Existence and linear independence theorem for linear fractional differential equations with constant coefficients (Q6557962)

In this paper, the authors analyze the following homogenous linear equations with constant coefficients \N\[\ND^\alpha u(x)-\sum_{i=1}^k p_i D^{\beta_i} u(x)=0,\ x>0\tag{1}\N\]\Nserving as a base for the theory of fractional differential equations, where \(l-1<\alpha\leq l,\ l\in \mathbb N,\ \alpha>\beta_i\geq0,\ p_i\in \mathbb R \) and \( i=1,2,\cdots,k\) and \(D^\alpha\) represents Caputo and Riemann-Liouville \(\alpha\)-th order fractional derivatives.\N\NSolving and finding the fundamental systems of solutions is critical in any further developments. The most advanced state of the art of the theory of fractional linear ordinary differential equations with constant coefficients is presented in the monograph [\textit{A. A. Kilbas} et al., Theory and applications of fractional differential equations. Amsterdam: Elsevier (2006; Zbl 1092.45003)]. Using the Laplace transform, Kilbas, Srivastava and Trujillo [loc. cit.] presented the solutions in the form of series over the generalized Wright functions, but did not prove the convergence of these series. The authors justified the linear independence of the obtained solutions under the constraint that all derivatives, except for at most one, are within the interval \((0, 1]\). This restriction is an essential gap in the existing theory for constant-coefficient equations.\N\NThe goal of this article is to fill in this gap. The authors introduce a different approach based on the search for solutions in the form of double and multi-sum fractional multi-step power series, which is new for the fractional analysis for the cases of both Riemann-Liouville and Caputo fractional derivatives. They achieve the results [loc. cit.], but, in addition, construct the linearly independent set of l solutions in the space \(C^l(0, \infty)\) allowing, unlike [loc. cit.], close proximity of derivatives \(\alpha\) and \(\beta\) without the above restriction. This approach enables to remove the previously required condition that, for linear independence, all derivatives have to be within the interval \([0, 1]\) with at most one exception. The number of linearly independent solutions is equal to the order of the equation. They also demonstrate the convergence of the series, which allows to prove the existence theorem.\N\NSection 2 addresses equations with Riemann-Liouville derivatives, whereas Section 3 analyses equations with Caputo derivatives. In both cases, they elaborate a new multi-sum fractional series form and prove the existence of l linearly independent continuous solutions. In order to obtain an additional estimate of the validity of the results, they plug the multi-series solution into the equation and see that the residual (error) is negligible as shown in the presented figures. The fractional derivatives are calculated using the substitution method [\textit{P. B. Dubovski} and \textit{J. A. Slepoi}, ``Dual approach as empirical reliability for fractional differential equations'', Preprint, \url{arXiv:2112.09258}].