In this paper, we consider spectral approximation of fractional differential equations (FDEs). A main ingredient of our approach is to define a new class of generalized Jacobi functions (GJFs), which is intrinsically related to fractional calculus and can serve as natural basis functions for properly designed spectral methods for FDEs. We establish spectral approximation results for these GJFs ...

It is known that at least ten equivalent definitions of the fractional Laplacian exist in an unbounded domain. Here we derive a further definition based on Mellin transform and it can be used when applied to radial functions. The main finding tested case space-fractional diffusion equation. one-dimensional also considered, such Riesz (namely symmetric Riesz–Feller) derivative established. This ...

The fractional variational calculus is a recent fifield, where classical problems are considered, but in the presence of derivatives. Since there several defifinitions derivatives, it logical to think different types optimality conditions. For this reason, order solve problems, two theorems necessary conditions well known: an Euler-Lagrange equation which involves Caputo and Riemann-Liouville o...

We study the existence and monotone iterative approximation of mild solutions fractional-order neutral differential equations involving a generalized fractional derivative order 0 < α 1 which can be reduced to Riemann–Liouville or Hadamard derivatives. The is obtained via fixed point techniq...

Some new necessary and sufficient conditions for the existence of analytic resolving families operators to linear equation with a distributed Riemann–Liouville derivative in Banach space are established. We study unique solvability natural initial value problem fractional derivatives corresponding inhomogeneous equations. These abstract results applied class boundary problems equations time pol...

Some power series representations of the modified Bessel functions (McDonald functions Kα) are derived using the little known formalism of fractional derivatives. The resulting summation formulae are believed to be new. 1 Fractional derivatives There are several non-trivial examples in mathematics when some quantity, originally defined as integer, can radically extend its original range and ass...

In this paper, we apply the concept of Caputo’s H-differentiability, constructed based on the generalized Hukuhara difference, to solve the fuzzy fractional differential equation (FFDE) with uncertainty. This is in contrast to conventional solutions that either require a quantity of fractional derivatives of unknown solution at the initial point (Riemann–Liouville) or a solution with increasing...

An optimal control problem for the variable-order fractional-integer mathematical model of vaccination Covid 19 is presented in this research, where order derivatives varies during course time interval, becoming fractional or classical when it more favourable. The are defined here using both integral Riemann–Liouville and Caputo derivative. existence, uniqueness, boundedness positivity solution...

This paper delves into the extension and characterization of radial positive definite functions non-integer dimensions. We provide a thorough investigation by employing Riemann–Liouville fractional integral Caputo derivatives, enabling comprehensive understanding these functions. Additionally, we introduce secondary based on Bernstein completely monotone The practical significance our study is ...