Summary In this paper, an efficient and accurate computational method based on the hybrid of block‐pulse functions and Taylor polynomials is proposed for solving a class of fractional optimal control problems. In the proposed method, the Riemann‐Liouville fractional integral operator for the hybrid of block‐pulse functions and Taylor polynomials is given. By taking into account the property of this operator, the solution of fractional optimal control problems under consideration is reduced to a nonlinear programming one to which existing well‐developed algorithms may be applied. The present method applies to both fractional optimal control problems with or without inequality constraints. The method is computationally very attractive and gives very accurate results. Easy implementation and simple operations are the essential features of the proposed hybrid functions. Illustrative examples are given to assess the effectiveness of the developed approximation technique.

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In this paper, we introduce the notions of almost bi-Γ-ideals and fuzzy almost bi-Γ-ideals of Γ-semigroups and give properties of them. Moreover, we investigate  relationship between almost bi-Γ-ideals and fuzzy almost bi-Γ-ideals.

Ideal theory plays an important role in studying in many algebraic structures, for example, rings, semigroups, semirings, etc. The algebraic structure Γ-semigroup is a generalization of the classical semigroup. Many results in semigroups were extended to results in Γ-semigroups. Many results in ideal theory of Γ-semigroups were widely investigated. In this paper, we first focus to study some novel ideals of Γ-semigroups. In Section 2, we define almost interior Γ-ideals and weakly almost interior Γ-ideals of Γ-semigroups by using the concept ideas of interior Γ-ideals and almost Γ-ideals of Γ-semigroups. Every almost interior Γ-ideal of a Γ-semigroup S is clearly a weakly almost interior Γ-ideal of S but the converse is not true in general. The notions of both almost interior Γ-ideals and weakly almost interior Γ-ideals of Γ-semigroups are generalizations of the notion of interior Γ-ideal of a Γ-semigroup S. We investigate basic properties of both almost interior Γ-ideals and weakly almost interior Γ-ideals of Γ-semigroups. The notion of fuzzy sets was introduced by Zadeh in 1965. Fuzzy set is an extension of the classical notion of sets. Fuzzy sets are somewhat like sets whose elements have degrees of membership. In the remainder of this paper, we focus on studying some novelties of fuzzy ideals in Γ-semigroups. In Section 3, we introduce fuzzy almost interior Γ-ideals and fuzzy weakly almost interior Γ-ideals of Γ-semigroups. We investigate their properties. Finally, we give some relationship between almost interior Γ-ideals [weakly almost interior Γ-ideals] and fuzzy almost interior Γ-ideals [fuzzy weakly almost interior Γ-ideals] of Γ-semigroups.

In this paper, we study picture N-structures and apply it to semigroups. Moreover, we define picture N-ideals in semigroups and investigate several properties of these ideals in semigroups.