In this article, we introduce a novel generalized Caputo fractional derivative, using a special type of function known as the Wright function in the definition of the classical Caputo fractional derivative. We also apply the Fourier, Laplace, and Mellin integral transform methods, which are very useful popular mathematical tools in various scientific fields, to the new generalized fractional derivative. Moreover, as illustrative examples, we calculate the new generalized fractional derivative of constant, power, exponential, sine, and cosine functions. Furthermore, we obtain the solutions of the generalized motion, harmonic vibration, and Bessel differential equations defined by the new generalized fractional derivative using the Fourier, Laplace, and Mellin integral transform methods. Finally, we obtain approximate behavior graphs for both the classical Caputo fractional derivative and the generalized Caputo fractional derivative using some specific data and present these graphs comparatively.
Ata, E. and Kıymaz, İ. O. (2026). A novel generalized type of the Caputo fractional derivative: integral transforms, illustrative examples, and solution of fractional-order generalized differential equations. Computational Methods for Differential Equations, (), -. doi: 10.22034/cmde.2026.61736.2682
MLA
Ata, E. , and Kıymaz, İ. O. . "A novel generalized type of the Caputo fractional derivative: integral transforms, illustrative examples, and solution of fractional-order generalized differential equations", Computational Methods for Differential Equations, , , 2026, -. doi: 10.22034/cmde.2026.61736.2682
HARVARD
Ata, E., Kıymaz, İ. O. (2026). 'A novel generalized type of the Caputo fractional derivative: integral transforms, illustrative examples, and solution of fractional-order generalized differential equations', Computational Methods for Differential Equations, (), pp. -. doi: 10.22034/cmde.2026.61736.2682
CHICAGO
E. Ata and İ. O. Kıymaz, "A novel generalized type of the Caputo fractional derivative: integral transforms, illustrative examples, and solution of fractional-order generalized differential equations," Computational Methods for Differential Equations, (2026): -, doi: 10.22034/cmde.2026.61736.2682
VANCOUVER
Ata, E., Kıymaz, İ. O. A novel generalized type of the Caputo fractional derivative: integral transforms, illustrative examples, and solution of fractional-order generalized differential equations. Computational Methods for Differential Equations, 2026; (): -. doi: 10.22034/cmde.2026.61736.2682
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