Rasyonel Üslü Cebirsel ve Üstel Eşleme Yaklaşımı ile Thomas-Fermi Denklemi için İkinci Derece Doğruluklu Sonlu Farklar Yöntemi

Doğa bilimlerine dayalı birçok problemin insanlığa hizmet etmesi için bilim insanları ve mühendisler tarafından çözülmeleri gerekir. Atomik dünyadaki iyi bilinen modellerden biri, bir denklemde yoğunlaşır ve bu denklem Thomas-Fermi denklemi olarak adlandırılır. Thomas-Fermi denklemi ağır, nötr atomların yük dağılımlarını tanımlayan ikinci dereceden bir diferansiyel denklemdir. Denklem için henüz tam bir analitik çözüm bulunamamıştır. Esasen, problemin güçlü nonlineer yapısı, tekil özellik sergilemesi ve sınırsız aralıklı tanım kümesi, yaklaşık sayısal bir çözüm elde etmede de büyük zorluklara yol açmaktadır. Bu makalede, Thomas-Fermi denklemi, sanki-doğrusallaştırma yöntemi ile birlikte ikinci dereceden doğruluklu bir sonlu farklar yöntemi kullanılarak çözülmüştür. Problemin yarı sonsuz aralığı, cebirsel ve üstel eşleme olarak adlandırılan iki farklı koordinat dönüşümü kullanılarak [0, 1) aralığına dönüştürülmüştür. Sayısal doğruluk mertebesi, sistematik ağ sıkılaştırma tekniği kullanılıp hesaplanan başlangıç eğim y'(0) değerlerinin karşılaştırılması ile kontrol edilmiştir. Başlangıç eğimi için hesaplanan sonuçların, literatürde verilen sonuçlarla iyi bir uyum içinde olduğu gösterilmiştir. Son olarak, Richardson ekstrapolasyonunun uygulanmasıyla çözümün doğruluk mertebesi arttırılmıştır.

Anahtar Kelimeler:

Thomas-Fermi Denklemi, nonlineer ADD, Yarı sonsuz aralık, Sonlu Farklar Metodu, Sanki-lineerleştirme, Aralık eşleme

Second Order Finite Difference Method for the Thomas-Fermi Equation via Fractional Order of Algebraic and Exponential Mapping Approach

Many problems based on natural sciences need to be solved by the scientists and engineers to serve the humanity. One of the well-known model in atomic universe is condensed into an equation, and called the Thomas-Fermi equation. It is a second order differential equation, which describes charge distributions of heavy, neutral atoms. No exact analytical solution has been found for the equation yet. In fact, strong nonlinearity, singular character and unbounded interval of the problem causes great difficulty to obtain an approximate numerical solution as well. In this paper, the Thomas-Fermi equation is solved using a second order finite difference method along with application of quasi-linearization method. Semi-infinite interval of the problem is converted into [0, 1) using two different coordinate transformations, namely algebraic and exponential mapping. Numerical order of accuracy has been checked using systematic mesh refinements and comparing the calculated initial slope y'(0). Calculated results for initial slope is found in good agreement with the results available in the literature. Lastly, accuracy is improved by the application of the Richardson extrapolation.

Keywords:

Thomas-Fermi equation, non-linear ODE, semi-infinite interval, finite difference method, quasi-linearization, interval mapping,

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