A semi-analytical method has been planned for the precise solution of the differential equation established on the New Homotopy Perturbation Method (NHPM), and to develop a generalized Burger-Huxley (BH) equation, in this paper. By employing NHPM, two case studies show the precise solution of the BH equation. It is shown that the NHPM is yield solution is convergent from with the easy computability term; NHPM is an effective and easy tool for cracking many real world difficulties. The three-dimension and two dimension graphical solutions of the BH equations are also provided to validate the mathematical models. MATLAB software is used to calculate the series obtained from HPM.

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We investigated the effect of Si substitution on structural, electronic and optical properties of [Formula: see text]-type [Formula: see text] alloys. The optimized lattice constants and internal cell parameters are in agreement with the available data. We found that the valence band is mainly dominated by Ni-3d states in the energy range 0–4 eV below the [Formula: see text], whereas conduction band is contributed by spin-down Ho-[Formula: see text] states and lies about 2 eV above the [Formula: see text]. Substitution of Si atoms for Ni decreases the total magnetic moment from 6.308 [Formula: see text]/f.u. [Formula: see text] to 4.052 [Formula: see text]/f.u. [Formula: see text], whereas magnetic moments on [Formula: see text]-ions increase from 3.92 [Formula: see text]/atom [Formula: see text] to 4 [Formula: see text]/atom [Formula: see text]. On the other hand, induced moment on Ni[Formula: see text]-ions decrease rapidly to a negligibly small value. By the use of charge density estimates, we found that Ho-5d, Ni-3d and Si-3p orbitals are mainly involved in bonding and there is a weak hybridization between Ni-3d and Si-3p orbitals which varies with substitution of Si for Ni. Furthermore, the complex role played by Ho-[Formula: see text] electrons is also investigated, they are found to be partly involved in metallic bondings as well as in the intra-atomic charge transfer from [Formula: see text] to [Formula: see text] states in [Formula: see text]-ions. The calculated interband optical conductivity spectra reproduce the main features of experimental spectra well and also reveal their metallic nature.

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