A Collocation Method Based on Morgan-Voyce Polynomials for Approximating Solutions to Nonlinear Pantograph Differential Equations

Abstract

This study proposes a novel collocation technique based on Morgan-Voyce polynomials to obtain approximate solutions for a certain class of nonlinear pantograph differential equations that are subject to initial and boundary conditions. The method utilizes specially selected collocation points to transform the original nonlinear differential problem into an equivalent nonlinear algebraic system. The solutions of this algebraic system correspond to the unknown coefficients in the approximate solution expression. The main advantage of the proposed approach lies in its ability to reduce the complexity of solving nonlinear differential equations by converting them into manageable algebraic systems, while maintaining a high level of accuracy. To validate the accuracy and efficiency of the technique, several benchmark problems are considered. Numerical experiments are carried out, and the absolute error functions are used to analyze the performance of the method. All numerical computations and graphical illustrations presented in this paper have been conducted using a program developed in MATLAB R2022b.

Keywords

• Nonlinear pantograph differential equation (NPDE)
• Pantograph equations
• Morgan-Voyce collocation method
• Collocation points
• Morgan-Voyce polynomials

Doğrusal Olmayan Pantograf Diferansiyel Denklemlerinin Yaklaşık Çözümleri için Morgan-Voyce Polinomlarına Bağlı bir Sıralama Yöntemi

Abstract

Bu çalışma, başlangıç ve sınır koşullarına tabi belirli bir sınıftaki doğrusal olmayan pantograf diferansiyel denklemler için yaklaşık çözümler elde etmek amacıyla Morgan-Voyce polinomlarına dayalı yeni bir sıralama tekniği önermektedir. Yöntem, özel olarak seçilen sıralama noktalarını kullanarak orijinal doğrusal olmayan diferansiyel problemi eşdeğer bir doğrusal olmayan cebirsel sisteme dönüştürür. Bu cebirsel sistemin çözümleri, yaklaşık çözüm ifadesindeki bilinmeyen katsayılara karşılık gelir. Önerilen yaklaşımın temel avantajı, doğrusal olmayan diferansiyel denklemlerin çözümündeki karmaşıklığı azaltarak bunları daha yönetilebilir cebirsel sistemlere dönüştürmesi ve aynı zamanda yüksek doğruluk düzeyini korumasıdır. Yöntemin doğruluğunu ve etkinliğini doğrulamak amacıyla çeşitli karşılaştırmalı test problemleri ele alınmıştır. Sayısal deneyler gerçekleştirilmiş ve yöntemin performansını analiz etmek için mutlak hata fonksiyonları kullanılmıştır. Bu çalışmada sunulan tüm sayısal hesaplamalar ve grafiksel gösterimler, MATLAB R2022b ortamında geliştirilen bir program aracılığıyla gerçekleştirilmiştir.

Keywords

• Doğrusal olmayan pantograf diferansiyel denklemler
• Pantograf Denklemler
• Morgan- Voyce Sıralama Yöntemi
• Sıralama noktaları
• Morgan-Voyce polinomları

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