Higher-dimensional physical models with multimemory indices: analytic solution and convergence analysis

Imad Jaradat; Marwan Alquran; Ruwa Abdel-Muhsen; Shaher Momani; Dumitru Baleanu

doi:10.1186/s13662-020-02822-7

Higher-dimensional physical models with multimemory indices: analytic solution and convergence analysis

Adv Differ Equ. 2020;2020(1):364. doi: 10.1186/s13662-020-02822-7. Epub 2020 Jul 16.

Authors

Imad Jaradat¹, Marwan Alquran¹, Ruwa Abdel-Muhsen¹, Shaher Momani^{2

3}, Dumitru Baleanu^{4

5

6}

Affiliations

¹ Department of Mathematics & Statistics, Jordan University of Science and Technology, Irbid, 22110 Jordan.
² Department of Mathematics and Sciences, College of Humanities and Sciences, Ajman University, Ajman, UAE.
³ Department of Mathematics, Faculty of Science, The University of Jordan, Amman, 11942 Jordan.
⁴ Department of Mathematics, Cankaya University, 06530 Balgat, Ankara Turkey.
⁵ Institute of Space Sciences, Magurele, Bucharest, Romania.
⁶ Department of Medical Research, China Medical University, Taichung, Taiwan.

Abstract

The purpose of this work is to analytically simulate the mutual impact for the existence of both temporal and spatial Caputo fractional derivative parameters in higher-dimensional physical models. For this purpose, we employ the γ̅-Maclaurin series along with an amendment of the power series technique. To supplement our idea, we present the necessary convergence analysis regarding the γ̅-Maclaurin series. As for the application side, we solved versions of the higher-dimensional heat and wave models with spatial and temporal Caputo fractional derivatives in terms of a rapidly convergent γ̅-Maclaurin series. The method performed extremely well, and the projections of the obtained solutions into the integer space are compatible with solutions available in the literature. Finally, the graphical analysis showed a possibility that the Caputo fractional derivatives reflect some memory characteristics.

Keywords: Analytic solution; Fractional PDEs; Memory index.