Theory of Approximation and Applications
Volume:13 Issue: 1, Winter and Spring 2019

Let R is a commutative ring whit Z(R) as the set of zero divisors. The total graph of R, denoted by T ((R)) is the (undirected) graph with all elements of R as vertices, and two distinct vertices are adjacent if their sum is a zero divisor. For a graph G = (V; E), a set S is a dominating set if every vertex in V n S is adjacent to a vertex in S. The domination number is equal |S|where |S| is minimum. For R-module M, an Nagata extension (idealization), denoted by R(+)M is a ring with identity and for two elements (r; m); (s; n) of R(+)M we have (r; m) + (s; n) = (r + s; m + n) and (r; m)(s; n) = (rs; rn + sm). In this paper, we seek to determine the bound for the domination number of total graph T ((R(+)M)).

In this paper we define fuzzy farthest points, fuzzy best approximation points and farthest orthogonality in fuzzy normed spaces and we will find some results. We prove some existence theorems, also we consider fuzzy Hilbert and show every nonempty closed and convex subset of a fuzzy Hilbert space has an unique fuzzy best approximation.It is well know that the conception of fuzzy sets, firstly defined by Zadeh in 1965. Fuzzy set theory provides us with a framework which is wider than that of classical set theory. Various mathematical structures, whose features emphasize the effects of ordered structure, can be developed on the theory. The theory of fuzzy sets has become an area of active research for the last forty years. On the other hand, the notion of fuzzyness has a wide application in many areas of science and engineering, chaos control, nonlinear dynamical systems, etc. In physics, for example, the fuzzy structure of space time is followed by the fat that in strong quantum gravity regime space time points are determined in a fuzzy manner.

Burgers' equation arises in various areas of applied mathematics,‎‎such as modeling of dynamics‎, ‎heat conduction‎, ‎and acoustic‎‎waves‎ Also‎, ‎this equation has a large variety of applications in‎‎the modeling of water in unsaturated soil‎, ‎dynamics of soil‎‎water‎, ‎models of traffic‎, ‎turbulence and fluid flow‎, ‎mixing and‎‎turbulent diffusion. Many researchers tried to find analytic and numerical solutions of‎‎ this equation by different methods.Sinc method is a powerful numerical tool for finding fast and‎‎accurate solution in various areas of problems.‎In this paper‎, ‎numerical solution of Burgers' equation‎‎is considered by applying Sinc method‎. ‎For this purpose‎, ‎we apply‎‎Sinc method in cooperative with a classic finite difference‎‎formula to‎ ‎Burgers'equation‎. ‎‎The purpose of this paper is to extend the application of the‎‎sinc method for solving Burgers'equation by considering stability‎‎analysis of the method.‎ Numerical examples are provided to verify the validity of proposed method

The two-stage data envelopment analysis models show the performance of individual processes and ‎thus, provide more information for decision-making compared with conventional one-stage models. ‎This article presents a set of additive models (optimistic and pessimistic) to measure inefficiency ‎slacks in which observations are shown with crisp numbers. In the concept of pessimistic efficiency, ‎DMU with balanced input and output data can be scored as efficient. Since pessimistic efficiency ‎represents the minimum efficiency that is guaranteed in any unfavorable conditions, the assessment ‎based on this efficiency is in compliance with our natural meaning, especially in risk-averse situations. ‎Therefore, pessimistic efficiency solely can play a useful role in the DMU ranking. However, it is not ‎a good idea to ignore optimistic efficiency. Hence, it is an inevitable necessity to integrate different ‎performance sizes in order to achieve an overall performance assessment for each DMU. An example ‎of resin manufacturer companies in Iran is presented to explain how to calculate the system and ‎process inefficiency slacks.‎

This paper, about the solution of fuzzy Volterra integral equation of fuzzy Volterra integral equation of second kind (F-VIE2) using spectral method is discussed. The parametric form of fuzzy driving term is applied for F-VIE2. Then three cases for (F-VIE2) are searched to solve them. This classifications are considered based on the sign of interval. The Gauss-Legendre points and Legendre weights for arithmetics in spectral method are used to solve (F-VIE2). Finally two examples are got to illustrate more.

In this paper, we have combined the ideas of the False Position (FP) and Artificial Bee Colony (ABC) algorithms to find a fast and novel method for solving nonlinear equations. Additionally, to illustrate the efficiency of the proposed method, several benchmark functions are solved and compared with other methods such as ABC, PSO and GA.

Fractional order partial differential equations are generalization of classical partialdifferential equations. Increasingly, these models are used in applications such as fluid flow, financeand others. In this paper we examine some practical numerical methods to solve a class of initial-boundary value fractional partial differential equations with variable coefficients on a finite domain. An approach based on the classical Crank-Nicolson method combined with spatial extrapolation is used to obtain temporally and spatially second-order accurate numerical estimates. Stability, consistency, and convergence of the method are examined. It is shown that the fractional Crank-Nicolson method based on the shifted Gr"{u}nwaldformula is unconditionally stable. Some numerical examples are presented and compared with the exact analytical solutionfor its order of convergence.Fractional order partial differential equations are generalization of classical partialdifferential equations. Increasingly, these models are used in applications such as fluid flow, financeand others. In this paper we examine some practical numerical methods to solve a class of initial-boundary value fractional partial differential equations with variable coefficients on a finite domain. An approach based on the classical Crank-Nicolson method combined with spatial extrapolation is used to obtain temporally and spatially second-order accurate numerical estimates. Stability, consistency, and convergence of the method are examined. It is shown that the fractional Crank-Nicolson method based on the shifted Gr"{u}nwaldformula is unconditionally stable. Some numerical examples are presented and compared with the exact analytical solutionfor its order of convergence.