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    A novel fractional structure of a multi-order quantum multi-integro-differential problem
    (Springer Science and Business Media Deutschland GmbH, 2020) Duc Phuong, Nguyen; Sakar, Fethiye Müge; Etemad, Sina; Rezapour, Shahram
    In the present research manuscript, we formulate a new generalized structure of the nonlinear Caputo fractional quantum multi-integro-differential equation in which such a multi-order structure of quantum integrals is considered for the first time. In fact, in the light of this type of boundary value problem equipped with the multi-integro-differential setting, one can simply study different cases of the existing usual integro-differential problems in the literature. In this direction, we utilize well-known analytical techniques to derive desired criteria which guarantee the existence of solutions for the proposed multi-order quantum multi-integro-differential problem. Further, some numerical examples are considered to examine our theoretical and analytical findings using the proposed methods.
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    Some results on a system of multiterm fractional integro-differential equations
    (TUBITAK, 2020) Rezapour, Shahram; Sakar, Fethiye Müge; Aydoğan, Seher Melike; Ravash, Elahe
    In present study, we investigate the existence of solution for a multiterm fractional integro-differential system with special boundary conditions under some different conditions. In this way, we provide different results for the existence of solutions for the system and also for obtaining unique solution for the system under different conditions. We also present three numerical examples in which by using the Legendre multiwavelet method, we approximate solutions of the system. These examples illustrate our main results. © This work is licensed under a Creative Commons Attribution 4.0 International License
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    A study on the existence of numerical and analytical solutions for fractional integrodifferential equations in hilfer type with simulation
    (American Institute of Mathematical Sciences, 2023) George, Reny; Aydoǧan, Seher Melike; Sakar, Fethiye Müge; Ghaderi, Mehran; Rezapour, Shahram
    Previous studies have shown that fractional derivative operators have become an integral part of modeling natural and physical phenomena. During the progress and evolution of these operators, it has become clear to researchers that each of these operators has special capacities for investigating phenomena in engineering sciences, physics, biological mathematics, etc. Fixed point theory and its famous contractions have always served as useful tools in these studies. In this regard, in this work, we considered the Hilfer-type fractional operator to study the proposed integrodifferential equation. We have used the capabilities of measure theory and fixed point techniques to provide the required space to guarantee the existence of the solution. The Schauder and Arzela-Ascoli theorems play a fundamental role in the existence of solutions. Finally, we provided two examples with some graphical and numerical simulation to make our results more objective.
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    Two hybrid and non-hybrid k-dimensional inclusion systems via sequential fractional derivatives
    (Springer Science and Business Media Deutschland GmbH, 2021) Aydoğan, Seher Melike; Sakar, Fethiye Müge; Fatehi, Mostafa; Rezapour, Shahram; Masiha, Hashem Parvaneh
    Some complicated events can be modeled by systems of differential equations. On the other hand, inclusion systems can describe complex phenomena having some shocks better than the system of differential equations. Also, one of the interests of researchers in this field is an investigation of hybrid systems. In this paper, we study the existence of solutions for hybrid and non-hybrid k-dimensional sequential inclusion systems by considering some integral boundary conditions. In this way, we use different methods such as α-ψ contractions and the endpoint technique. Finally, we present two examples to illustrate our main results.