Journal of Mathematical Extension
Volume:18 Issue: 10, Oct 2024

A numerical approximation combining the fast finite difference method in time and the finite element method in space is studied to solve the distributed-order time and Riesz space fractional Schr\"{o}dinger equation. In this work, a fast evaluation of the distributed-order time fractional derivative based on graded time mesh is applied to the time approximation of this equation. Also, the finite element method is used for space approximation. Moreover, the stability and convergence of the resulting discrete scheme are discussed. Finally, some numerical examples are presented to confirm the theoretical results.

Using the Atanassov's intuitionistic fuzzy set, the concept of intuitionistic fuzzy deductive system and intuitionistic fuzzy filter in Sheffer stroke Hilbert algebras is introduced, several properties are investigated.  The conditions under which an intuitionistic fuzzy set can be an intuitionistic fuzzy filter are explored, and  characterizations of an intuitionistic fuzzy filter are considered.  The process of making an intuitionistic fuzzy filter through the collection of filters is displayed, and the union and intersection of intuitionistic fuzzy filters are discussed. Several properties are investigated in relation to the homomorphsm of Sheffer stroke Hilbert algebras, and finally the relationship between an intuitionistic fuzzy filter and an intuitionistic fuzzy deductive system is established.

The problem of characterizing the maximal left algebras of Toeplitz matrices with quaternion entries is a complex as well as a harder problem that has not received much attention until now. In the current paper, we introduce certain families of maximal left algebras of Toeplitz matrices with entries from an algebra of quaternions that cover various classes of the left algebras of quaternion Toeplitz matrices.

In this article, we study the Local-global Principles for the Artinianness of ordinary local cohomology modules and the finiteness of general local cohomology modules. Let R be a Noetherian ring, Φ be a system of ideals of R and N be an R-module. Assume that S is a Serre subcategory of Mod(R) satisfying the condition CΦ and the Residual F ields condition (briefly RF condition) and let SA be the class of Artinian R-modules. For t ∈ N0, we first show that the general local cohomology modules Hi Φ(N) ∈ S for every i < t if and only if Hi b (N) ∈ S for any b ∈ Φ and every i < t. Then, for a finite R-module N, we conclude that if Hi Φ(N) ∈ SA for every i < t, thus Hi Φ(N) ∈ S for every i < t. Consequently, we show that the least non-negative integer i in which Hi Φ(N) is not Artinian, is a lower bound for all S-depthΦ(N). Finally, we prove that if n ∈ N0 is such that N is in dimension < n and ( AssR(H h n Φ(N) Φ (N))) ≥n is a finite set, then f n Φ (N) = h n Φ(N).

In the current article, fuzzy Sumudu transform iterative method is defined and used to obtain an analytical fuzzy triangular solution of the time-fractional Cauchy reaction-diffusion equation under generalized Hukuhara partial differentiability. On this basis, we prove some properties of fuzzy Sumudu transform. The merits and applicability of the proposed theory are validated through numerical simulation.

‎In this paper‎, ‎we investigated boundedness of difference of general polynomial weighted differentiation composition operators from Cauchy transform spaces into function spaces $S=\{f:\ \ f'\in H^1\}$ and $S^2=\{f:\ \ f'\in H^2\}$ with derivative in Hardy spaces‎. ‎‎‎We also obtained an exact formula for the norm of this operator and prove that there is no composition isometry from the Cauchy transform spaces into $S$ and $S^2$ ‎spaces.‎