The Laplace-Adomian Decomposition Method (LADM) and Homotopy Perturbation Method (HPM) are both utilized in this research in order to obtain an approximate analytical solution to the nonlinear Schrödinger equation with harmonic oscillator. Accordingly, nonlinear Schrödinger equation in both one and two dimensions is provided to illustrate the effects of harmonic oscillator on the behavior of the wave function. The available literature does not provide an exact solution to the problem presented in this paper. Nevertheless, approximate analytical solutions are provided in this paper using LADM and HPM methods, in addition to comparing and analyzing both solutions.

The structural, elastic, electronic, magnetic, and thermoelectric properties of MgEu2X4 (X = S and Se) spinel compounds are investigated computationally. Calculations are performed using the full‐potential linearized augmented plane wave (FP‐LAPW) method within the Perdew, Burke, and Ernzerhof generalized gradient approximation (PBE‐GGA), GGA + U, and modified Becke–Johnson (mBJ‐GGA) approximations. The band structure and density of states results from the three exchange‐correlation approximation methods (mBJ, GGA + U, and PBE) show that these spinel compounds are fully spin‐polarized. Also, they possess a half‐metallic character in the spin‐down channel with a direct bandgap (Γ–Γ) of about 3.44, 2.712, and 2.472 eV for MgEu2S4 and 2.89, 2.285, and 2.017 eV for MgEu2Se4, respectively. The formation of both compounds is energetically favorable based on the results of the total energy and cohesive energy calculations. Furthermore, the two compounds are chemically and mechanically stable, as concluded from cohesive energy and elastic calculations. The elastic calculations reveal that both spinel compounds are ductile materials. The ionic bonds are predominant. The quasi‐harmonic model has also been used to investigate the influences of temperature and pressure on thermal characteristics. The thermoelectric behavior is studied using the BoltzTraP code. Both systems show good thermoelectric properties for the spin‐down channel.

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The fractional form of the electromagnetic Lagrangian density is presented using the Riemann-Liouville fractional derivative. Agrawal procedure is employed to obtain Maxwell's equations in fractional form. The Hamilton equations of motion resulting from the electromagnetic Lagrangian density are obtained. Conserved quantities, such as energy density, momentum, and Poynting's vector, are also derived using fractional Noether's theorem.

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