حل معادله کلین–گوردن کسری زمانی فازی با استفاده از ترکیب دو روش تفاضلات متناهی و تبدیل دیفرانسیل کسری

نوع مقاله : مقاله پژوهشی

نویسندگان

گروه ریاضی، دانشکده ریاضی، دانشگاه سیستان و بلوچستان، زاهدان، ایران.

چکیده

در این مقاله یک روش ترکیبی برای حل معادله دیفرانسیل کلین ـ گوردن کسری زمانی فازی ارائه می‌شود. روش مذکور، مبتنی بر ترکیب روش‌های تبدیل دیفرانسیل و تفاضلات متناهی است، به این ترتیب که با استفاده از تکنیک نیمه گسسته‌سازی، ابتدا شاخه پایینی معادله فازی کلین ـ گوردن به یک دستگاه معادلات دیفرانسیل معمولی کسری تبدیل می‌شود، سپس با استفاده از روش تبدیل دیفرانسیل کسری، دستگاه معادلات دیفرانسیل حاصل حل می‌شود و جواب شاخه پایینی به دست می‌آید. در نهایت، با بکارگیری این روند برای شاخه بالایی معادله، جواب تقریبی شاخه بالایی و به دنبال آن، جواب تقریبی معادله کلین ـ گوردن کسری فازی به دست می‌آید. همگرایی روش مورد بررسی قرار می‌گیرد و با ارائه مثال‌های عددی، نشان داده می‌شود که این ایده، برای حل معادله فازی کسری از کارایی و دقت قابل قبولی برخوردار است. همچنین زمان اجرای محاسبات نشان می‌دهد، هزینه محاسباتی آن پایین است.

کلیدواژه‌ها

موضوعات


عنوان مقاله [English]

Solving the fuzzy time-fractional Klein-Gordon equation using the finite difference and differential transform methods

نویسندگان [English]

  • Zahra Sahraye
  • Maryam Arabameri
Department of Mathematics, Faculty of Mathematics, University of Sistan and Baluchestan, Zahedan, IranIran.
چکیده [English]

In this paper, a hybrid method is presented to solve the fuzzy time-fractional Klein-Gordan equation. This method is a combination of finite difference and fractional differential transform methods. In this way, using the semi-discretization technique, the lower branch of the fuzzy Klein-Gordan equation is converted to a system of ordinary differential equations. Then, the obtained system of differential equations is solved by the fractional differential transform method, and the solution of the lower branch is obtained. Finally, by using the same procedure for the upper branch, the approximate solution of the upper branch, and the approximate solution of the fuzzy fractional Klein-Gordan equation is obtained. The convergence is examined, also, the efficiency and accuracy of this method are approved by solving some examples using the proposed method. Also, the CPU time shows the computational cost of this algorithm is low.

کلیدواژه‌ها [English]

