مجله مدل‌سازی پیشرفته ریاضی

مجله مدل‌سازی پیشرفته ریاضی

بررسی عددی و تخمین خطا برای معادلات انتشار کسری-زمانی چندگانه‌ی مبتنی بر عملگر کسری جدید

نوع مقاله : مقاله پژوهشی

نویسندگان
1 گروه ریاضی کاربردی، دانشکده علوم ریاضی، دانشگاه شهرکرد، شهرکرد، ایران
2 گروه ریاضی دانشگاه پیام نور، صندوق پستی 4697-19395، تهران ایران
چکیده
در این مقاله، روشی عددی برای حل معادلات انتشار کسری-زمانی چندجمله‌‌ای مرتبط با یک عملگر کسری جدید ارائه شده است. بر اساس روش تفاضل متناهی، یک طرح نیمه-گسسته در مسیر زمان به‌دست آورده شده و سپس برای گسسته سازی مکانی روش تقریب طیفی چبیشف استفاده شده است. همچنین آنالیز پایداری و خطای روش طیفی پیشنهادی بررسی شده است. بیشتر، توسعه‌‌ی معادله‌ی انتشار کسری چندجمله‌‌ای به معادله‌ی مرتبه‌ی توزیعی در نظر گرفته شده و روی آن تجزیه و تحلیل عددی صورت گرفته شده است. در پایان، با استفاده از برخی مثال‌های عددی‌، نتایج تئوری مورد تایید واقع شده است.
کلیدواژه‌ها
موضوعات

عنوان مقاله English

Numerical investigation and error estimate for multi-term time-fractional diffusion equations based on new fractional operator

نویسندگان English

Mojtaba Fardi 1
ٍٍEbrahim Amini 2
1 Department of Applied Mathematics, Faculty of Mathematical Science, Shahrekord University, Shahrekord, P. O. Box 115, Iran.
2 Department of Mathematics, Payme Noor University, P. O. Box 19395-4697 Tehran, Iran.
چکیده English

In this paper‎, ‎a numerical method is provided for solving multi-term time-fractional diffusion equations associated with a new fractional operator‎. ‎A semi-discrete scheme is obtained in temporal direction based on the finite difference method afterwards‎, ‎a Chebyshev-spectral method is used for spatial discretization‎. ‎Also‎, ‎the stability and error analysis are investigated‎. ‎Moreover‎, ‎the multi-term time-fractional diffusion equation is extended to a distributed order diffusion equation and numerical analysis has been done on it‎. ‎Finally‎, ‎the theoretical results are confirmed using some numerical examples.

کلیدواژه‌ها English

Time fractional diffusion equation
Spectral Approximation
Stability analysis
Error analysis
[1] D. A. Benson, R. Schumer, M. M. Meerschaert, and S. W. Wheatcraft, Fractional dispersion, Levy motion, and the MADE tracer tests, Transport in porous media, 42 (2001), 211-240.
[2] B. OShaughnessy and I. Procaccia, Analytical solutions for diffusion on fractal objects, Physical review letters, 54 (1985), 455.
[3] M. R. Ubriaco, A simple mathematical model for anomalous diffusion via Fisher’s information theory, Physics Letters, 373 (2009), 4017-4021.
[4] M. Fardi, A kernel-based pseudo-spectral method for multi-term and distributed order time-fractional diffusion equations, Numerical Methods Partial Differential Equations, 39(2023), 2630-2651.
[5] M. Fardi, S.-K.-Q. Al-Omari, and S. Araci, S. A pseudo-spectral method based on reproducing kernel for solving the time-fractional diffusion-wave equation, Advance in Continuous Discrete Models, 54 (2022), 1-14.
[6] M. Fardi, M. A. Zaky, and A. S. Hendy, Nonuniform difference schemes for multi-term and distributed-order fractional parabolic equations with fractional Laplacian, Mathematic and Computers in Simulation, 206 (2023), 614–635.
[7] M. Fardi, Y. Khan, A fast difference scheme on a graded mesh for time-fractional and space distributedorder diffusion equation with nonsmooth data, International Journall of Modern Physics B, 36 (2022), 2250076.
[8] M. Fardi, and M. Ghasemi, A numerical solution strategy based on error analysis for time-fractional mobile/immobile transport model, Soft Computing, 25 (2021), 11307–11331.
[9] S. Mohammadi, M. Ghasemi, and M. Fardi, A fast Fourier spectral exponential time-differencing method for solving the time- fractional mobile-immobile advection-dispersion equation, Computational and Applied Mathematics, 41 (2022), 264.
[10] M. Fardi and E. Amini, Numerical investigation of a new difference scheme on a graded mesh for solving the time-space fractional sub-diffusion equations with nonsmooth solutions, Journal of Advanced Mathematical Modeling, 12 (2022), 212-231.
[11] E. Amini, Optimal Control Problems: Convergence and Error Analysis in Reproducing Kernel Hilbert Spaces, Control and Optimization in Applied Mathematics, 6 (2021), 53-77.
[12] Z. Hao and W. Cao, An improved algorithm based on finite difference schemes for fractional boundary value problems with nonsmooth solution, Journal of Scientific Computing, 73 (2017), 395-415.
[13] M. M. Meerschaert and C. Tadjeran, Finite difference approximations for fractional advectiondispersion flow equations, Journal of computational and applied mathematics, 172 (2004), 65-77.
[14] Z. Zhao, Y. Zheng , and P. Guo, A Galerkin finite element method for a class of time-space fractional differential equation with nonsmooth data, Journal of Scientific Computing, 70 (2017), 386-406.
[15] Z. Hao, M. Park, G. Lin and Z. Cai, Finite element method for two-sided fractional differential equations with variable coefficients: Galerkin approach, Journal of Scientific Computing, 79 (2019), 700-717.
[16] F. Zeng, Z. Mao, and G. E. Karniadakis, A generalized spectral collocation method with tunable accuracy for fractional differential equations with end-point singularities, SIAM Journal on Scientific Computing, 39(2017), A360-A383.
[17] Z. Zhang, F. Zeng, and G. E. Karniadakis, Optimal error estimates of spectral Petrov-Galerkin and collocation methods for initial value problems of fractional differential equations, SIAM Journal on Numerical Analysis, 53 (2015), 2074-2096.
[18] Q. Xu and J. S. Hesthaven, Discontinuous Galerkin method for fractional convection-diffusion equations, SIAM Journal on Numerical Analysis, 52(2014), 405-423.
[19] A. Simmons, Q. Yang, and T. Moroney, A finite volume method for two-sided fractional diffusion equations on non-uniform meshes, Journal of Computational Physics, 335 (2017), 747-759.
[20] M. Caputo, and M. Fabrizio, A new definition of fractional derivative without singular kernel, Progr. Fract. Differ. Appl, 1 (2015), 1-13.
[21] J. Losada, and J. J. Nieto, Properties of a new fractional derivative without singular kernel, Progr. Fract. Differ. Appl, 1 (2015), 87-92.
[22] M. Caputo, and M. Fabrizio, Applications of new time and spatial fractional derivatives with exponential kernels, Progr. Fract. Differ. Appl, 2 (2016), 1-11.
[23] M. A. Dokuyucu, E. Celik, H. Bulut, and H. M. Baskonus, Cancer treatment model with the Caputo-Fabrizio fractional derivative, The European Physical Journal Plus, 133(2018), 1-6.
[24] S. Bushnaq, S. A. Khan, K. Shah, and G. Zaman, Mathematical analysis of HIV/AIDS infection model with Caputo-Fabrizio fractional derivative, