article · AIMS Mathematics
This research introduces a novel mathematical formulation termed the Hilfer cotangent fractional derivative. This formulation combines key features from both the Riemann-Liouville cotangent fractional derivative and the Caputo cotangent fractional derivative. The essential properties of this newly defined derivative are established and examined. Using this framework, the study investigates a class of nonlinear fractional differential problems subject to nonlocal initial conditions. It proves that this differential problem is mathematically equivalent to a cotangent Volterra integral equation. By applying fixed-point theorems, the existence and uniqueness of solutions to these equations are demonstrated. Finally, two illustrative examples are presented to confirm the analytical findings and demonstrate the behaviour of the solutions under the new derivative.
Fractional calculus provides tools for modelling complex systems with memory effects and non-standard rates of change. By establishing a unified derivative that merges two standard mathematical approaches, this work broadens the analytical methods available to mathematicians. It also ensures that equations formulated with this derivative possess well-defined, unique solutions, offering reliable theoretical foundations for future mathematical modelling.
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<abstract><p>In this work, a novel Hilfer cotangent fractional derivative is presented. This derivative combines the characteristics of the Riemann-Liouville cotangent fractional derivative and the Caputo cotangent fractional derivative. The essential properties of the newly introduced derivative are discussed. By utilizing this derivative, a nonlinear fractional differential problem with a nonlocal initial condition is investigated, and its equivalence to a cotangent Volterra integral equation is demonstrated. The uniqueness and existence of solutions are established by employing fixed-point theorems. Additionally, two illustrative examples are provided to illustrate the obtained results.</p></abstract>
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DOI: 10.3934/math.20231450
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