چندجمله‌ای‌های لژاندر بهبود یافته کسری در حل معادلات دیفرانسیل با مشتقات جزیی کسری با تأخیر زمانی

نویسندگان
1 دانشگاه بین‌المللی امام خمینی(ره)، گروه ریاضی کاربرد
2 دانشگاه شیخ زاید دانشکدۀ تعلیم و تربیه خوست افغانستان
3 دانشگاه بین‌المللی امام خمینی (ره)، مرکز آموزش عالی فنی و مهندسی بوئین زهرا
چکیده
در این مقاله ماتریس‌های شبه عملیاتی مشتق صحیح و کسری برای توابع لژاندر بهبود یافته کسری برای اولین بار معرفی ‌می‌شوند. این ماتریس‌های عملیاتی به همراه روش هم‌ محلی گاوس- لژاندر برای حل دسته‌ای از معادلات دیفرانسیل با مشتقات جزئی کسری با تاخیر زمانی به کار برده می‌شوند. معادلات مورد بررسی به کمک ماتریس‌های عملیاتی و شبه عملیاتی به دستگاه معادلات جبری با ضرایب مجهول بسط لژاندر بهبود یافته کسری تبدیل می‌شوند. در پایان کارایی روش با در نظر گرفتن چند مثال عددی نشان داده می شود.




