Solving of Some Random Partial Differential Equations by Using Differential Transformation Method and Laplace- Padé Method

Yıl 2019, , 108 - 118, 15.01.2019

Halil Anaç Mehmet Merdan Zafer Bekiryazıcı Tülay Kesemen

https://doi.org/10.17714/gumusfenbil.404332

Cited By: 1

Öz

In this study, the solutions of
random partial differential equations are examined. The parameters and the
initial conditions of the random component partial differential equations are
investigated with Beta distribution. A few examples are given to illustrate the
efficiency of the solutions obtained with the random Differential
Transformation Method (rDTM). Functions for the expected values and the
variances of the approximate analytical solutions of the random equations are
obtained. Random Differential Transformation Method is applied to examine the
solutions of these partial differential equations and MAPLE software is used
for the finding the solutions and drawing the figures. Also the Laplace-
Padé Method is used to improve the convergence of the solutions. The
results for the random component partial differential equations with Beta
distribution are analysed to investigate effects of this distribution on the
results. Random characteristics of the equations are compared with the results
of the deterministic partial differential equations. The efficiency of the
method for the random component partial differential equations is investigated
by comparing the formulas for the expected values and variances with results
from the simulations of the random equations.

Anahtar Kelimeler

Expected Value, Random Component Partial Differential Equation, Random Differential Transformation Method

Bazı Rastgele Kısmi Diferansiyel Denklemlerin Diferansiyel Dönüşüm Metodu ve Laplace- Padé Metodu Kullanarak Çözümü

Öz

Bu çalışmada, rastgele kısmi diferansiyel denklemlerin çözümleri incelenmiştir. Rastgele bileşenli kısmi diferansiyel denklemlerin başlangıç şartları ve parametreleri Beta dağılımı ile incelenmiştir. Rastgele Diferansiyel dönüşüm yöntemi ile elde edilen çözümlerin etkinliği birkaç örnekle verilmiştir. Rastgele denklemlerin yaklaşık analitik çözümlerinin beklenen değerleri ve varyansları için fonksiyonlar elde edilmiştir. Rastgele Diferansiyel dönüşüm yöntemi, bu kısmi diferansiyel denklemlerin çözümlerini incelemek için uygulanmış ve MAPLE programı, çözümleri bulmak ve grafikleri çizmek için kullanılmıştır. Ayrıca çözümlerin yakınsaklığını iyileştirmek için Laplace-Padé metodu kullanılmıştır. Beta dağılımı ile rastgele bileşenli kısmi diferansiyel denklemlerin sonuçları, bu dağılımın sonuçlara etkilerini incelemek amacıyla analiz edilmiştir. Denklemlerin rastgele karakteristikleri ile rastgele olmayan kısmi diferansiyel denklemlerin sonuçları karşılaştırılmıştır. Rastgele bileşenli kısmi diferansiyel denklemler için yöntemin etkinliği, rastgele denklemlerin simülasyonlarından elde edilen sonuçlarla beklenen değerlerin ve varyansların formüllerini karşılaştırarak incelenmiştir. MAPLE programı, rastgele bileşenli kısmi diferansiyel denklemlerin sonuçlarını simüle etmek için kullanılmıştır ve bu simülasyon sonuçlarından standart sapma, güven aralığı gibi diğer karakteristiklerler elde edilmiştir.

Anahtar Kelimeler

Beklenen değer, Rastgele bileşenli kısmi diferansiyel denklemler, Rastgele diferansiyel dönüşüm metodu

Kaynakça

• Abassy, T.A., El-Tawil, M.A. and El-Zoheiry, H., 2007. Exact Solutions of Some Nonlinear Partial Differential Equations Using the Variational Iteration Method Linked with Laplace Transforms and the Pade Technique. Computers and Mathematics with Applications, 54, 940-954.
• Bildik, N., Konuralp, A., Bek, F.O. and Küçükarslan, S., 2006. Solution of Different Type of the Partial Differential Equation by Differential Transform Method and Adomian’s Decomposition Method. Applied Mathematics and Computation, 172(1), 551-567.
• Eugene, N., Lee, C. and Femoye, F., 2002. Beta-Normal Distribution and Its Applications. Communications in Statistics-Theory and Methods, 31(4), 497-512.
• Hadizadeh, M. and Moatamedi, N., 2007. A New Differential Transformation Approach for Two-Dimensional Volterra Integral Equations, International Journal of Computer Mathematics, 84(4), 515-526.
• Jang, M.J., Chen, C.L. and Liu, Y.C., 2001. Two-Dimensional Differential Transform for Partial Differential Equations. Applied Mathematics and Computation, 121(2-3), 261-270.
• Kangalgil, F. and Ayaz, F., 2009. Solitary Wave Solutions for the KdV and mKdV Equations by Differential Transform Method. Chaos, Solitons and Fractals, 41, 464-472.
• Kanth, R.A.S.V. and Aruna, K., 2009. Two-Dimensional Differential Transform Method for Solving Linear and Non-linear Schrödinger Equations. Chaos, Solitons and Fractals, 41, 2277-2281.
• 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 Adomian Decomposition Method. Applied Mathematical Sciences, 5, 2521-2535.
• 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.
• Merdan, M., 2010. A New Application of Modified Differential Transformation Method for Modelling the Pollution of a System of Lakes. Selçuk Journal of Applied Mathematics, 11(2), 27-40.
• Pukhov, G.E., 1982. Differential Transforms and Circuit Theory. Circuit Theory and Applications. 10, 265-276.
• Tari, A., Rahimi, M.Y., Shahmorad, S. and Talati, F., 2009. Solving a Class of Two-Dimensional Linear and Nonlinear Volterra Integral Equaitons by the Differential Transform Method. Journal of Computational and Applied Mathematics, 228, 70-76.
• Villafuerte, L. and Chen-Charpentier, B.M., 2012. A Random Differential Transform Method: Theory and Applications, Applied Mathematics Letters, 25(10), 1490-1494.
• Yüzbaşi, Ş. and Ismailov, N., 2017. Differential Transform Method to Solve Two-Dimensional Volterra Integral Equations with Proportional Delays. New Trends in Mathematical Sciences, 5(4), 65-71.
• Zhou, J.K., 1986. Differential Transformation and Its Applications for Electrical Circuits, Huazhong University Press, Wuhan.
• Ziyaee, F. and Tari, A., 2015. Differential Transform Method for Solving the Two-Dimensional Fredholm Integral Equations. Applications and Applied Mathematics, 10(2), 852-863.