Operator-Based Approach for the Construction of Solutions to (<sup><i>C</i></sup><b>D</b><sup>(1/<i>n</i>)</sup>)<sup><sup><i>k</i></sup></sup>-Type Fractional-Order Differential Equations

Operator-Based Approach for the Construction of Solutions to (<sup><i>C</i></sup><b>D</b><sup>(1/<i>n</i>)</sup>)<sup><sup><i>k</i></sup></sup>-Type Fractional-Order Differential Equations

A novel methodology for solving Caputo <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mfenced separators="" open="(" close=")"><mrow><mmultiscripts><mi...

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Bibliographic Details
Main Authors:	Inga Telksniene, Zenonas Navickas, Romas Marcinkevičius, Tadas Telksnys, Raimondas Čiegis, Minvydas Ragulskis
Format:	Article
Language:	English
Published:	MDPI AG 2025-04-01
Series:	Mathematics
Subjects:	fractional differential equation operator calculus fractional power series expansion
Online Access:	https://www.mdpi.com/2227-7390/13/7/1169
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Summary:	A novel methodology for solving Caputo <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mfenced separators="" open="(" close=")"><mrow><mmultiscripts><mi mathvariant="bold">D</mi><none></none><mrow><mo>(</mo><mn>1</mn><mo>/</mo><mi>n</mi><mo>)</mo></mrow><mprescripts></mprescripts><none></none><mi>C</mi></mmultiscripts></mrow></mfenced><mi>k</mi></msup></semantics></math></inline-formula>-type fractional differential equations (FDEs), where the fractional differentiation order is <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>k</mi><mo>/</mo><mi>n</mi></mrow></semantics></math></inline-formula>, is proposed. This approach uniquely utilizes fractional power series expansions to transform the original FDE into a higher-order FDE of type <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mfenced separators="" open="(" close=")"><mrow><mmultiscripts><mi mathvariant="bold">D</mi><none></none><mrow><mo>(</mo><mn>1</mn><mo>/</mo><mi>n</mi><mo>)</mo></mrow><mprescripts></mprescripts><none></none><mi>C</mi></mmultiscripts></mrow></mfenced><mrow><mi>k</mi><mi>n</mi></mrow></msup></semantics></math></inline-formula>. Significantly, this perfect FDE is then reduced to a <i>k</i>-th-order ordinary differential equation (ODE) of a special form, thereby allowing the problem to be addressed using established ODE techniques rather than direct fractional calculus methods. The effectiveness and applicability of this framework are demonstrated by its application to the fractional Riccati-type differential equation.
ISSN:	2227-7390