Theory of Approximation and Applications
Volume:12 Issue: 2, Summer and Autumn 2018

In recent years, Fuzzy differential equations are very useful indifferent sciences such as physics, chemistry, biology and economy. It should be noted, that if the equations that appear to be uncertain, then take help of fuzzy logic at these equations. Considering that most of the time analytic solution of such equations and finding an exact solution has either high complexity or cannot be solved, we applied numerical methods for the solution. The topics of fuzzy differential equations have been rapidly growing in recent years. So far, many methods have been presented to solve the first-order differential equations. Not many studies have been conducted for numerical solution of high-order fuzzy differential equations. In this research, first, the equation by reducing time, we become the first-order equation. Then we have applied Adam-Moulton multi-step methods for the initial approximation of one order differential equations. Finally, we examine the accuracy of method by presenting examples.

The Rényi entropy is a generalization of Shannon entropy to a one-parameter family of entropies. Tsallis entropy too is a generalization of Shannon entropy. The measure for Tsallis entropy is non-logarithmic. After the introduction of Shannon entropy , the conditional Shannon entropy was derived and its properties became known. Also, for Tsallis entropy, the conditional entropy was introduced and its properties were shown. But no specific definition has been given for the conditional Rényi entropy. Several authors have used some definitions of the conditional Rényi entropy, to find their properties and relations among them, but there is no general agreement on any specific definition In this paper, we focus on the definitions of the conditional Rényi entropy, and select one of them on the basis of a relation between Rényi and Tsallis entropies, and show that the chain rule holds generally for the case of conditional Rényi entropy. Then, using this definition, we show some of the properties of conditional Rényi entropy. Finally, we show the relations among Rényi, Shannon and Tsallis entropies.

In this paper, we will obtain first integral, integrating factor and λ-symmetry of third-order ODEs u ⃛=F(x,u,u ̇,u ̈). Also we compare Prelle -Singer (PS) method and λ-symmetry method for third-order differential equations.

In this paper, a numerical implementation of an expansion method is developed for solving Abel's integral equations of the first and second kind. The solution of such equations may demonstrate a singular behaviour in the neighbourhood of the initial point of the interval of integration. The suggested method is based on the use of Taylor series expansion to overcome the singularity which leads to approximating the unknown function and it's derivatives in terms of Chebyshev polynomials of the first kind. The proposed method, transforms the Abel's integral equations of the first and second kind into a system of linear algebraic equations which can be solved by Gaussian elimination algorithm. Finally, some numerical examples are included to clarify the accuracy and applicability of the presented method which indicate that proposed method is computationally very attractive. In this paper, all numerical computations were carried out on a PC executing some programs written in maple software.

Let H be an innite dimensional Hilbert space and K(H) be the set of all compact
operators on H. We will adopt spectral theorem for compact self-adjoint operators, to investigate of
higher derivation and higher Jordan derivation on K(H) associated with the following Cauchy-Jensen
type functional equation
2f((T + S)/2+ R) = f(T ) + f(S) + 2f(R)
for all T, S, R are in K(H).

Data Envelopment Analysis (DEA) is actually to obtain the efficiency using inputs and outputs, which can determine efficient and inefficient units with the help of performance calculations such that the efficiency for efficient DMUs is one and less than one for inefficient DMUs [1]. In some cases, the ranking of the decision-making units are not important for decision-makers, and they are only looking to obtain the most efficient DMUs, so that they can directly achieve the most efficient DMU from all existing DMUs. In a number of papers regarding this subject, several steps were taken to find the most efficient DMU, [2,3], which later examined the problems of these models and other models were announced by the researchers to resolve them. Some of the problems that can be mentioned:1. Solving the model took place in two steps and could not directly reach the final answer.
2. Many unnecessary conjunctions were used in the models.
Other models were proposed to solve the problems that could eliminate unnecessary conjunctions and solve the problem in two phases [5].
Therefore, in this paper, it has been tried to provide a model that avoids unnecessary conjunctions, and most importantly, maximizes the distance between the other DMUs of an efficient DMU [6].

Charles Dodgson (1866) introduced a method to calculate matrices determinant, in a simple way. The method was highly attractive, however, if the sub-matrix or the main matrix determination is divided by zero, it would not provide the correct answer. This paper explains the Dodgson method's structure and provides a solution for the problem of "dividing by zero" called "virtual center".

This paper discusses about the solution of fuzzy Volterra integral equation of first-kind (F-VIE1) using spectral method. The parametric form of fuzzy driving term is applied for F-VIE1, then three classifications for (F-VIE1) are searched to solve them. These classifications are considered based on the interval sign of the kernel. The Gauss-Legendre points and Legendre weights for arithmetics in spectral method are used to solve (F-VIE1). Finally, two examples are got to illustrate more. However, accuracy and efficiency are shown in tables.