TY - JOUR
T1 - Existence and linear independence theorem for linear fractional differential equations with constant coefficients
AU - Dubovski, Pavel B.
AU - Slepoi, Jeffrey A.
N1 - Publisher Copyright:
© 2024 Walter de Gruyter GmbH, Berlin/Boston 2024.
PY - 2024
Y1 - 2024
N2 - We consider the l-th order linear fractional differential equations with constant coefficients. Here l ∈ ℕ {l\in\mathbb{N}} is the ceiling for the highest derivative of order α, l - 1 < α ≤ l {l-1<\alpha\leq l}. If β i < α {\beta_{i}<\alpha} are the other derivatives, the existing theory requires α - max { β i } ≥ l - 1 {\alpha-\max\{\beta_{i}\}\geq l-1} for the existence of l linearly independent solutions. Thus, at most one derivative may have order greater than one, but all other derivatives must be between zero and one. We remove this essential restriction and construct l linearly independent solutions. With this aim, we remodel the series approaches and elaborate the multi-sum fractional series method in order to obtain the existence and linear independence results. We consider both Riemann-Liouville or Caputo fractional derivatives.
AB - We consider the l-th order linear fractional differential equations with constant coefficients. Here l ∈ ℕ {l\in\mathbb{N}} is the ceiling for the highest derivative of order α, l - 1 < α ≤ l {l-1<\alpha\leq l}. If β i < α {\beta_{i}<\alpha} are the other derivatives, the existing theory requires α - max { β i } ≥ l - 1 {\alpha-\max\{\beta_{i}\}\geq l-1} for the existence of l linearly independent solutions. Thus, at most one derivative may have order greater than one, but all other derivatives must be between zero and one. We remove this essential restriction and construct l linearly independent solutions. With this aim, we remodel the series approaches and elaborate the multi-sum fractional series method in order to obtain the existence and linear independence results. We consider both Riemann-Liouville or Caputo fractional derivatives.
KW - Constant-coefficient fractional differential equations
KW - double fractional series
KW - linear independence
KW - multi-sum fractional series
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U2 - 10.1515/jaa-2023-0009
DO - 10.1515/jaa-2023-0009
M3 - Article
AN - SCOPUS:85182576889
SN - 1425-6908
JO - Journal of Applied Analysis
JF - Journal of Applied Analysis
ER -