**Commenced**in January 2007

**Frequency:**Monthly

**Edition:**International

**Paper Count:**5

# Explicit Solution Related Publications

##### 5 Solution of S3 Problem of Deformation Mechanics for a Definite Condition and Resulting Modifications of Important Failure Theories

**Authors:**
Ranajay Bhowmick

**Abstract:**

Analysis of stresses for an infinitesimal tetrahedron leads to a situation where we obtain a cubic equation consisting of three stress invariants. This cubic equation, when solved for a definite condition, gives the principal stresses directly without requiring any cumbersome and time-consuming trial and error methods or iterative numerical procedures. Since the failure criterion of different materials are generally expressed as functions of principal stresses, an attempt has been made in this study to incorporate the solutions of the cubic equation in the form of principal stresses, obtained for a definite condition, into some of the established failure theories to determine their modified descriptions. It has been observed that the failure theories can be represented using the quadratic stress invariant and the orientation of the principal plane.

**Keywords:**
principal stress,
failure criterion,
Explicit Solution,
cubic equation,
stress invariant,
trigonometric

##### 4 Study of Explicit Finite Difference Method in One Dimensional System

**Authors:**
Azizollah Khormali,
Seyyed Shahab Tabatabaee Moradi,
Dmitry Petrakov

**Abstract:**

One of the most important parameters in petroleum reservoirs is the pressure distribution along the reservoir, as the pressure varies with the time and location. A popular method to determine the pressure distribution in a reservoir in the unsteady state regime of flow is applying Darcy’s equation and solving this equation numerically. The numerical simulation of reservoirs is based on these numerical solutions of different partial differential equations (PDEs) representing the multiphase flow of fluids. Pressure profile has obtained in a one dimensional system solving Darcy’s equation explicitly. Changes of pressure profile in three situations are investigated in this work. These situations include section length changes, step time changes and time approach to infinity. The effects of these changes in pressure profile are shown and discussed in the paper.

**Keywords:**
Numerical Simulation,
pressure distribution,
Explicit Solution,
Petroleum reservoir

##### 3 Explicit Solutions and Stability of Linear Differential Equations with multiple Delays

**Authors:**
Felix Che Shu

**Abstract:**

We give an explicit formula for the general solution of a one dimensional linear delay differential equation with multiple delays, which are integer multiples of the smallest delay. For an equation of this class with two delays, we derive two equations with single delays, whose stability is sufficient for the stability of the equation with two delays. This presents a new approach to the study of the stability of such systems. This approach avoids requirement of the knowledge of the location of the characteristic roots of the equation with multiple delays which are generally more difficult to determine, compared to the location of the characteristic roots of equations with a single delay.

**Keywords:**
Lyapunov exponents,
delay differential equation,
multiple delays,
Explicit Solution,
Exponential
Stability

##### 2 Data Envelopment Analysis with Partially Perfect Objects

**Authors:**
Alexander Y. Vaninsky

**Abstract:**

This paper presents a simplified version of Data Envelopment Analysis (DEA) - a conventional approach to evaluating the performance and ranking of competitive objects characterized by two groups of factors acting in opposite directions: inputs and outputs. DEA with a Perfect Object (DEA PO) augments the group of actual objects with a virtual Perfect Object - the one having greatest outputs and smallest inputs. It allows for obtaining an explicit analytical solution and making a step to an absolute efficiency. This paper develops this approach further and introduces a DEA model with Partially Perfect Objects. DEA PPO consecutively eliminates the smallest relative inputs or greatest relative outputs, and applies DEA PO to the reduced collections of indicators. The partial efficiency scores are combined to get the weighted efficiency score. The computational scheme remains simple, like that of DEA PO, but the advantage of the DEA PPO is taking into account all of the inputs and outputs for each actual object. Firm evaluation is considered as an example.

**Keywords:**
Data Envelopment Analysis,
Explicit Solution,
Perfect Object,
Partially
perfect object,
Partial efficiency,
Simplified
algorithm

##### 1 Environmental Performance of the United States Energy Sector: A DEA Model with Non-Discretionary Factors and Perfect Object

**Authors:**
Alexander Y. Vaninsky

**Abstract:**

**Keywords:**
Environmental Performance,
energy sector,
DEA with Non Discretionary Factors,
Explicit Solution,
Perfect Object