Analytical and Numerical Solutions for Nonlinear Equations
Volume:7 Issue: 1, Summer and Autumn 2022

Let G be a group and N be a normal subgroup of G. In this paper, we provide some results on c-covers of a pair of groups. Moreover, we prove that every c-perfect pair of groups (G, N) admits at least one c-cover and also we show that a c-cover of a pair of finite groups has a unique domain up to isomorphism under some assumptions.

‎In this paper‎, ‎the Tau method based on shifted Legendre polynomials has been introduced to approximate the numerical solutions of a class of fractional integro-differential equations with a weakly singular kernel‎ . ‎By using operational matrices we reduce the problem to a set of algebraic equations‎ . ‎Also the upper bound of the error of the shifted Legendre expansion is investigated‎. ‎Finally‎, ‎several numerical examples are given to illustrate the high accuracy of the method‎.

In this paper, we propose an improved cuckoo optimization algorithm (ICOA) to determine unknown function u(x) in the Fredholm integral equations of the second kind. To show utility and capability of the ICOA, we solve some Fredholm integral equations of the second kind using the ICOA and Adomian decomposition method and compare results each other. Also, by using the parallelization technique the running time of the algorithm was reduced significantly. Results obtained by implementing the ICOA on a computer with 2.2 GHz CPU and 14Gb RAM.

Connectivity has been used in the past to describe the stability of graphs. If two graphs, have the same connectivity, then it dose not distinguish between these graphs. That is, the connectivity is not a good measure of graph stability. Then we need other graph parameters to describe the stability. Suppose that two graphs have the same connectivity and the order (the number of vertices or edges) of the largest components of these graphs are not equal. Hence, we say that these graphs must be different in respect to stability and so we can define a new measure which distinguishes these graphs. In this paper, the Weak-Tenacity of graph G is introduced as a new measure of stability in this sense and it is defined asTw(G) = minS⊆V(G) { (|S| + me (G-S)) / ω(G-S) : ω(G-S) > 1},Where me(G-S) denotes the number of, edges of the largest component of G-S. At last, We give the Weak-Tenacity of graphs obtained via various operations.

Let P be a group property. A group G is called a locally Pgroup if each finite subset of G is contained in a P-subgroup of G. In this paper some relations between the central factor groups and commutator subgroups in locally nilpotent and locally finite groups are investigated.

‎Let G be a finite group. The notion of ‎‎‎n‎-Engel degree of G,‎ denote by dn(G), is the probability of ‎two ‎randomely chosen elements ‎‎‎x‎‎, ‎‎y‎ ∈ G ‎satisfy‎ the ‎‎n‎-Engel condition [y,nx]=1‎. ‎The case n=1 is the known commutativity degree of G‎. ‎The aim of this ‎paper,‎ is to define ‎and ‎investigate‎‎ the relative 2-Engel degree of a ‎‎‎subgroup H of G ‎as the probability of ‎two ‎randomely chosen elements ‎‎‎x∈G‎‎‎ and ‎y∈H‎‎ ‎satisfy‎ the ‎‎‎2‎‎‎-Engel condition [y,‎x,‎x]=1‎. ‎‎‎

In this paper, we study the notion of c-isoclinism for the pairs of Filippov algebras. Also, we give an equivalent condition for pairs of Filippov algebras to be c-isoclinic. In particular, it is shown that two Filippov algebras are c-isoclinic if and only if then each of them can be constructed from another by using the operations of forming direct sums, taking subalgebras, and factoring Filippov algebras. Moreover, we introduce the concept of c-perfect pair of Filippov algebras and obtain some relations between c-isoclinic and c-perfect pairs of Filippov algebras.

A proper coloring of a graph G is called a dominated coloring whenever each color class is dominated by at least one vertex. The minimum number of colors among all dominated colorings of G is called its dominated chromatic number, denoted by χ_{dom}(G). We define a parameter related to dominated coloring, namely dominated chromatic covering. For a minimum dominated coloring of G, a set of vertices S is called a dominated chromatic covering if each color class is dominated by a vertex of S. The minimum cardinality of a dominated chromatic covering of G is called its dominated chromatic covering number, denoted by θ_{χ_{dom}}(G). It is clear that θ_{χ_{dom}}(G) ≤ χ_{dom}(G). In this paper, we obtain the dominated chromatic number and θ_{χ_{dom}}(G) when G is middle and total graph of paths and cycles.

For the feasibility of using analytical and numerical studies and findings on fractional integral equations for integro-differential of the fractional order, in this work, the equivalence of a fractional Volterra integro-differential equation of the Hammerstein type with a fractional integral equation is investigated in the Banach space. For this purpose, we use the mutual properties of the fractional order derivative and integral on each other.

A dominating set D of a graph G = (V, E) is a nonsplitdominating set if the induced graph hV − Di is connected. The nonsplitdomination number γns(G) is the minimum cardinality of a nonsplitdomination set. The purpose of this paper is to initiate the investigationof those graphs which are critical in the following sense: A graph G iscalled vertex domination critical if γ(G − v) < γ(G) for every vertex vin G. A graph G is called vertex nonsplit critical if γns(G −v) < γns(G)for every vertex v in G. Initially we test whether some particular classesof graph are γns-critical or not and then we have shown that thereis no existence of 2-γns-critical graph. Then 3-γns-critical graphs arecharacterized.

The aim of this paper is to study the existence of infinitely many solutions for discrete fourth-order boundary value problem with four parameters involving oscillatory behaviors of nonlinearity at infinity. The approach is based on variational methods. In addition, one example is presented to illustrate the feasibility and effectiveness of the main result.

The purpose of this paper is to use the decision making techniques of Data Envelopment Analysis (DEA) in order to evaluate the existing Internet networks to select the most desirable networks.To achieve this goal, we first begin by simulating a specific Internet network called Differentiated Service (DS) network that provides the quality of service to the user through the mechanism of Call Admission Control (CAC). We then evaluate and rank the networks by proposing a novel DEA model in the literature of undesirable outputs. Finally, by using the results of DEA model, we select the optimal Internet network.