AN EFFICIENT NUMERICAL SOLUTION METHOD FOR NONLINEAR KLEIN-GORDON EQUATION VIA TAYLOR WAVELET METHOD

Abstract

The Klein–Gordon equation is of fundamental importance in mathematical physics, particularly due to its extensive applications in the analysis of solitonic phenomena, condensed matter systems, and the behavior of nonlinear wave dynamics. In this study, we develop a highly accurate numerical algorithm based on Taylor wavelets combined with the collocation technique, to approximate the solutions of nonlinear Klein-Gordon equations. An integration operational matrix is constructed and employed to transform the nonlinear Klein-Gordon initial–boundary value problem into an equivalent system of algebraic equations. One of the advantages of this method is that it does not require any restriction on domain discretization. This study also provides valuable insights into the underlying theoretical properties of the proposed method. To verify the reliability and accuracy of the proposed Taylor wavelet-based algorithm, a convergence analysis is performed. The method is then applied to four benchmark problems to further assess its effectiveness and computational performance. The comparison between the numerical and exact solutions demonstrates that the proposed method yields highly accurate results with minimal errors. All computations have been executed using MATLAB-2023b programming language.

Keywords

• Numerical solution
• Taylor Wavelet
• Nonlinear wave
• Error
• Convergence

DOĞRUSAL OLMAYAN KLEIN–GORDON DENKLEMİ İÇİN TAYLOR DALGACIK YÖNTEMİYLE ETKİLİ BİR SAYISAL ÇÖZÜM YÖNTEMİ

Abstract

Klein–Gordon denklemi, solitonik olayların analizi, yoğun madde sistemleri ve doğrusal olmayan dalga dinamiklerinin incelenmesi gibi geniş uygulama alanları nedeniyle matematiksel fizikte temel bir öneme sahiptir. Bu çalışmada, doğrusal olmayan Klein–Gordon denklemlerinin çözümlerini yaklaştırmak amacıyla Taylor dalgacıkları ile kolokasyon tekniğini birleştiren yüksek doğruluklu bir sayısal algoritma geliştirilmiştir. Oluşturulan integrasyon işlemsel matrisi, başlangıç–sınır değer problemindeki doğrusal olmayan Klein–Gordon denklemini eşdeğer bir cebirsel denklem sistemine dönüştürmek için kullanılmaktadır. Önerilen yöntemin önemli avantajlarından biri, tanım alanının ayrıklaştırılmasına yönelik herhangi bir kısıtlama gerektirmemesidir. Çalışma ayrıca yöntemin temelinde yer alan teorik özelliklere ilişkin değerli bilgiler sunmaktadır. Önerilen Taylor dalgacığı tabanlı algoritmanın doğruluğunu ve güvenilirliğini değerlendirmek amacıyla yakınsama analizi yapılmıştır. Yöntem daha sonra dört örnek problem üzerinde uygulanarak etkinliği ve hesaplama performansı test edilmiştir. Sayısal ve tam çözümlerin karşılaştırılması, yöntemin çok küçük hata değerleriyle yüksek doğruluk sağladığını göstermektedir. Tüm hesaplamalar MATLAB-2023b programlama ortamında gerçekleştirilmiştir.

Keywords

• Sayısal çözüm
• Taylor Dalgacığı
• Doğrusal olmayan dalga
• Hata
• Yakınsama

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