کلیدواژه‌ها

1. Baillie R. T. "Long memory processes and fractional integration in econometrics", J. Econ., 73 (1996) 5-59. ## 2. Hall M. G., Barrick T. R., "From diffusion-weighted MRI to anomalous diffusion imaging", Med, Reson. MAGN., 59, (2008) 447-455. ## 3. Povstenko Y. Z. "Signaling problem for time-fractional diffusion-wave equation in a half-space in the case of angular symmetry", Nonl, Dyn., 55 (2010) 593-605. ## 4. Engheta N., "On fractional calculus and fractional multipoles in electromagnetism", IEEE T. Antenn.Propag., 44 (1996) 554-566. ## 5. Oldham K. B., "Fractional differential equation in electro chemsitory", Adv. Eng. Soft., 41 (2010) 9-12. ## 6. Lederman C., Roquejoffre J. M., Wolanski, "Mathematical justification of a nonlinear integrodifferential equation for the propagation of spherical flames", Annali di. Matematica pura ed Applicata., 183 (2004) 173-239. ## 7. Mainardi F., "Fractional calculus: Some basic problems in continuum and statistical mechanics. In: Carpinteri, A. Mainardi, F (eds.) Fractals and Fractional Calculusin Continuum Mechanics, Springer", New York., (1997) 291-348. ## 8. Rossihin Y. A., Shitikova M. V., "Application of fractional calculus to dynamic problems of linear and nonlinear hereditary mechanics of solids", Appl. Mech. Rev., 50, (1997) 15-67. ## 9. He J. H., "Some applications of nonlinear fractional differential equation and their approximations", Bull. Sci. Technol., 15 (1999) 86-90. ## 10. Suarez L., Shokooh A., "An eigenvector expansion method for the solution of motion containing fractional derivative", J. Appl. Mech., 64 (1997) 629-635. ## 11. Abdulazi O., Hashim I., Momani S., "Solving systems of fractional differential equations by homotopy-perturbation method", Phys. Lett, A., 372 (2009) 451-459. ## 12. Odibat Z., Momani S. "Application of variational iteration method to nonlinear differential equations of fractional order", int. J. Nonl. Sci. Numer. Simul., 1(7) (2006) 15-27. ## 13. Momani S., Odibat Z., "Numerical comparison of methods for solving linear differential equations of fractional order", Chaos Solitons Fractals., 31 (2007) 1248-1255. ## 14. Daftardar-Gejj V., Jafari H. "Solving a multi-order fractional differential equation using decompostion", Appl. Math. Comput, 189 (2007) 541-548. ## 15. Meershaer M., Tadjeran C. "Finite difference approximations for two-sided space-fractional partial differential equations", App. Numer. Math., 56 (2006) 80-90. ## 16. Dubois F., Mengue S., "Mixed collocation for fractional differential equations", Numer. Algor., 34 (2003) 303-311. ## 17. Saadatmandi A., Dhghan M., "A legendre collocation method for fractional integro-differential equations", J. Vib. Control, 17 (2011) 2050-2058. ## 18. Bhrawy A. H., Tharwat M. M., Yildirim A., "A new formula for fractional integrals of Chebyshev polynomials: application for solving mulit-term fractional differentials", J. Appl. Math. Model, 37 (6) (2013) 4245-4252. ## 19. Saadatmandi A., "Bernstein operational matrix of fractional derivatives and its applications", Apple. Math. Model, 38 (4) (2014) 1365-1372. ## 20. Rehman M., Ur Ali Khan R., "The Legendre wavelet method for solving fractional differential equations", Comnun. Nonl. Sci. Numer. Simu., 16 (2011) 4163-4173. ## 21. Rahimkhani P., Ordokhani Y., Lima P., "An improved composite collocation method for distributed-order fractional differential equations based on fractional Chelyshkov wavelets", Appl. Numer. Math., (145 ) (2010) 1-27. ## 22. Keller A.A., "Contribution of the delay differential equations to the complex economic macrodynamics", WSEAS. Trans Syst., 9 (4) (2010) 358-371. ## 23. Ockhendon J. R., Tayler A. B., "The dynamics of a current collection system for an electric locomotive", Proc. R. Soc. Lond. Ser, A., 322 (1971) 447-468. ## 24. Ajello W. G., Freedom H. I., Wu J., "A model of stage structured population growth with density depended time delay", SIAM. J. Appl. Math., 52 (1992) 855-869. ## 25. Sedaghat S., Nemati S., Orokhani Y., "Application of the hybrid functions to solve neutral delay functional differential equations", Int. J. Comput. Math., 94 (2017) 503-514. ## 26. Tohidi E., Bhrawy A. H., Erfani K., "A collocation method based on Bernoulli operational matrix for numerical solution of generalized pantograph equation", Appl. Math. Model, 37 (2012) 4283-4294. ## 27. Yu Z. H., "Variational iteration method for solving the mutli-pantoraph delay equation", Phys. Lett, A., 372 (2008) 6475-6479. ## 28. Wang W. S., Li S. F. "On the one-leg-method for solving nonlinear neutral functional differential equations", Appl. Math. Comput., 193 (2007) 285-301. ## 29. Evans D. J., Raslan K. R. "The Adomian decomposition method for solving delay differential equation", Int. J. comput. Math., 82 (2005) 49-54. ## 30. Marzban H. R., Razzaghi M., "Solution of multi-delay systems using hybrid of block-pulse functions and Taylor series", Sound. Vib., 292 (2006) 954-963. ## 31. Sadeghi Hafshejani M., Karimi Vanani S., Sedighi Hafshejani J., "Numerical solution of delay differential equations using Legendre wavelet method", World Appl. Sci., 13 (2011) 27-33. ## 32. Saeed U., Rehman M. U., "Hermiter Wavelet method for fractional delay differential equation", J. Diff. Equa. (2014) 1-8. ## 33. Zubik-Kowal B., "Chebyshev pseudo spectral method and waveform relaxation for differential and differential–functional parabolic equations", Apple, Numer Math., 34 (2-3) (2000) 309-328. ## 34. Yang Y., Huang Y., "Spectral-collocation methods for fractional pantograph delay-integrodifferential equations", Advan. Math. Phys. (2013) 1-14. ## 35. Sakar M. G., Uludag F., Erdogan F., "Numerical solution of time-fractional nonlinear PDEs with proportional delays by homotopy perturbation method", Apple. Math Model, (2016) 40 (13-14) 6639-6649. ## 36. Khader M. M., Hendy A. S., "The approximate and exact solutions of the fractional-order delay differential equations using Legendre seudo spectral method", Int. J. pure. Apple. Math., 74, (2012) 287-297. ## 37. Wang Z., "A numerical method for delayed fractional-order differential equations", J. Apple. Math. (2013) 1-7. ## 38. Rahimkhani P., Ordokhani Y., Babolian E., "A new operational matrix based on Bernoulli wavelets for solving fractional delay differential equations", Numer. Algorithms, 74 (2016) 223-250. ## 39. Saeed U., Ur Rahman M., Iqbal M. A., "Modified Chebyshev wavelet methods for fractional delay-type equations", Apple. Math Compute, 264 (2015) 431-442. ## 40. Moghaddam B. H., Mostaghim Z. S., "A numerical method based on finite difference for solving fractional delay differential equations", J.Taibah Univ, Sci., 7 (2013) 120-127. ## 41. Rahimkhani P., Ordokhani Y., "Numerical Solution of Volterra–Hammerstein Delay Integral Equations", Iranian Journal of Science and Technology, Transactions A: Science, 44 (2) (2020) 445-457. ## 42. Marzban H., Razzaghi M., "Hybrid Functions Approach for Linearly Constrained Quadratic Optimal Control Problems", Apple. Math. Model, 27, (2003) 471-485. ## 43. Razzaghi M., Elnagar G., "Linear quadratic optimal control problems via shifted Legendre state paramerization", Int. J. Syst. Sci., 25 (1994) 393-399. ## 44. Bazm S., Hoseseini A., "Numerical solution of nonlinear integral equation using alternative Legendre polynomials", J. Appl. Math. Comput, (2016) 1-27. ## 45. Meng Z., Yi M., Huang J., Song L., "Numerical solutions of nonlinear fractional differential equations by alternative Legendre polynomials", Applied Mathematics and Computation, 336 (2018) 454-464. ## 46. Rahimkhani P., Ordokhani Y., "Approximate solution of nonlinear fractional and integro-differential equation using fractional alternative Legendre function", J. Comput. Appl. Math., 36 (2020) 1-150. ## 47. Podlubny I., "Fractional Differential Equations", Academic Press, San Diego (1999). ## 48. Abbasbandy S., Kazem S., Alhuthali M. S., Alsulami H. H., "Application of the operational matrix of fractional-order Legendre functions for solving the time-fractional convection–diffusion equation", Appl. Math Comput, 266 (2015) 31-40. ## 49. Chelyshkov V. S., "Alternative orthogonal polynomials and quadrature", Trans. Numer. Anal., 25 (7) (2006) 17-26. ## 50. Pimenov V. G., Hendy A. S., "A numerical solution for a class of time fractional diffusion equations with delay", Int J Appl. Math Comput. Sci., 27 (3) (2017) 477-488. ##