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An efficient computational framework for predicting the behavior of multi-term Caputo fractional differential equations with a delay | ||
| Computational Methods for Differential Equations | ||
| مقالات آماده انتشار، پذیرفته شده، انتشار آنلاین از تاریخ 17 شهریور 1405 اصل مقاله (1.24 M) | ||
| نوع مقاله: Research Paper | ||
| شناسه دیجیتال (DOI): 10.22034/cmde.2026.71068.3570 | ||
| نویسندگان | ||
| Payam Moradpour؛ Khosro Sayevand* ؛ Iman Masti | ||
| Faculty of Mathematical Sciences, Malayer University, P. O. Box 16846-13114, Malayer, Iran. | ||
| چکیده | ||
| Developing robust numerical techniques for approximating solutions to fractional-order differential equations holds significant value, as exact analytical solutions are often unattainable for this class of problems. In this paper, a novel and efficient numerical method based on modified linear B-spline basis functions has been developed for solving multi-term Caputo fractional delay differential equations (FDDEs). The proposed technique employs operational matrices for fractional integration and delay transformation, incorporating the Caputo fractional derivative \(\CD{\alpha}\) along with both the initial conditions at \(t=0\) and the history function \(g(t)\) for \(t<0\) (which provides the necessary prehistory for the delay terms), converting the original FDDE into an equivalent algebraic system (linear or nonlinear) that can be solved by standard linear solvers or iterative schemes. A rigorous convergence analysis is provided, establishing an optimal algebraic convergence rate (order $O(2^{-2J})$, equivalently $O(N^{-2})$ for $N=2^J+1$ basis functions) for sufficiently smooth solutions, while the method retains the simplicity and sparsity of finite-element approaches. Several numerical examples including linear, nonlinear, system and multi-term FDDEs have been solved to illustrate the efficiency and reliability of the approach. The results demonstrate excellent agreement with exact solutions and, for the considered smooth test problems, attain errors close to machine precision with a relatively small number of basis functions, consistent with the theoretical algebraic rate. Comparisons with existing methods confirm that the present scheme is both more accurate and computationally economical. | ||
| کلیدواژهها | ||
| multi-term Caputo fractional delay differential equations؛ linear B-spline؛ operational matrix؛ convergence analysis | ||
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