  • Fuzzy fractional Klein-Gordan equation
  • Fractional differential transform method
  • Finite difference method
  • Approximate solution
  • Convergence
[1] H. Moradi Farahani, J. Asgari, and M. Zakeri, “A Surveying on Type-2 Fuzzy Logic: Its Genesis and Its Application,” Soft Comput. J., vol. 2, no. 1, pp. 22-43, 2013, dor: 20.1001.1.23223707.1392.2.1.58.2 [In Persian].
[2] A. Akrami and M. Parsamanesh, “Investigation of a mathematical fuzzy epidemic model for the spread of coronavirus in a population,” Soft Comput. J., vol. 11, no. 1, pp. 2-9, 2022, doi: 10.22052/scj.2022.246053.1045 [In Persian].
[3] R. Akhoondi and R. Hosseini, “A Novel Fuzzy-Genetic Differential Evolutionary Algorithm for Optimization of A Fuzzy Expert Systems Applied to Heart Disease Prediction,” Soft Comput. J., vol. 6, no. 2, pp. 32-47, 2018, dor: 20.1001.1.23223707.1396.6.2.3.7 [In Persian].
[4] M. Alshammari, W.W. Mohammed, and M. Yar, “Novel analysis of fuzzy fractional Klein-Gordon model via semianalytical method,” Function Spaces, pp. 1-9, 2022, doi: 10.1155/2022/40020269.
[5] S. Arshad and V. Lupulescu, “On the fractional differential equations with uncertainty,” Nonlinear Anal. Theory Methods Appl., vol. 74, no. 11, pp. 3685-3693, 2011, doi: 10.1016/j.na.2011.02.048.
[6] S. Arshad and V. Lupulescu, “Fractional differential equation with the fuzzy initial condition,” Electron. J. Differ. Equ., vol. 2011, no. 34, pp. 1-8, 2011.
[7] A. Ahmadian, M. Suleiman, S. Salahshour, and D. Baleanu, “Jacobi operational matrix for solving a fuzzy linear fractional differential equation,” Adv. Differ. Equ., vol. 2013, no. 1, pp. 1-29, 2013, doi: 10.1186/1687-1847-2013-104.
[8] E. Khodadadi and E. Celik, “The variational iteration method for fuzzy fractional differential equations with uncertainty,” Fix. Point Theory Appl., vol. 2013, pp. 1-7, 2013, doi: 10.1186/1687-1812-2013-13.
[9] S. Arshad, “On existence and uniqueness of solution of fuzzy fractional differential equations,” J. Fuzzy Syst., vol. 10, no. 6, pp. 137-151, 2013.
[10] D. Takaci, A. Takaci, and A. Takaci, “On the operational solutions of fuzzy fractional differential equations,” Fract. Calc. Appl. Anal., vol. 17, pp. 1100-1113, 2014, doi: 10.2478/s13540-014-0216-y.
[11] A. Rivaz, O.S. Fard, and T.A. Bidgoli, “Solving fuzzy fractional differential equations by a generalized differential transform method,” SeMA J., vol. 73, pp. 149-170, 2016, doi: 10.1007/s40324-015-0016-x.
[12] N.A. Rahman and M.Z. Ahmad, “Solving fuzzy fractional differential equations using fuzzy Sumudu transform,” J. Nonlinear Sci. Appl., vol. 10, no. 5, pp. 2620-2632, 2017, doi: 10.22436/jnsa.010.05.28.
[13] S. Tomasiello and J.E. Macias-Diaz, “Note on a Picard-like method for Caputo fuzzy fractional differential equations,” Appl. Math. Inf. Sci., vol. 11, no. 1, pp. 281-287, 2017, doi: 10.18576/amis/110134.
[14] M. Alaroud, M. Al-Smadi, R.R. Ahmad, and U.K.S. Din, “Computational optimization of residual power series algorithm for certain classes of fuzzy fractional differential equations,” J. Differ. Equ., vol. 2018, pp. 1-11, 2018, doi: 10.1155/2018/8686502.
[15] A. Harir, S. Melliani, and L.S. Chadli, “Fuzzy space-time fractional telegraph equations,” J. Math. Trends Technol., vol. 64, no. 2, pp. 101-108, 2018, doi: 10.1155/2019/5734190.
[16] A. Armand, T. Allahviranloo, S. Abbasbandy, and Z. Gouyandeh, “The fuzzy generalized Taylor’s expansion with application in fractional differential equations,” Int. J. Fuzzy Syst., vol. 16, no. 2, pp. 57-72, 2019, doi: 10.22111/IJFS.2019.4542.
[17] K. Shah, A.R. Seadawy, and M. Arfan, “Evaluation of one dimensional fuzzy fractional partial differential equations,” Alexandria Eng. J., vol. 59, no. 5, pp. 3347-3353, 2020, doi: 10.1016/j.aej.2020.05.005.
[18] O.A. Arqub and M. Al-Smadi, “Fuzzy conformable fractional differential equations: novel extended approach and new numerical solutions,” Soft Comput., vol. 24, no. 16, pp. 12501-12522, 2020, doi: 10.1007/s00500-020-04687-0.
