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Investigation of solution behavior of Differential Equations by Sumudu methods of random complex Partial Differential Equations

Yıl 2024, Cilt: 45 Sayı: 3, 562 - 570, 30.09.2024

Mehmet Merdan , Merve Merdan , Rıdvan Şahin

https://doi.org/10.17776/csj.1256101

Öz

In this study, solutions of random complex partial differential equations were found using the two-dimensional Sumudu transformation method(STM). The initial conditions of a deterministic equation or the non-homogeneous part of the equation are transformed into random variables to obtain a random complex partial differential equation. With the help of the properties of two-dimensional Sumudu and inverse Sumudu transformation, an approximate analytical solution of a complex partial differential equation with random constant coefficients was obtained by selecting a random variable with an initial condition of Normal and Gamma distribution. The probability characteristics of the resulting solutions, such as expected value and variance, were obtained and graphically shown with the help of the Maple package program.

Anahtar Kelimeler

Two-dimensional Sumudu and Inverse Sumudu Transformation Complex Differential Equation Expected Value Variance and Confidence Interval.

Kaynakça

  • [1] Düz M., Solution of complex differential equations with variable coefficients by using reduced differential transform, Mis. Math. Not., 21(1) (2020) 161–170.
  • [2] Düz M., Application of Elzaki Transform to first order constant coefficients complex equa ions, Bul. Int. math. Virt. inst., 7 (2017) 387–393.
  • [3] Düz M., On an application of Laplace transforms, NTMSCI., 5(2) (2017) 193–198.
  • [4] Düz M., Solution of complex equations with Adomian Decomposition method, TWMS J. App. Eng. Math.,7(1) (2017) 66–73.
  • [5] Watugala G. K., Sumudu transform: a new integral transform to solve differential equations and control engineering problems, Int. J. Math. Educ. Sci. Technol., 24(1) (1993) 35–43.
  • [6] Anac H., Merdan M., Kesemen T., Solving for the random component time-fractional partial differential equations with the new Sumudu transform iterative method, SN App. Sci., 2 (2020) 1112 .
  • [7] Weerakoon S., Application of Sumudu transform to partial differential equations, Int. J. Math. Educ. Sci. Technol., 25(2) (1994) 277–283.
  • [8] Weerakoon S., Complex inversion formula for Sumudu transform, Int. J. Math. Educ. Sci. Technol. 29(4) (1998) 618–621.
  • [9] Demiray S.T., Bulut H., Belgacem F.B.M., Sumudu transform method for analytical solutions of fractional type ordinary differential equations, Math Prob Eng., (2015) https ://doi.org/10.1155/2015/13169 0
  • [10] Kumar M., Daftardar-Gejji V., Exact solutions of fractional partial differential equations by Sumudu transform iterative method., (2018) arXiv :1806.03057 v1
  • [11] Rahman N.A.A., Ahmad M.Z., Solving fuzzy fractional differential equations using fuzzy Sumudu transform, J. Nonl. Sci. Appl., 10(5) (2017) 2620–2632
  • [12] Prakash A., Kumar M., Baleanu D., A new iterative technique for a fractional model of nonlinear Zakharov–Kuznetsov equations via Sumudu transform, Appl. Math. Comput., 334 (2018) 30–40.
  • [13] Soong, T.T., Random Differential Equations in Science and Engineering, Academic Press, 327, (1973).
  • [14] Belgacem F.B.M, Karaballi A.A, Sumudu Transform Fundamental Properties Investigations and Applications, J. Appl. Math. Stoch. Analy., (2006) Article ID 91083 1–23
  • [15] Eltayeb H., Kılıçman A., A Note on the Sumudu Transforms and Differential Equations, Applied Math. Sci., 4(22) (2010) 1089 –1098.
  • [16] Watugala, G.K. The Sumudu transform for functions of two variables, Mathematical Engin. in Indus., 8(4)(2002) 293–302.
  • [17] Kılıçman A., Eltayeb H.,Agarwal R.P., On Sumudu Transform and System of Differential Equations, Abstract Appl. Analy., (2010), Article ID 598702, 11 pages.
  • [18] Merdan M., Anac H., Bekiryazici Z., Kesemen, T., Solving of Some Random Partial Differential Equations by Using Differential Transformation Method and Laplace-Padé Method, J. Gumushane Univ. Inst. Sci. Tech., 9(1) (2019) 108-118.
  • [19] Merdan M., Atasoy N., On Solutions Of Random Partial Differential Equations With Laplace Adomian Decomposition, Cumhuriyet Sci. J., 44(1) (2023) 160-169.
  • [20] Merdan M., Şişman Ş., Analysıs of Random Discrete Tıme Logistic Model, Sigma J. Eng. Nat. Sci., 38(3) (2020) 1269-1298.
  • [21] Merdan M., Altay Ö., Bekiryazici Z., Investigation of the Behaviour of Volterra Integral Equations with Random Effects, J. Gumushane Univ. Inst. Sci. Tech., 10(1) (2020) 205-216.
  • [22] Merdan M., Ordinary and partial complex differential equations with random effects. Master's Thesis, Gumushane University, Institute of Science, (2020).
  • [23] Merdan, M., Merdan, M., ve Şahin, R., Investigation of Behavior on Solutions of Lane–Emden Complex Differential Equations by a Random Differential Transformation Method, Compl., (2023) 3713454.
  • [24] Rogers, L.C.G., Williams, David., Diffusions, Markov Processes and Martingales, Vol 2: Ito Calculus (2nd ed., Cambridge Mathematical Library ed.). Cambridge University Press.(2000).
  • [25] Musiela, M., Rutkowski, M., Martingale Methods in Financial Modelling, 2nd Edition, Springer Verlag, Berlin, (2004).
  • [26] Øksendal, Bernt K., Stochastic Differential Equations: An Introduction with Applications. Berlin: Springer, (2003).
  • [27] Kunita, H., Stochastic Differential Equations Based on Lévy Processes and Stochastic Flows of Diffeomorphisms. In: Rao, M.M. (eds) Real and Stochastic Analysis. Trends in Mathematics. Birkhäuser, (2004).
  • [28] Imkeller P., Schmalfuss B., The Conjugacy of Stochastic and Random Differential Equations and the Existence of Global Attractors. J. Dyn. Diff. Equ., 13 (2) (2001) 215–249.
  • [29] Michel E., Stochastic calculus in manifolds. Springer Berlin, Heidelberg, (1989).
  • [30] Brzeźniak Z., Elworthy K.D., Stochastic differential equations on Banach manifolds, Methods Funct. Anal. Top. 6(1) (2000) 43-84.
  • [31] Feller W., An Introduction to Probability Theory and Its Applications, volume 1, 3rd edition. New York: John Wiley & Sons. (1968)
  • [32] Khaniyev, T., Ünver, İ., Küçük, Z., ve Kesemen T. (2017). Olasılık Kuramında Çözümlü Problemler, Nobel Akademik Yayıncılık.

