the second-order coefficient b starts sharing effectively the resistivity–temperature dependence. Furthermore, it has been shown that the range of temperature in which the first-order coefficient a is dominating slightly increases after slow cooling twice to the room temperature in two successive runs. Correlation of a and b with the lattice rigidity of the alloy under investigation was established. Quenching and slow cooling affect strongly the observed correlation. All measurements in the present investigation were taken under non-isothermal conditions.

Characterization of the different precipitates developed in supersaturated Al–1.12Mg2Si–0.35Si (mass%) alloy by thermoelectric power (TEP) and electrical resistivity (ER) measurements was considered. The effect of precipitation of coherent, semi-coherent and noncoherent phases on the TEP was found impressive in describing different precipitates. Upon growing and coherency loss of b0 particles, the TEP increases to reach a maximum value. TEP begins to approach a stable value by the formation of the equilibrium b (Mg2Si) precipitates and then stabilizes with the complete growth of more stable precipitates b phase and Si particles. The first-order coefficient a of the ER–temperature dependence was found dominating in the temperature range 300–635 K. Above this temperature range, the second-order coefficient b starts sharing effectively the resistivity–temperature dependence. Furthermore, it has been shown that the range of temperature in which the first-order coefficient a is dominating slightly increases after slow cooling twice to the room temperature in two successive runs. Correlation of a and b with the lattice rigidity of the alloy under investigation was established. Quenching and slow cooling affect strongly the observed correlation. All measurements in the present investigation were taken under non-isothermal conditions.