Kesirli telegraf kısmi diferansiyel denklemlerin fark şeması metodu ile nümerik çözümü

Abstract

Bu çalışmada, özellikle mühendislik, finans, fizik ve sismoloji gibi pek çok bilim dalında uygulamalara haiz başlangıç değer koşullarına sahip kesirli telegraf kısmi diferansiyel denklemi ele alındı.  Caputo kesirli kısmi türevli denklemin tanımı vasıtasıyla ele alınan kesirli telegraf kısmi diferansiyel denkleminin sonlu farklardaki ifadesi oluşturuldu. Aynı şekilde, ele alınan denklemin abstract formu ifade edildi.  Abstract formda verilen bu denklem için sonlu fark şemaları oluşturuldu.  Hilbert uzayı üzerinde tanımlanan norma göre denklemin oluşturulan bu sonlu fark şemalar için kararlılık kestirimleri gösterildi.  Kararlılık kestirimini ifade eden Teorem ispatıyla birlikte ifade edildi.  Sonlu fark şeması metodu kullanılarak α=0.1,0.5,0.9 un farklı değerleri için Caputo kesirli türevi vasıtası ile tanımlanan kesirli telegraf kısmi diferansiyel denkleminin nümerik çözümü elde edildi. Burada, kullanılan örnek problemlerin nümerik çözümleri Matlab programı kullanılarak oluşturuldu.  Laplace metodu veya geleneksel metotlar yardımıyla elde edilen tam çözüm ile yaklaşık çözümler mukayese edilerek hata analizi yapıldı.  Hata analizi tablosundan elde edilen çıkarsamaya göre önerilen metodun ne kadar etkili ve tutarlı olduğu gözlemlendi.

Keywords

• Kesirli telegraf kısmi diferansiyel denklem,kararlılık,fark şeması,hata analizi

Numerical solution of fractional telegraph partial differential equations by difference scheme method

Abstract

In this study, fractional telegraph partial differential equation with initial value condition having applications in physics, engineering, finance, seismology and other disciplines is discussed. By applying definition of Caputo fractional, difference scheme for fractional telegraph partial differential equation is obtained.  The abstract form of the considered equation is also stated.  The finite difference schemes of the abstract form for fractional telegraph partial differential equation are constructed.  The stability estimates of this finite difference scheme is proved with respect to the norm defined on the Hilbert space.  The proof of our main theorem determining the stability estimates is given in detail.  By using difference scheme method defined by Caputo fractional derivative, numerical solution of fractional telegraph partial differential equation is obtained  for different values of α=0.1,0.5,0.9.  Numerical solutions of our example is tested by using Matlab programming.  Error analysis was performed by comparing approximate solutions with exact solution obtained by Laplace or other traditional methods.  It is obvious that the proposed method is effective and consistent according to error analysis. 

Keywords

• Fractional telegraph partial differential equation,stability,difference schemes,error analysis

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