Investigation of the Behaviour of Volterra Integral Equations with Random Effects

Yıl 2020, , 205 - 216, 15.01.2020

Mehmet Merdan , Özge Altay Zafer Bekiryazıcı

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

Cited By: 3

Öz

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.

Anahtar Kelimeler

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

Volterra İntegral Denklemlerinin Rastgele Etkilerle Davranışlarının İncelenmesi

Öz

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.

Anahtar Kelimeler

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

Kaynakça

• Aksoy, Y.,1983. İntegral Denklemler. Yıldız Üniversitesi Yayınları, Cilt:1,Sayı:166.
• Arikoglu, A. and Ozkol, I., 2005. Solution of boundary value problems for integro differential equations by using differential transform method. Appl Math Comput, 168, 1145-1158.
• Arikoglu, A. and Ozkol, I., 2008. Solution of integral and integro-differential equation systems by using differential transform method. Comput Math Appl, 56, 2411-2417.
• Calbo, G., Cortés, J.C. and Jódar, L., 2010. Mean Square Power Series Solution Of Random Linear Differential Equations. Computers And Mathematics With Applications, 59, 559-572.
• Cherruault, Y., Saccomandi, G. and Some, B., 1993. New results for convergence of Adomian’s method applied to integral equations, Mathl. Comput. Modelling, 16(2), 85–93.
• Chiles, J. and Delfiner, P.,1999. Geostatistics: Modelling Spatial Uncertainty. John Wiley, New York.
• Cortés, J.C., Jódar, L. and Villafuerte, L., 2007. Mean Square Numerical Solution Of Random Differential Equations: Facts And Possibilities. Computers And Mathematics With Applications, 53, 1098-1106.
• Cortés, J.C., Jódar, L. and Villafuerte, L., 2007. Numerical Solution Of Random Differential Equations: A Mean Square Approach. Mathematical And Computer Modelling, 45, 757-765.
• Cortes, J.C., Jodar, L. and Villafuerte, L., 2009. Random Linear-Quadratic Mathematical Models: Computing Explicit Solutions and Applications. Mathematics and Computers in Simulation, 79, 2076-2090. Cortés, J.C., Jódar, L., Villafuerte, L. and Company, R., 2011. Numerical Solution Of Random Differential Models. Mathematical And Computer Modelling, 54, 1846-1851.
• Cortés, J.C., Jódar, L., Villanueva, R.-J. and Villafuerte, L., 2010. Mean Square Convergent Numerical Methods For Nonlinear Random Differential Equations. Lecture Notes İn Computer Science, 5890, 1-21.
• Fakharzadeh, J., Hesamaeddini, E. and Soleimanivareki, M., 2015. Multi-step Stochastic Differential Transformation Method for solving Some Class of Random Differential Equations. Applied Mathematics in Engineering, Management and Technology, 3(3), 115–123.
• Feller W., 1968. An Introduction to Probability Theory and Its Applications, volume 1, 3rd edition. New York: John Wiley & Sons.
• Golmankhaneh, A.K., Porghoveh, N.A. and Baleanu, D., 2013. Mean Square Solutions of Second-Order Random Differential Equations by Using Homotopy Analysis Method. Romanian Reports in Physics, 65(2), 350–362.
• Khalaf, S.L., 2011. Mean Square Solutions of Second-Order Random Differential Equations by Using Homotopy Perturbation Method. International Mathematical Forum, 6, 2361-2370.
• Khudair, A.R., Ameen, A.A. and Khalaf, S.L., 2011. Mean Square Solutions of Second-Order Random Differential Equations by Using Variational Iteration Method. Applied Mathematical Sciences, 5, 2505-2519.
• Khudair, A. R. , Haddad, S. A.M. and Khalaf, S. L., 2016. Mean Square Solutions of Second-Order Random Differential Equations by Using the Differential Transformation Method, Open Journal of Applied Sciences, 6, 287-297.
• Kythe, P. and Puri, P., 2011. Computational methods for linear integral equations. Springer Science & Business Media.
• Lovitt, W.V., 1950 Linear Integral Equations, Dover Publications, Inc.: New York.
• Merdan, M., Anac, H., Bekiryazici, Z. and Kesemen, T. 2019. Solving of Some Random Partial Differential Equations by Using Differential Transformation Method and Laplace-Padé Method, Gumushane Universitesi Fen Bilimleri Enstitusu Dergisi, 9(1), 108-118.
• Mohyud-Din, S. T., Yildirim, A. and Gülkanat, Y., 2010. Analytical solution of Volterra’s population model, J. King Saud Univ. - Sci., 22(4), 247–250.
• Soong, T.T., 1973. Random Differential Equations İn Science And Engineering. Academic Press, New York.
• Villafuerte, L., Braumann, C.A., Cortés, J.C. and Jódar, L., 2010. Random Differential Operational Calculus: Theory And Applications. Computers & Mathematics With Applications, 59, 115-125.
• Wazwaz, A.M., 1997. A First Course in Integral Equations, World Scientific: Singapore.
• Wazwaz, A. M., 1999. Analytical approximations and Padé approximants for Volterra’s population model, Appl. Math. Comput., 100(1), 13–25.
• Wazwaz, A. M., 2011. Linear and nonlinear integral equations (Vol. 639). Heidelberg: Springer.