Linear α −Differential Equations


  • Mohammed Shehata Bilbeis Higher Institute for Engineering, Ministry of Higher Education, Egypt


Differential equations, α calculus, fractional calculus, Riemann-Liouville and Caputo fractional derivatives


y α −differential equations, we mean that the branch studies differential equations containing fractional or real order derivatives. In [1], Shehata has overcome the problem of multiple previous definitions of
fractional calculus by putting an accurate definition of the α− fractional calculus using the normal way. He
concluded from this definition that the α fractional calculus is a complex-valued function that depends on
the principal root of the real number. As an extension of the study of fractional calculus and its importance
in applications, in this paper, we study differential equations that contain fractional differentials based on the
Shehata definition. We define and study the so-called linear α− differential equations of the first extension,
higher extension, and system of the first extension. We give the closed formula for each case. To illustrate our
result, we give some numerical examples of fractional differential equations and their solutions.



How to Cite

Shehata, M. (2024). Linear α −Differential Equations. Journal of Fractional Calculus and Nonlinear Systems, 5(1), 12–31.