The Heat-Balance Integral Method (HBIM) of Goodman under classic prescribed temperature boundary conditions has been studied towards it optimization. Because the parabolic profile satisfies both the boundary conditions and the heat-balance integral at any value of the exponent the calibration is of a primary importance in generation of the approximate solution. The simple 1-D heat conduction problem, enabling one to demonstrate the HBIM performance with the entropy generation minimization (EGM) concept in calibration of a parabolic temperature profile with unspecified exponents, has been developed. The EGM concept provides constraints that impose addition boundary conditions at the approximate parabolic profile. Additionally, entire domain optimizations based on the mean-squared error concept has been performed in two versions -the method Myers and through a similarity transformed diffusion equation.

Fractional calculus has performed an important role in the fields of mathematics, physics, electronics, mechanics, and engineering in recent years. The modeling methods involving fractional operators have been continuously generalized and enhanced, especially during the last few decades. Many operations in physics and engineering can be defined accurately by using systems of differential equations containing different types of fractional derivatives.This book is a result of the contributions of scientists involved in the special collection of articles organized by the journal Fractal and Fractional (MDPI), most of which have been published at the end of 2017 and the beginning 2018. In accordance with the initial idea of a Special Issue, the best published have now been consolidated into this book.The articles included span a broad area of applications of fractional calculus and demonstrate the feasibility of the non-integer differentiation and integration approach in modeling directly related to pertinent problems in science and engineering. It is worth mentioning some principle results from the collected articles, now presented as book chapters, which make this book a contemporary and interesting read for a wide audience:The fractional velocity concept developed by Prodanov [1] is demonstrated as tool to characterize Hölder and in particular, singular functions. Fractional velocities are defined as limits of the difference quotients of a fractional power and they generalize the local notion of a derivative. On the other hand, their properties contrast some of the usual properties of derivatives. One of the most peculiar properties of these operators is that the set of their nontrivial values is disconnected. This can be used, for example, to model instantaneous interactions, such as Langevin dynamics. In this context, the local fractional derivatives and the equivalent fractional velocities have several distinct properties compared to integer-order derivatives.The classical pantograph equation and its generalizations, including fractional order and higher order cases, is developed by Bhalekar and Patade [2]. The special functions are obtained from the series solution of these equations. Different properties of these special functions are established, andtheir relations with other functions are developed.The new direction in fractional calculus involving nonsingular memory kernels, developed in the last three years following the seminar articles of Caputo and Fabrizio in 2015 [3], is hot research topic. Two studies in the collection clearly demonstrate two principle directions: operators with nonsingular exponential kernels, i.e., the so-called Caputo-Fabrizio derivatives [4,5] (Hristov, 2016, Chapter 10), and operators with nonsingular memory kernels based on the Mittag-Leffler function [6,7] (Atangana, Baleanu, 2016; Baleanu, Fennandez, 2018).Yavuz and Ozdemir [8] demonstrate a novel approximate-analytical solution method, called the Laplace homotopy analysis method (LHAM), using the Caputo-Fabrizio (CF) fracti...

An En tropy Gen er a tion Ap proach in Op ti mal Pro file De ter mi na tion by Jor dan HRISTOV Orig i nal sci en tific pa per UDC: 536.75:517.95The heat-bal ance in te gral method of Good man is stud ied with two sim ple 1-D heat con duc tion prob lems with pre scribed tem per a ture and flux bound ary con di tions. These clas si cal prob lems with well known ex act so lu tions en able to dem on strate the heat-bal ance in te gral method performance by a par a bolic pro file and the en tropy gen er a tion minimization con cept in def i ni tion of the ap pro pri ate pro file ex po nent. The ba sic as sump tion gen er at ing the ad di tional con straints needed to per form the so lu tion is based on the re quire ment to min i mize the dif fer ence in the lo cal ther mal en tropy gen er a tion rates cal cu lated by the ap prox i mate and the ex act pro file, respec tively. This con cept is eas ily ap pli ca ble since the gen eral con cept has sim ple im ple men ta tion of the con di tion re quir ing the ther mal en tropy generations cal culated through both pro files to be the same at the bound ary. The en tropy minimization gen er a tion ap proach au to mat i cally gen er ates the ad di tional re quirement which is de fi cient in the set of con di tions de fined by the heat-bal ance in te gral method con cept.

A comprehensive understanding of fractional systems plays a pivotal role in practical applications [...]

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