Yıl 2024, Cilt: 45 Sayı: 3, 562 - 570, 30.09.2024

Mehmet Merdan , Merve Merdan , Rıdvan Şahin

https://doi.org/10.17776/csj.1256101

Öz

Kaynakça

  • [1] Düz M., Solution of complex differential equations with variable coefficients by using reduced differential transform, Mis. Math. Not., 21(1) (2020) 161–170.
  • [2] Düz M., Application of Elzaki Transform to first order constant coefficients complex equa ions, Bul. Int. math. Virt. inst., 7 (2017) 387–393.
  • [3] Düz M., On an application of Laplace transforms, NTMSCI., 5(2) (2017) 193–198.
  • [4] Düz M., Solution of complex equations with Adomian Decomposition method, TWMS J. App. Eng. Math.,7(1) (2017) 66–73.
  • [5] Watugala G. K., Sumudu transform: a new integral transform to solve differential equations and control engineering problems, Int. J. Math. Educ. Sci. Technol., 24(1) (1993) 35–43.
  • [6] Anac H., Merdan M., Kesemen T., Solving for the random component time-fractional partial differential equations with the new Sumudu transform iterative method, SN App. Sci., 2 (2020) 1112 .
  • [7] Weerakoon S., Application of Sumudu transform to partial differential equations, Int. J. Math. Educ. Sci. Technol., 25(2) (1994) 277–283.
  • [8] Weerakoon S., Complex inversion formula for Sumudu transform, Int. J. Math. Educ. Sci. Technol. 29(4) (1998) 618–621.
  • [9] Demiray S.T., Bulut H., Belgacem F.B.M., Sumudu transform method for analytical solutions of fractional type ordinary differential equations, Math Prob Eng., (2015) https ://doi.org/10.1155/2015/13169 0
  • [10] Kumar M., Daftardar-Gejji V., Exact solutions of fractional partial differential equations by Sumudu transform iterative method., (2018) arXiv :1806.03057 v1
  • [11] Rahman N.A.A., Ahmad M.Z., Solving fuzzy fractional differential equations using fuzzy Sumudu transform, J. Nonl. Sci. Appl., 10(5) (2017) 2620–2632
  • [12] Prakash A., Kumar M., Baleanu D., A new iterative technique for a fractional model of nonlinear Zakharov–Kuznetsov equations via Sumudu transform, Appl. Math. Comput., 334 (2018) 30–40.
  • [13] Soong, T.T., Random Differential Equations in Science and Engineering, Academic Press, 327, (1973).
  • [14] Belgacem F.B.M, Karaballi A.A, Sumudu Transform Fundamental Properties Investigations and Applications, J. Appl. Math. Stoch. Analy., (2006) Article ID 91083 1–23
  • [15] Eltayeb H., Kılıçman A., A Note on the Sumudu Transforms and Differential Equations, Applied Math. Sci., 4(22) (2010) 1089 –1098.
  • [16] Watugala, G.K. The Sumudu transform for functions of two variables, Mathematical Engin. in Indus., 8(4)(2002) 293–302.
  • [17] Kılıçman A., Eltayeb H.,Agarwal R.P., On Sumudu Transform and System of Differential Equations, Abstract Appl. Analy., (2010), Article ID 598702, 11 pages.
  • [18] Merdan M., Anac H., Bekiryazici Z., Kesemen, T., Solving of Some Random Partial Differential Equations by Using Differential Transformation Method and Laplace-Padé Method, J. Gumushane Univ. Inst. Sci. Tech., 9(1) (2019) 108-118.
