Document Type : Research Article
Authors
1 Department of Applied Mathematics, Imam Khomeini International University, Qazvin, 34149-16818, Iran.
2 Khawarizmi Laboratory of Mathematics and Applications (KALMA), Faculty of Science, Lebanese University, Hadith-Beirut, Lebanon.
Abstract
In this paper, we investigate the existence and uniqueness of solutions for a high-order boundary value problem involving non-integer derivatives, specifically utilizing the Caputo fractional derivative. The problem is subject to non-local boundary conditions. To tackle this, we introduce the fractional Green's function as an analytical tool. The Banach contraction fixed-point theorem serves as the fundamental method to establish our main results. To support the theoretical findings, we provide illustrative examples. Furthermore, we develop a numerical semi-analytical approach to approximate the unique solution with the desired accuracy.
Highlights
- Proved existence and uniqueness for a broad class of high-order boundary value problems (BVPs) involving Caputo fractional derivatives.
- Introduced a nonlocal-capable fractional Green’s function as a core analytical tool.
- Applied a fixed-point framework in well-posed function spaces.
- Provided explicit and verifiable sufficient solvability conditions.
- Developed a flexible, semi-analytical numerical scheme with proven convergence properties.
- Demonstrated high-precision numerical convergence and stability.
- Presented illustrative cases that validate the theory and show robustness to variations in ξ, boundary weights, and nonlinearity strength.
- Supplied error bounds for the semi-analytical approximations and cross-validation with high-precision solvers.
- Offered practical guidance for implementation in scientific computing workflows.
- Expanded applicability to applied sciences and engineering domains.
Keywords
- Caputo fractional derivatives
- High-order boundary value problems
- Nonlocal boundary conditions
- Fractional Green’s function
- Fixed point theory
Main Subjects
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