Investigation of the Behaviour of Volterra Integral Equations with Random Effects

Year 2020, Volume: 10 Issue: 1, 205 - 216, 15.01.2020

Mehmet Merdan , Özge Altay Zafer Bekiryazıcı

https://doi.org/10.17714/gumusfenbil.586796

Cited By: 3

Abstract

In this study, random Volterra integral equations obtained by transforming components of deterministic Volterra integral equations to random variables are analysed. Beta, Normal (Gaussian), Gamma, Geometric and Uniform distributions are used to investigate the random behaviour of the solutions for Volterra integral equations under random effects. The random version of Differential Transformation Method (RDTM) is used to obtain an approximation to the solution of the random Volterra integral equation. Using the approximate solutions, approximate expected values and approximate variances are calculated. Some integro-differential equations, obtained by using random components with the above mentioned distributions, are solved as numerical examples. Results are obtained in MAPLE and shown in graphs. It is seen that random Differential Transformation Method is effective for the examination of random Volterra integral equations. Comparison of the solutions is given to underline the accuracy of the method.

Keywords

Diferansiyel Dönüşüm Yöntemi, Beklenen Değer, Modifiye DTM, Varyans, Volterra İntegral Denklemi

Volterra İntegral Denklemlerinin Rastgele Etkilerle Davranışlarının İncelenmesi

Abstract

Bu çalışmada deterministik Volterra integral denklemlerinin bileşenlerinin rastgele değişkenlere dönüştürülmesi ile elde edilen rastgele Volterra integral denklemleri incelenmektedir. Volterra integral denklemlerinin rastgele etkiler altındaki rastgele davranışlarını incelemek için Beta, Normal, Gamma, Geometrik ve Düzgün dağılımlar kullanılmaktadır. Rastgele Volterra integral denkleminin çözümüne bir yaklaşım elde etmek için Diferansiyel Dönüşüm Yöntemi’nin rastgele versiyonu (RDTM) kullanılmaktadır. Yaklaşık çözüm kullanılarak yaklaşık beklenen değerler ve yaklaşık varyanslar hesaplanmaktadır. Bahsedilen dağılımlara sahip rastgele bileşenler kullanılarak elde edilen bazı integro-diferansiyel denklemler sayısal örnek olarak kullanılmaktadır. Sonuçlar MAPLE’da elde edilmiş ve grafiklerle gösterilmiştir. Rastgele Diferansiyel Dönüşüm Yöntemi’nin rastgele Volterra İntegral Denklemleri’nin incelenmesinde etkili bir araç olduğu görülmektedir. Yöntemin doğruluğunu göstermek için sonuçların karşılaştırmalarına yer verilmiştir.

Keywords

Differential Transformation Method, Expected Value, Modified DTM, Variance, Volterra Integral Equation

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