The Application of Spline Function Method to the Modeling of Linear Boundary Value Problems with Delay

• Datum: 2026
• Hauptverfasser: Жолтовський, Олексій, Черевко, Ігор
• Format: Artikel
• Sprache: Ukrainisch
• Veröffentlicht: Кам'янець-Подільський національний університет імені Івана Огієнка 2026
• Online Zugang: https://mcm-math.kpnu.edu.ua/article/view/360573
• Назва журналу: Mathematical and computer modelling. Series: Physical and mathematical sciences

The article introduces a set of algorithms for finding approximate solutions to linear delay boundary value problems, as exact solutions are obtainable only in the most trivial cases. In the scientific literature, methods such as colocation, projection-iteration methods, and numerically-analytical algorithms have been proposed for finding approximate solutions to delay boundary value problems; however, these approaches are quite complex to implement. It should also be noted that solutions to delay boundary value problems may exhibit discontinuities in their derivatives, which complicates the application of finite-difference methods. The spline method has been proven to be an effective approach for finding the approximate solution of delay boundary value problems. The article considers two approaches to applying the method, namely basis cubic splines and an iterative scheme using cubic splines of defect 2. The first approach is suitable for approximating smooth solutions of delay boundary value problems, while the second accounts for possible derivative discontinuities. For numerical modeling of linear boundary value problems involving differential-difference equations, an application was developed using C++, Lua, and Vulkan graphics and computing API. Numerical experiments were performed on test model examples, and a comparative analysis was conducted.