Global Analysis and Discrete Mathematics
Volume:8 Issue: 1, Summer and Autumn 2023

This paper introduces two families of modified Householder’s method (HM) that are optimal in line with Kung-Traub conjecture given in [4]. The modification techniques employed involved approximation of the function derivatives in the HM with divided difference operator, a polynomial function approximation and the modified Wu function approximation in [17]. These informed the formation of two families of methods that that are optimal and do not or require function derivative evaluation. The both families do not breakdown when f(·) ≈ 0 as in the case with the HM and many existing modified HM. From the convergence investigation carried out on the methods, the sequence of approximations produced by the methods, converged to solution of nonlinear equation with order four. The implementation of the methods was illustrated and numerical results obtained were compared with that of some recently developed methods.

In this paper, we deal with the existence of a positive solution for the following fractional discrete boundary-value problemT+1∇αk(k∇α0(UK)))=λƒ(K,U(K)), k∈[1,T]N0,u(0)=u (T+1)=0,where 0<α<1 and k∇α0 is the left nabla discrete fractional difference and T+1∇αk is the right nabla discrete fractional difference ƒ:[1,T]N0×(0,+∞)→R may be singular at t=0  and may change sign and λ>0 is a parameter. The technical method is variational approach for differentiable functionals. An example is included to illustrate the main results.

Let G be a molecular graph. The eccentric connectivity index, ξ(G) , is defined as, ξ(G)=∑deg(u).ecc(u) , where deg(u) denotes the degree of vertex u and ecc(u) is the largest distance between u and any other vertex v of G. In this paper, an exact formula for the eccentric connectivity index of nanostar dendrimer NS3[n] is given.

Assume we have a set of k colors and to each vertex of a graph G we assign an arbitry of these colors. If we require that each vertex to set is assigned has in its closed neighborhood all k colors, then this is called the generalized k-rainbow dominating function of a graph G. The corresponding γgkr, which is the minimum sum of numbers of assigned colores over all vertices of G, is called the gk-rainbow domination number of G. In this paper we present a linear algorithms for determining a minimum generalized 2-rainbow dominating set of a tree and on GP(n,2).

A group G is called n-capable if for a suitable group H we have G ≅H/Zn(H). In this article, we impose some conditions to an n-capable group G and find a group H with the mentioned condition such that G ≅ H/Zn(H).

Corruption is a slowly decaying poison in Nigeria. Corruption is a global problem that individuals in a community can be exposed to. This paper developed the dynamics of corruption and the compartments were divided into six sections: Susceptible, Exposed, Corrupt, Honest, Punished and Recovered. The paper was designed to deal with the stability of corrupt individuals and, using the homotopy perturbation technique, the model equations are solved for simulations to performed numerically. The analysis findings demonstrate that the corruption-free equilibrium is locally asymptotically stable if R0<1, indicating that there is corruption in the population. disappears and if R0>1, means that the number of corruption rises per-capital in a society. Also from the results, the homotopy perturbation method shows accuracy and convergence very quickly for numerical simulations despite it require perturbation for convergent. The observations and suggestions are outlined to have a corruption-free society.

In this research paper, a numerical method for one- and two- dimensional heat equation with nonlinear diffusion conductivity and source terms is proposed. In this work, the numerical technique is based on the polynomial differential quadrature method for discretization of the spatial domain. The resulting nonlinear system time depending ordinary differential equations is discretized by using the second order Runge–Kutta methods. The Chebyshev-Gauss-Lobatto points in this paper are used for collocation points in spatial discretization. We study accuracy in terms of L_∞ error norm and maximum absolute error along time levels. Finally, several test examples demonstrate the accuracy and efficiency of the proposed schemes. It is shown that the numerical schemes give better solutions. Moreover, the schemes can be easily applied to a wide class of higher dimension nonlinear diffusion equations.

Many interesting structures are arising from the Tower of Hanoi puzzle. Some of them increase the number of pegs and some of the others relax the Divine Rule. But all of them accept discs of different diameters. In this paper, we increased the number of available pegs and changed the Divine Rule by considering similar discs, that is, all discs have the same size diameter. From this point of view, the Tower of Hanoi puzzle becomes the distributing of n identical discs (objects) into k distinct labeled pegs (boxes). We modify Lucas’s legend to justify these variations. Each distribution of n discs on kpegs is a regular state. In a Diophantine Graph, every possible regular state is represented by a vertex. Two vertices are adjacent in a Diophantine Graph if their corresponding states differ by one move. The Diophantine Graphs have shown to possess attractive structures. Since it can be embedded as a subgraph of a Hamming Graph, the Diophantine Graph may find applications in faulttolerant computing.

It has been conjectured there is a Hamiltonian cycle in every Cayley graph. Interest in this and other closely related questions has grown in the past few years. There have been many papers on the topic, but it is still an open question whether every connected Cayley graph has a Hamiltonian cycle. In this paper, we survey the results, techniques, and open problems in the field.

Let E and F be two Riesz spaces. An operator T : E→ F between two Riesz spaces is said to be unbounded order-to-order continuous whenever x∝→ 0 in E implies Tx∝ → 0 in F for each net (x∝)⊆ E. This paper aims to investigate several properties of a novel class of operators and their connections to established operator classifications. Furthermore, we introduce a new class of operators, which we refer to as order-to-unbounded order continuous operators. An operator T : E→ F rightarrow F between two Riesz spaces is said to beorder-to-unbounded order continuous (for short, ouo-continuous), if x∝→ 0 in E implies Tx∝ → 0 in F for each net (x∝)⊆ E.In this manuscript, we investigate the lattice properties of a certain class of objects and demonstrate that, under certain conditions, order continuity is equivalent to unbounded order-to-order continuity of operators on Riesz spaces. Additionally, we establish that the set of all unbounded order-to-order continuous linear functionals on a Riesz space E forms a band of E∼.

This study aims to develop a robust numerical algorithm for solving parabolic partial differential equations (PDEs) arising in the domain of financial mathematics. The proposed approach leverages the finite difference method (FDM) to discretize the temporal and spatial domains of the problem. To approximate the unknown solution, we employ a polynomial interpolation technique, ensuring high accuracy and stability in the numerical solution. The effectiveness and efficiency of our method are demonstrated through comprehensive numerical experiments, showcasing its potential for practical applications in financial modeling.

Suppose that G is a connected graph constructed from pairwise disjoint connected graphs G1,... ,Gt by selecting a vertex of G1, a vertex of G2, and identifying these two vertices. Then continue in this manner inductively. The graphs G1,... ,Gk are the primary subgraphs of G. Some particular cases of these graphs are important in chemistry which we consider them in this paper and study their elliptic Sombor index.