In this manuscript, the methods, Optimal Homotopy Asymptotic Method (OHAM), Homotopy Perturbation Method (HPM) and Homotopy Analysis Method (HAM) are applied to solve various order boundary value problems. These techniques give solutions in the form of a series. Their solutions and graphs are compared with their exact solutions and graphs. All these give profitable results, but the Optimal Homotopy Asymptotic Method is more precise, whose convergence is restrained optimizations and the convergence area can be accustomed affording to the problem concerned. The results show that the suggested scheme is more active and relaxed to routine.

In [1], Gyamfi et al. described homological properties in relation to Nakayama Algebras with projectives that satisfied condition Extn (M,N) = 0 for n ≫ 0 ⇐⇒ Extn (N,M) = 0 for n ≫ 0, [1]. The purpose of this paper is to give a similar characterization of Nakayama algebras. In particular, we present Ext-groups of the Nakayama algebras with projectives that do not satisfy the condition Extn (M,N) = 0 for n ≫ 0 ⇐⇒ Extn (N,M) = 0 for n ≫ 0. To do this, we consider the Ext-groups of Nakayama algebra with projectives of lengths 3n and 4n using combinations of modules of different lengths.

The concept of ‘Ψ - Caputo’ fractional derivative is discussed in this article. This method is based on the fractional derivative in Caputo sense of a function with respect to another function Ψ , called kernel. The kernel function Ψ , is any increasing function such that . Experimental studies are used to support the fact that fractional approach of solving differential equations is often better than the classical ordinary approach. The solution to two exponential decay models and one exponential growth model are built using the classical approach and the kernel approach. Several kernel functions are considered and their performances evaluated.

Quadratic forms in four variables over the field 2 are sorted first of all with respect to permutation symmetry. Thereafter it is shown that any such form is equivalent to one of seven such canonical forms. The orthogonal group of each one of these seven forms is obtained. The paper closes with some remarks about quadratic forms in three variables.

The determination of the number of representations of a positive integer by certain octonary quadratic forms are given in the literature. For instance, the formulas N(1^{2a}, 2^{2b}, 3^{2c}, 6^{2d}; n) for the nine octonary quadratic forms are given with a = 1, b = 2, c = 3 and d = 6. Moreover, the formulas for N(1^{a}, 3^{b}, 9^{c}; n) for several octonary quadratic forms have been given by Alaca. Here, by using MAGMA, we determine the formulas, for N(1^{a}, 5^{b}, 25^{c}; n) by Eisenstein series and eta quotients for several octonary quadratic forms whose theta functions are in M_{4} (Γ_{0} (100) , χ) .