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Volume 9
Issue 5
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Article

Approximation of Fractional Caputo Derivative of Variable Order and Variable Terminals with Application to Initial/Boundary Value Problems

by
Paulina Stempin
and
Wojciech Sumelka
*
Institute of Structural Analysis, Poznan University of Technology, Piotrowo 5 Street, 60-965 Poznań, Poland
*
Author to whom correspondence should be addressed.
Fractal Fract. 2025, 9(5), 269; https://doi.org/10.3390/fractalfract9050269
Submission received: 25 March 2025 / Revised: 18 April 2025 / Accepted: 19 April 2025 / Published: 22 April 2025
(This article belongs to the Special Issue Contemporary Methods of Fractional Order Differential and Differential-Operator Equations)
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Abstract

This article presents a method for the approximate calculation of fractional Caputo derivatives, including a crucial aspect of the ability to handle arbitrary—even variable—terminals and order. The proposed method involves rearranging the fractional operator as a series of higher-order derivatives considered at a specific point. We demonstrate the effect of the number of terms included in the series expansion on the solution accuracy and error analysis. The advantage of the method is its simplicity and ease of implementation. Additionally, the method allows for a quick estimation of the fractional derivative by using a few first terms of the expansion. The elaborated algorithm is tested against a comprehensive series of illustrative examples, providing very good agreement with the exact/reference solutions. Furthermore, the application of the proposed method to fractional boundary/initial value problems is included.
Keywords: fractional calculus; fractional differential equation; fractional boundary value problem; fractional initial value problem; fractional derivative approximation; fractional variable order; variable terminal of fractional derivative fractional calculus; fractional differential equation; fractional boundary value problem; fractional initial value problem; fractional derivative approximation; fractional variable order; variable terminal of fractional derivative

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MDPI and ACS Style

Stempin, P.; Sumelka, W. Approximation of Fractional Caputo Derivative of Variable Order and Variable Terminals with Application to Initial/Boundary Value Problems. Fractal Fract. 2025, 9, 269. https://doi.org/10.3390/fractalfract9050269

AMA Style

Stempin P, Sumelka W. Approximation of Fractional Caputo Derivative of Variable Order and Variable Terminals with Application to Initial/Boundary Value Problems. Fractal and Fractional. 2025; 9(5):269. https://doi.org/10.3390/fractalfract9050269

Chicago/Turabian Style

Stempin, Paulina, and Wojciech Sumelka. 2025. "Approximation of Fractional Caputo Derivative of Variable Order and Variable Terminals with Application to Initial/Boundary Value Problems" Fractal and Fractional 9, no. 5: 269. https://doi.org/10.3390/fractalfract9050269

APA Style

Stempin, P., & Sumelka, W. (2025). Approximation of Fractional Caputo Derivative of Variable Order and Variable Terminals with Application to Initial/Boundary Value Problems. Fractal and Fractional, 9(5), 269. https://doi.org/10.3390/fractalfract9050269

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