TY - JOUR
T1 - Analysis of solutions of some multi-term fractional Bessel equations
AU - Dubovski, Pavel B.
AU - Slepoi, Jeffrey
N1 - Publisher Copyright:
© 2021 Diogenes Co., Sofia.
PY - 2021/10/1
Y1 - 2021/10/1
N2 - We construct the existence theory for generalized fractional Bessel differential equations and find the solutions in the form of fractional or logarithmic fractional power series. We figure out the cases when the series solution is unique, non-unique, or does not exist. The uniqueness theorem in space Cp is proved for the corresponding initial value problem. We are concerned with the following homogeneous generalized fractional Bessel equation i=1mdix α iD α iu(x)+(x β-ν 2)u(x)=0,αi> - >0,β> - >0, which includes the standard fractional and classical Bessel equations as particular cases. Mostly, we consider fractional derivatives in Caputo sense and construct the theory for positive coefficients di. Our theory leads to a threshold admissible value for ν2, which perfectly fits to the known results. Our findings are supported by several numerical examples and counterexamples that justify the necessity of the imposed conditions. The key point in the investigation is forming proper fractional power series leading to an algebraic characteristic equation. Depending on its roots and their multiplicity/complexity, we find the system of linearly independent solutions.
AB - We construct the existence theory for generalized fractional Bessel differential equations and find the solutions in the form of fractional or logarithmic fractional power series. We figure out the cases when the series solution is unique, non-unique, or does not exist. The uniqueness theorem in space Cp is proved for the corresponding initial value problem. We are concerned with the following homogeneous generalized fractional Bessel equation i=1mdix α iD α iu(x)+(x β-ν 2)u(x)=0,αi> - >0,β> - >0, which includes the standard fractional and classical Bessel equations as particular cases. Mostly, we consider fractional derivatives in Caputo sense and construct the theory for positive coefficients di. Our theory leads to a threshold admissible value for ν2, which perfectly fits to the known results. Our findings are supported by several numerical examples and counterexamples that justify the necessity of the imposed conditions. The key point in the investigation is forming proper fractional power series leading to an algebraic characteristic equation. Depending on its roots and their multiplicity/complexity, we find the system of linearly independent solutions.
KW - characteristic equation
KW - existence
KW - fractional power series
KW - generalized fractional Bessel equation
KW - logarithmic fractional series
KW - threshold value
KW - uniqueness
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U2 - 10.1515/fca-2021-0059
DO - 10.1515/fca-2021-0059
M3 - Article
AN - SCOPUS:85119075161
SN - 1311-0454
VL - 24
SP - 1380
EP - 1408
JO - Fractional Calculus and Applied Analysis
JF - Fractional Calculus and Applied Analysis
IS - 5
ER -