Variational Calculus (Springer Monographs In Ma...
Rockafellar contributed to nonsmooth analysis by extending the rule of Fermat, which characterizes solutions of optimization problems, to composite problems using subgradient calculus and variational geometry and thereby bypassing the implicit function theorem. The approach broadens the notion of Lagrange multipliers to settings beyond smooth equality and inequality systems. In his doctoral dissertation and numerous later publications, Rockafellar developed a general duality theory based on convex conjugate functions that centers on embedding a problem within a family of problems obtained by a perturbation of parameters. This encapsulates linear programming duality and Lagrangian duality, and extends to general convex problems as well as nonconvex ones, especially when combined with an augmentation.
Variational Calculus (Springer Monographs in Ma...
The paper is devoted to the development of a comprehensive calculus for directional limiting normal cones, subdifferentials and coderivatives in finite dimensions. This calculus encompasses the whole range of the standard generalized differential calculus for (non-directional) limiting notions and relies on very weak (non-restrictive) qualification conditions having also a directional character. The derived rules facilitate the application of tools exploiting the directional limiting notions to difficult problems of variational analysis including, for instance, various stability and sensitivity issues. This is illustrated by some selected applications in the last part of the paper.
The VIM solution of the fractional semi-derivative equation was developed by Das [31]. Other methods applied to this equation are available in [32] and the monographs [33, 34] in the fractional calculus.
In fact, the algorithm obtained in [189] can be viewed as a generalisation of variational calculus, where variation along time is permitted [170, 190]. In this new approach, the energy equation is not defined as in [189] but results naturally from the variation of the action in time direction. In the present paper, this second approach is adopted.
The content of this thematic series will contain the latest and the most significant results in fractional differential equations and their real world applications. The main aim is to highlight recent advances in this field as well as to bring together the best researchers in the field of fractional calculus and its applications. In the last sixty years, fractional calculus has emerged as a powerful and efficient mathematical tool in the study of several phenomena in science and engineering. As a result, hundreds of research papers, monographs and international conference papers, have been published. Research in fractional differentiation is inherently multi-disciplinary and its application is done in various contexts: elasticity, continuum mechanics, quantum mechanics, signal analysis, biomedicine, bioengineering, social systems, management, financial systems, turbulence, pollution control, landscape evolution, population growth and dispersal, complex systems, medical imaging, and finance, and some other branches of pure and applied mathematics. This special issue aims at promoting the exchange of novel and important theoretical and numerical results, as well as computational methods, to study fractional order systems, and to spread new trends in the area of fractional calculus and its real world applications. 041b061a72