Ali Youssef

Sohag University · Department of Mathematics

The diversity of problems about the theory of partial differential equations, which we wish to study requires a number of distinct methods from potential theory, mathematical physics, discrete function theory, differential geometry, etc. Thus even though the individual parts of the research are intimately connected, most of the goals are methodically largely independent of each other.

The higher transcendental functions (special functions) have an increasing and recognised role in mathematical physics, fractional calculus, theory of differential equations, quantum mechanics, approximation theory and many branches in science. Special functions have a long history that can be traced back to the past three centuries where the problems of terrestrial and celestial mechanics that were solved in the eighteenth and nineteenth centuries, the boundary value problems of electromagnetism and heat in the nineteenth and the eigenvalue problems of quantum mechanics in the twentieth. The study of higher transcendental functions (special functions) grew up with the calculus and is consequently one of the oldest branches of analysis. The discoveries of new special functions and their applications to new areas of mathematics have initiated a resurgence of interest in this field. Much attention has been devoted over the past decades to generalizing the Theory of Special Functions within two fundamental directions: a fractional calculus and a higher-dimensional setting, mainly motivated by the problems coming from the fields ranging from applied mathematics to classical problems of physics and engineering. Accordingly, the main goal of this work is studying some new interesting higher transcendental functions, investigating some important convergence properties of the in the sense of fractional calculus, and hence provide with some methods to solve different kind of fractional differential equations which are much close to some interesting applications.