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Some results of KKAT transform in fractional calculus | ||
| Journal of University of Anbar for Pure Science | ||
| Volume 20, Issue 2, August 2026, Pages 380-385 PDF (309.89 K) | ||
| Document Type: Research Paper | ||
| DOI: 10.37652/juaps.2025.164955.1676 | ||
| Authors | ||
| Rifaat Saad Abdul Jabbar* 1; Omar Kareem Ali2; Mustafa Akram Saeed2; Ali Kadhim Yaqoob3 | ||
| 1Department of Mathematics, College of Science, University of Anbar, Iraq | ||
| 2Department of Mathematics, College of Science, University of Anbar, Ramadi, Iraq | ||
| 3Department of Mathematics, College of Science, University of Anbar, Ramadi, Iraq. | ||
| Abstract | ||
| This paper advances the study of the Karry-Kalim-Adnan Transform (KKAT) within fractional calculus by extending its use to fractional derivatives and integrals and examining its fundamental properties. The framework for fractional orders is clarified, and the main structural properties of KKAT, including linearity, stability under order variation, and compatibility with standard fractional operators, are established under conditions that ensure a well-defined transform. Building on this foundation, a modified solution procedure based on KKAT is formulated for solving fractional integro-differential equations. The proposed workflow is designed for straightforward implementation using common symbolic or numerical routines, with emphasis on transparency in the intermediate steps. The performance of the method is assessed using several benchmark problems with known or high-accuracy reference solutions, documenting convergence behavior and computational effort. The results indicate that the KKAT-based method can streamline the solution process and improve interpretability compared with direct time-domain techniques. It may also reduce computational steps compared with other integral-transform approaches for solving linear and nonlinear fractional integro-differential equations. Future directions include parameter identification in data-driven models and applications in viscoelasticity, anomalous diffusion, and control systems, where fractional dynamics naturally arise. Further work may also explore modified transforms derived from existing integral transforms, such as the Aboodh, Elzaki, Mohand, Al-Zughair, Mahgoub, and SEE transforms. These transforms are generally related to the Laplace transform through suitable modifications adapted to specific mathematical problems and physical phenomena. | ||
| Keywords | ||
| Convolution theorem; Fractional derivative and integral; KKAT transform | ||
| References | ||
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