For fractional derivatives and time-fractional differential equations, we construct a framework on the basis of operator theory in Sobolev spaces. Our provides feasible extension classical Caputo Riemann–Liouville within spaces orders, including negative ones. approach enables unified treatment for calculus equations. We formulate initial value problems ordinary equations boundary partial to pr...

Necessary and sufficient conditions are explored for the asymptotic stability instability of linear two-dimensional autonomous systems fractional-order differential equations with Caputo derivatives. Fractional-order-dependent fractional-order-independent properties fully characterized, in terms main diagonal elements systems' matrix, as well its determinant.

This study investigates the problem of finite-time boundedness a class neural networks Caputo fractional order with time delay and uncertain terms. New sufficient conditions are established by constructing suitable Lyapunov functionals to ensure that addressed fractional-order stable. Criteria for considered models also achieved. The obtained results based on newly developed property derivative...

Abstract: In this paper, we study the approximation of analytical solution for Fractional Reaction Diffusion model which describes the evaluation of bacterium Bacillus, which grows on the surface of thin agar plates, by using homotopy analysis transform method. The fractional derivatives are described by caputo sense. A comparative study between the homotopy analysis transform method and the cl...

in this paper, a new numerical method for solving fractional optimal control problems (focps) is presented. the fractional derivative in the dynamic system is described in the caputo sense. the method is based upon biorthogonal cubic hermite spline multiwavelets approxima-tions. the properties of biorthogonal multiwavelets are first given. the operational matrix of fractional riemann-lioville i...

Here we define a Caputo like discrete nabla fractional difference and we produce discrete nabla fractional Taylor formulae for the first time. We estimate their remaiders. Then we derive related discrete nabla fractional Opial, Ostrowski, Poincaré and Sobolev type inequalities .

When a particle distributes at rate that deviates from the classical Brownian motion model, fractional space derivatives have been used to simulate anomalous diffusion or dispersion. derivative substitutes second-order in dispersion amplified occurs (named super-diffusion). The proposed approach this paper allows seeing physical background of newly defined Caputo space-time-fractional and indic...