Finite Difference Approach for Fourth-Order Impulsive Sturm-Liouville Boundary Value Problems

Year 2025, Volume: 18 Issue: 1, 1 - 9, 28.03.2025

Şerife Faydaoğlu

https://doi.org/10.18185/erzifbed.1593935

Abstract

This paper presents a finite difference method to solve a novel type fourth-order boundary value problem with impulsive conditions. These differential equations, which model deflections in beams, provide insights into various applications in fields such as civil, mechanical, and aeronautical engineering. Analytical solutions to boundary value problems are often challenging to derive, highlighting the need for robust numerical methods. In this study, a formula for finite difference approximation is derived by using Taylor series expansions at selected grid points. By transforming differential equations into algebraic systems, the unknown solutions are determined based on the grid points. The proposed method is validated through a numerical example involving a fourth-order impulsive linear boundary value problem, and the results demonstrate its effectiveness.

Keywords

The finite difference method, fourth order boundary value problem, impulse conditions, approximate solutions

Dördüncü Mertebe İmpulsiv Sturm-Liouville Sınır Değer Problemi için Sonlu Fark Yaklaşımı

Abstract

Bu makale, impulsiv koşullara sahip yeni bir dördüncü dereceden sınır değer problemini çözmek için sonlu farklar yöntemini sunmaktadır. Kirişlerdeki sapmaları modelleyen bu diferansiyel denklemler, inşaat, makine ve havacılık mühendisliği gibi alanlardaki çeşitli uygulamaların aydınlatılmasını sağlar. Sınır değer problemlerine yönelik analitik çözümlerin elde edilmesi çoğu zaman zorlayıcıdır ve bu durum sağlam sayısal yöntemlere olan ihtiyacı vurgulamaktadır. Bu çalışmada, seçili grid noktalarında Taylor serisi açılımları kullanılarak sonlu farklar yaklaşımı için bir formül ortaya çıkarılmıştır. Diferansiyel denklemler cebirsel denklem sistemlere dönüştürülerek, bilinmeyen çözümler grid noktalarına göre belirlenmiştir. Önerilen yöntem, dördüncü dereceden impulsiv doğrusal sınır değer problemini içeren sayısal bir örnek üzerinden doğrulanmış ve sonuçlar yöntemin etkinliğini göstermiştir.

Keywords

Sonlu fark yöntemi, dördüncü mertebe sınır değer problemi, impuls şartlar, yaklaşık çözümler

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