A double ambiguity has been charged against Rawls’s Difference principle (DP). Is it Maximin, Leximin, or something else? Usually, following A. Sen, scholars identify DP with the so-called Leximin. One argues here that one has to distinguish 1° the Leximin, 2° the Maximin (as rule of justice formally analogous to the maximin rule of decision), represented by the figure in L of the perfectly substitutable goods, and 3° the genuine DP. When the augmentation of inequality benefits the worse off, only Pareto-strong improvements are permitted. Leximin would also permit Pareto-weak improvements too (after the first maximum D), where only the richest improves: from (2, 3) to (2, 5), say. This is forbidden by DP. With two classes, unlike Maximin, DP has no curve of inDifference and is always decisive, as Leximin is. For undecisive Rules of Justice, which admit indifferent curves, I propose to add a lexically secondary rule, to break ties. That move is able to clarify the links and the Differences between on the one hand Maximin alone, with its typical inDifference curves in L, and on the other hand, the DP properly understood and the Leximin, which both have no indifferent curves. With two classes of persons (best off/worse off), DP appears more egalitarian than Leximin, because it's secondary rule is MinIn (Minimization of Inequality). But the intuition behind the distinction is that it cannot possible “fair” that only the best off improves in a productive social cooperation.

Introduction: In this paper, we have introduced a new Method for solving a class of the partial integro-differential equation with the singular kernel by using the finite Difference Method. One of the best subjects in the numerical analysis is a finite Difference Method (FDM). We used (FDM) to solve problems in mathematical physics, integral equations, and engineering, such as electromagnetic potential, fluid flow, radiation heats transfer, laminar boundary-layer theory and mass transport, Abel integral equations, and problem of mechanics or physics. Also in some physical problems such as fluid flow and heat transfer problems, the Laplace equations and the Poisson equations are describe by (FDM). In real life most phenomena are modelled by partial differential equations. Material and Methods: First, we employing an algorithm for solving the problem based on the Crank-Nicholson scheme with given conditions. Furthermore, we discrete the singular integral for solving of the problem. Also, the numerical results obtained here can be compared with the cubic B-spline Method. Results and discussion: In addition, solving some examples demonstrates the validity and applicability of the approached Method, so that the results are reported in the tables and their figures are shown. The high speed of the calculations, and the assurance of having an approximate solution are obtain by proving the stability of the Method. Conclusion: The following conclusions were drawn from this research. Coefficients of the approximate function via Crank-Nicholson scheme are found very easily and therefore many calculations are reduced. The numerical results obtained here can be compared with the cubic B-spline Method The assurance of having an approximate solution are obtain by proving the stability of the Method.

Nowadays analysis of mechanical seals is of great importance since it causes improvement in the performance of mechanical systems and it enables one to reduce the leakage in the seal. In this paper a MTM 30 mechanical seal has been analyzed. The analysis has been conducted in both thermal and isothermal modes using finite Difference Method. The pressure distribution as well as film thickness has been predicted and compared to the published results available in the literature. The results show that considering thermal effects results in an increase in pressure for 16.55% and a reduction in film thickness of 33.4% compared to the situations in which isothermal analysis is conducted. A parametric study has been conducted which evaluates the effect of speed, viscosity, and load on the performance of the mechanical seal. Film thickness, pressure distribution, and film thickness has been studied in this parametric study. The results in this Method have been shown to be in good comparison with other results.

This manuscript suggests an efficient scheme to find an approach for a class of differential equations arsing in the quantum calculus. The present scheme considers the solution in the form of a truncated Taylor series near zero with unknown coefficients. Then, by placing this approach into the problem and collocating the relation which is obtained at some nodes, a system of algebraic equations is achieved. The solution of this algebraic system is the unknown coefficients of the series. The ability of present Method is examined by some examples.

In financial markets, dynamics of underlying assets are often specified via stochastic differential equations of jump-diffusion type. In this paper, we suppose that two financial assets evolved by correlated Brownian motion. The value of a contingent claim written on two underlying assets under jump diffusion model is given by two-dimensional parabolic partial integro-differential equation ( P I D E ), which is an extension of the Black-Scholes equation with a new integral term. We show how basket option prices in the jump-diffusion models, mainly on the Merton model, can be approximated using finite Difference Method. To avoid a dense linear system solution, we compute the integral term by using the Trapezoidal Method. The numerical results show the efficiency of proposed Method.