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Generalized Bernstein-like basis functions and their applications to fractional differential equations | ||
| Computational Methods for Differential Equations | ||
| مقالات آماده انتشار، پذیرفته شده، انتشار آنلاین از تاریخ 30 مرداد 1405 اصل مقاله (1.47 M) | ||
| نوع مقاله: Research Paper | ||
| شناسه دیجیتال (DOI): 10.22034/cmde.2026.69904.3451 | ||
| نویسندگان | ||
| Shahad Adil Taher1؛ Jamshid Saeidian* 2 | ||
| 1Faculty of Computer Science and Mathematics , AL-Kufa University, Najaf, Iraq. | ||
| 2Faculty of Mathematical Sciences and Computer, Kharazmi University, Tehran, Iran. | ||
| چکیده | ||
| This paper addresses the longstanding challenge of finding precise solutions for fractional differential equations by introducing and thoroughly examining Generalized Bernstein-like basis functions (GBFs). We propose two distinct subclasses of GBFs and elucidate their properties, which include recursive definitions, first and second derivatives, linear independence, and generalization of the partition of unity property. Delving deeper, we explore the definite integrals of the GBFs and introduce computational methodologies that prioritize both stability and efficiency. A significant achievement in our research is the development of computational schemes for Caputo fractional derivatives of Generalized Bernstein-like basis functions. Utilizing these approaches, we develop collocation techniques for a range of fractional differential equation types, including initial-value multi-order FDEs and two-point fractional boundary value problems. A central focus of this development is the comprehensive integration and understanding of both left-sided and right-sided fractional derivatives. The efficacy of our approach is further underscored by its successful application to intricate challenges like the Volterra-Fredholm fractional integro-differential equation. Our research reaches its zenith with an analytical evaluation of convergence rates using GBFs. | ||
| کلیدواژهها | ||
| Caputo fractional derivative؛ Collocation method؛ Convergence rates؛ Fractional equation؛ Generalized Bernstein-like basis functions | ||
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آمار تعداد مشاهده مقاله: 40 تعداد دریافت فایل اصل مقاله: 54 |
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