[19] Z. Alijani, D. Baleanu, B. Shiri, and G.C. Wu, “Spline collocation methods for systems of fuzzy fractional differential equations,” Chaos Solitons Fractals, vol. 131, p. 109510, 2020, doi: 10.1016/j.chaos.2019.109510.
[20] A. El Mfadel, S. Melliani, and M.H. Elomari, “A note on the stability analysis of fuzzy nonlinear fractional differential equations involving the Caputo fractional derivative,” J. Math. Math. Sci., vol. 2021, pp. 1-6, 2021, doi: 10.1155/2021/7488524.
[21] E.U. Haq, Q.M. U. Hassan, J. Ahmad, and K. Ehsan, “Fuzzy solution of system of fuzzy fractional problems using a reliable method,” Alexandria Eng. J., vol. 61, no. 4, pp. 3051-3058, 2022, doi: 10.1016/j.aej.2021.08.034.
[22] J.J. Sakurai and E.D. Commins, Modern Quantum Mechanics, Revised ed. Reading, MA, USA: Addison-Wesley, 1995, doi: 10.1119/1.17781.
[23] M. Mulimani and S. Kumbinarasaiah, “A numerical study on the nonlinear fractional Klein–Gordon equation,” J. Umm Al-Qura Univ. Appl. Sci., pp. 1-22, 2023, doi: 10.1007/s43994-023-00091-0.
[24] A.K. Golmankhaneh, A.K. Golmankhaneh, and D. Baleanu, “On nonlinear fractional Klein-Gordon equation,” Signal Process., vol. 91, no. 3, pp. 446-451, 2011, doi: 10.1016/j.sigpro.2010.04.016.
[25] K.A. Gepreel and M.S. Mohamed, “Analytical approximate solution for nonlinear spacetime fractional Klein-Gordon equation,” Chin. Phys. B, vol. 22, no. 1, p. 010201, 2013, doi: 10.1088/1674-1056/22/1/010201.
[26] K. Hosseini, P. Mayeli, and R. Ansari, “Modified Kudryashov method for solving the conformable time-fractional Klein-Gordon equations with quadratic and cubic nonlinearities,” Optik, vol. 130, pp. 737-742, 2017, doi: 10.1016/j.ijleo.2016.10.136.
[27] H. Singh, D. Kumar, J. Singh, and C.S. Singh, “A reliable numerical algorithm for the fractional Klein-Gordon equation,” Eng. Trans., vol. 67, no. 1, pp. 21-34, 2019, doi: 10.24423/EngTrans.910.20190214.
[28] R.M. Ganji, H. Jafari, M. Kgarose, and A. Mohammadi, “Numerical solutions of timefractional Klein-Gordon equations by clique polynomials,” Alexandria Eng. J., vol. 60, no. 5, pp. 4563-4571, 2021, doi: 10.1016/j.aej.2021.03.026.
[29] V.R. Nikam, S.B. Gaikwad, S.A. Tarate, and K.A. Kshirsagar, “Fuzzy Laplace-Adomian Decomposition Method for Approximating Solutions of Time Fractional Klein-Gordon Equations in a Fuzzy Environment,” Eur. Chem. Bull., vol. 12, no. 8, pp. 5926-5943, 2023, doi: 10.1155/2022/3864053.
[30] S.H. Hashemi Mehne, “Differential transform method: A comprehensive review and analysis,” Iran. J. Numer. Anal. Optim., vol. 12, no. 3, pp. 629-657, 2022, doi: 10.22067/IJNAO.2022.77130.1153.
[31] V.S. Erturk, S. Momani, and Z. Odibat, “Application of generalized differential transform method to multi-order fractional differential equations,” Commun. Nonlinear Sci. Numer. Simul., vol. 13, no. 8, pp. 1642-1654, 2008, doi: 10.1016/j.cnsns.2007.02.006.
[32] Z. Odibat, S. Kumar, N. Shawagfeh, A. Alsaedi, and T. Hayat, “A study on the convergence conditions of generalized differential transform method,” Math. Methods Appl. Sci., vol. 40, pp. 40-48, 2016, doi: 10.1002/mma.3961.
[33] Z. Sahraee and M. Arabameri, “A semi-discretization method based on finite difference and differential transform methods to solve the time-fractional telegraph equation,” Symmetry, vol. 15, no. 9, p. 1759, 2023, doi: 10.3390/sym15091759.
[34] H. Porki, M. Arabameri, and R. Gharechahi, “Numerical solution of nonlinear fractional Riccati differential equations using compact finite difference method,” Iran. J. Numer. Anal. Optim., vol. 12, no. 3, pp. 585-606, 2022, doi: 10.22067/IJNAO.2022.76489.1129.
[35] Z. Avazzadeh, H. Hassani, P. Agarwal, S. Mehrabi, M.J. Ebadi, and M.K. Hosseini Asl, “Optimal study on fractional fascioliasis disease model based on generalized Fibonacci polynomials,” Math. Methods Appl. Sci., vol. 46, no. 8, pp. 9332-9350, 2023, doi: 10.1002/mma.9057.
[36] Z. Avazzadeh, H. Hassani, M. J. Ebadi, P. Agarwal, M. Poursadeghfard, and E. Naraghirad, “Optimal approximation of fractional order brain tumor model using generalized Laguerre polynomials,” Iran. J. Sci., vol. 47, no. 2, pp. 501-513, 2023, doi: 10.1007/s40995-022-01388-1.