الفهرس 
INTRODUCTIONOnly 14 pages are availabe for public view
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Abstract
Differential equations have been a major branch of pure and applied mathematics since their inauguration in the mid 17th century. While their history has been well studied, it remains a vital field of on-going investigation, with the emergence of new connections with other parts of mathematics, fertile interplay with applied subjects, interesting reformulation of basic problems and theory in various periods, new vistas in the 20th century, and so on. In this meeting we considered some of the principal parts of this story, from the launch with Newton and Leibniz up to around 1950.
Differential equations began with Leibniz, the Bernoulli brothers and others
from the 1680s, not long after Newton‟s „fluxional equations‟ in the 1670s. Applications were made largely to geometry and mechanics; isoperimetrical
problems were exercises in optimization.
Most 18th-century developments consolidated the Leibnizian tradition, extending its multi-variate form, thus leading to partial differential equations. Generalization of isoperimetrical problems led to the calculus of variations. New figures appeared, especially Euler, Daniel Bernoulli, Lagrange and Laplace. Development of the general theory of solutions included singular ones, functional solutions and those by infinite series. Many applications were made to mechanics, especially to astronomy and continuous media.
In the 19th century: general theory was enriched by development of the understanding of general and particular solutions, and of existence theorems. More types of equation and their solutions appeared; for example, Fourier analysis and special functions. Among new figures, Cauchy stands out. Applications were now made not only to classical mechanics but also to heat
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theory, optics, electricity and magnetism, especially with the impact of Maxwell. Later Poincare’s introduced recurrence theorems, initially connection with the three-body problem.
In the 20th century: general theory was influenced by the arrival of set theory in mathematical analysis; with consequences for the orisation, including further topological aspects. New applications were made to quantum mathematics, dynamical systems and relativity theory [1].
There is an increasing interest in the study of dynamic systems of fractional order. Extending derivatives and integrals from integer to non-integer order has a firm and long standing theoretical foundation. Leibniz mentioned this concept in a letter to LHopital over three hundred years ago. Following LHopital’s and Leibniz’s first inquisition, fractional calculus was primarily a study reserved to the best minds in mathematics.
Euler [2], Fourier [3] and Laplace [4,5]are among the many that contributed to the development of fractional calculus. Along the history, many found, using their own notation and methodology, definitions that fit the concept of a non-integer order integral or derivative. The most famous of these definitions among mathematicians that have been popularized in the literature of fractional calculus are the ones of Riemann-Liouville and Grunwald-Letnikov. On the other hand, the most intriguing and useful applications of fractional derivatives and integrals in engineering and science have been found in the past one hundred years. In some cases, the mathematical notations evolved in order to be better meet the requirements of physical reality. The best example of this is Caputo fractional derivative, nowadays the most popular fractional operator among engineers and applied scientists, obtained by reformulating the “classical” definition of Riemann-Liouville derivative in order to be possible to solve fractional initial value problems with standard initial conditions.