ANALYSIS AND COLLOCATION APPROXIMATIONS USING SHIFTED LEGENDRE GALERKIN ALGORITHM FOR SYSTEMS OF FRACTIONAL INTEGRODIFFERENTIAL DELAY PROBLEMS REGARDING HILFER FRACTIONAL TYPE

Hind Sweis, Omar Abu Arqub, Nabil Shawagfeh and Marwan Abukhaled
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Hind Sweis: Department of Data Science, Faculty of Data Science, Arab American University, Ramallah P600, Palestine
Omar Abu Arqub: ��Department of Mathematics, Faculty of Science, Al-Balqa Applied University, Salt 19117, Jordan
Nabil Shawagfeh: ��Department of Mathematics, Faculty of Science, The University of Jordan, Amman 11942, Jordan
Marwan Abukhaled: �Department of Mathematics and Statistics, American University of Sharjah, Sharjah 26666, United Arab Emirates

FRACTALS (fractals), 2025, vol. 33, issue 08, 1-26

Abstract: Delay problems are a special type of equation that includes values of the dependent variable at earlier times; it is widely used in modeling many natural and engineering phenomena that involve delays. This pioneering analysis initially addressed the existence and uniqueness attitude (EUA) of solutions for systems of fractional integro differential delay problems (FIDDPs) using the Hilfer fractional type. At first, we explored the properties of fundamental definitions concerning fractional attitudes, leveraging their characteristics to convert the system of FIDDP into an equivalent system of fractional volterra differential problem (FVDS). Employing the axiom of contraction mapping, we established the EUA of a solution for the resulting delays set. Thereafter, for approximation aims, the Galerkin scheme with shifted legendre orthogonal polynomials (SLOPs) was implemented. This approach transforms the system of FIDDP into a series of algebraic equations, enabling the approximation of the prompted solutions. A significant advantage of this algorithm lies in its ability to get accurate results by using fewer iterations compared to the difficulty of delays. To authenticate the workability of the theoretical–numerical achievements, we performed extensive tests on various delay problems of numerous natures. The attained achievements, presented with figures and tables, showcase and conduct the algorithm’s superior accuracy. The conclusion summarizes diverse highlights and proposes potential areas for future research.

Keywords: FIDDP; EUA; SLGA; SLOP; HFD (search for similar items in EconPapers)
Date: 2025
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DOI: 10.1142/S0218348X2540184X

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