  • [19] Merdan M., Atasoy N., On Solutions Of Random Partial Differential Equations With Laplace Adomian Decomposition, Cumhuriyet Sci. J., 44(1) (2023) 160-169.
  • [20] Merdan M., Şişman Ş., Analysıs of Random Discrete Tıme Logistic Model, Sigma J. Eng. Nat. Sci., 38(3) (2020) 1269-1298.
  • [21] Merdan M., Altay Ö., Bekiryazici Z., Investigation of the Behaviour of Volterra Integral Equations with Random Effects, J. Gumushane Univ. Inst. Sci. Tech., 10(1) (2020) 205-216.
  • [22] Merdan M., Ordinary and partial complex differential equations with random effects. Master's Thesis, Gumushane University, Institute of Science, (2020).
  • [23] Merdan, M., Merdan, M., ve Şahin, R., Investigation of Behavior on Solutions of Lane–Emden Complex Differential Equations by a Random Differential Transformation Method, Compl., (2023) 3713454.
  • [24] Rogers, L.C.G., Williams, David., Diffusions, Markov Processes and Martingales, Vol 2: Ito Calculus (2nd ed., Cambridge Mathematical Library ed.). Cambridge University Press.(2000).
  • [25] Musiela, M., Rutkowski, M., Martingale Methods in Financial Modelling, 2nd Edition, Springer Verlag, Berlin, (2004).
  • [26] Øksendal, Bernt K., Stochastic Differential Equations: An Introduction with Applications. Berlin: Springer, (2003).
  • [27] Kunita, H., Stochastic Differential Equations Based on Lévy Processes and Stochastic Flows of Diffeomorphisms. In: Rao, M.M. (eds) Real and Stochastic Analysis. Trends in Mathematics. Birkhäuser, (2004).
  • [28] Imkeller P., Schmalfuss B., The Conjugacy of Stochastic and Random Differential Equations and the Existence of Global Attractors. J. Dyn. Diff. Equ., 13 (2) (2001) 215–249.
  • [29] Michel E., Stochastic calculus in manifolds. Springer Berlin, Heidelberg, (1989).
  • [30] Brzeźniak Z., Elworthy K.D., Stochastic differential equations on Banach manifolds, Methods Funct. Anal. Top. 6(1) (2000) 43-84.
  • [31] Feller W., An Introduction to Probability Theory and Its Applications, volume 1, 3rd edition. New York: John Wiley & Sons. (1968)
  • [32] Khaniyev, T., Ünver, İ., Küçük, Z., ve Kesemen T. (2017). Olasılık Kuramında Çözümlü Problemler, Nobel Akademik Yayıncılık.
Toplam 32 adet kaynakça vardır.

Ayrıntılar

Birincil Dil İngilizce
Konular Matematik
Bölüm Natural Sciences
Yazarlar

Mehmet Merdan 0000-0002-8509-3044

Merve Merdan 0000-0002-6045-6531

Rıdvan Şahin 0000-0002-4597-7320

Yayımlanma Tarihi 30 Eylül 2024
Gönderilme Tarihi 27 Şubat 2023
Kabul Tarihi 13 Ağustos 2024
Yayımlandığı Sayı Yıl 2024Cilt: 45 Sayı: 3

Kaynak Göster

APA Merdan, M., Merdan, M., & Şahin, R. (2024). Investigation of solution behavior of Differential Equations by Sumudu methods of random complex Partial Differential Equations. Cumhuriyet Science Journal, 45(3), 562-570. https://doi.org/10.17776/csj.1256101
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