Summary
The mathematical sciences, and particularly probabilistic and statistical methods, are key to understanding the dependencies of the systems. The purpose of this paper is to encourage a wider recognition by engineers of a new generalized principle which in its mathematical form is a powerful instrument for the solution of practical problems. Generalized probability density function was introduced to permit analysis without pre‐knowledge of the source of the data. The fundamental principles are extended to apply the most related engineering applications without the need to know the type of source generating the data. The generalized model presented eliminates preliminary work in engineering problems. The proposed model introduces an exponential density function to produce a direct solution to randomly varying data. The exponential density function is fully compatible with applications containing randomly distributed data. The success of the generalized model presented is due to the calculated parameters in the exponential density function. The method is applied to various problems chosen from the field of engineering with great success.
In recent studies, papers related to the multiplicative based numerical methods demonstrate applicability and efficiency of these methods. Numerical root-finding methods are essential for nonlinear equations and have a wide range of applications in science and engineering. Therefore, the idea of root-finding methods based on multiplicative and Volterra calculi is self-evident. Newton-Raphson, Halley, Broyden, and perturbed root-finding methods are used in numerical analysis for approximating the roots of nonlinear equations. In this paper, Newton-Raphson methods and consequently perturbed root-finding methods are developed in the frameworks of multiplicative and Volterra calculi. The efficiency of these proposed root-finding methods is exposed by examples, and the results are compared with some ordinary methods. One of the striking results of the proposed method is that the rate of convergence for many problems are considerably larger than the original methods.
Summary
In this paper, generalized model based on the multiplicative least square method presented and showed that it is convenient to approximate observation in the electrical system analysis. The introduced model is exponential based and relies on parametric values. The advantage of the method is due to exponential derivation process within multiplicative calculus and has the flexibility to represent widely used functions such as Gaussian and exponentials. In spite of numerous results on the best fitting model, we study the robustness of the method by making direct comparisons with Matlab built‐in functions. The presented model is challenging because modern electrical circuits and systems are faced with different types of inputs that require near exact representation for accurate processing. Some real applications of exponential‐based data were selected to demonstrate the applicability and efficiency of the proposed representation.
Generalizations of the usual definition of saddle point and equilibrium point are introduced in this paper. The existence of these points is shown to be related to a class of functions that we call perturbed convex functions. First and second order conditions regarding the existence of these points are also proved. 2004 Elsevier Inc. All rights reserved.
The exponential‐based probability density functions (PDFs) such as Gaussian, exponential, and Rayleigh are widely used in lifetime modeling, survival analysis, and engineering. Recently, the generalized probability density functions consisting a well‐known exponential PDFs were introduced and successfully applied in electrical and electronic engineering. After encouraging results of generalized PDF, further theories and applications are considered within this paper. The parameters of the new (GE) PDF were successfully derived by using the maximum likelihood estimation method. The applications in the paper show that the new generalized probability density function based on maximum likelihood parameter estimators provides more effective probabilistic representation. In this study, statistical properties of new GE distribution is discussed. Some reliability characteristics such as hazard function, mean residual lifetime (MRL) function, and mean past lifetime (MPL) functions are obtained based on special functions. Also, r$r$th moment of the new GE random variable is obtained in analytical form. Maximum likelihood estimates (MLEs) of parameters are also discussed. Lastly, some illustrative examples in engineering and survival analysis are provided based on different real data sets.
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