A fast and reliable numerical approach for solving lake pollution problem

Öz

Just like air, water is an essential component of the environment and deterioration of water sources threatens all living organisms. Hence, investigating the water pollution problem is highly important. Lakes constitutes a big portion of water sources. The aim of this study is to analyze the dynamics of pollution in a system of three interconnected lakes using the Gegenbauer wavelet method. The problem is modeled by three ordinary linear differential equations representing the rate of pollution in each lake with respect to time. Time derivates are approximated by the truncated Gegenbauer wavelet series and the system of differential equations are converted into a system of algebraic equations. We show that the proposed technique is reliable and fast by comparing the numerical results with other numerical results available in the literature. We also show that the method is highly accurate and hence can also be used to solve other ecological phenomena as well.

Anahtar Kelimeler

• Lake contamination
• Water Pollution
• Gegenbauer wavelets

Göl kirliliği probleminin çözümü için hızlı ve güvenilir bir sayısal yaklaşım

Öz

Su, hava gibi çevrenin temel bileşenlerinden biridir ve su kaynaklarının bozulması tüm canlı organizmaları tehdit etmektedir. Bu nedenle su kirliliği sorununu araştırmak büyük önem taşımaktadır. Göller, su kaynaklarının büyük bir bölümünü oluşturmaktadır. Bu çalışmanın amacı, üç adet birbiriyle bağlantılı gölden oluşan bir sistemdeki kirlilik dinamiklerini Gegenbauer dalgacık yöntemi ile analiz etmektir. Problem, her bir göldeki kirlilik oranının zamana göre değişimini temsil eden üç doğrusal diferansiyel denklem sistemi ile modellenmiştir. Zaman türevlerine, kesikli Gegenbauer dalgacık serisiyle yaklaşılmış ve diferansiyel denklem sistemi, cebirsel denklem sistemine dönüştürülmüştür. Elde edilen sayısal sonuçlar, literatürde mevcut diğer sayısal sonuçlarla karşılaştırılarak önerilen tekniğin güvenilir ve hızlı olduğu gösterilmiştir. Ayrıca, yöntemin yüksek doğruluk sağladığı ve bu nedenle diğer ekolojik olayların çözümünde de kullanılabileceği ortaya konmuştur.

Anahtar Kelimeler

• Göl kirlinmesi
• Su kirliliği
• Gegenbauer dalgacıkları

Kaynakça

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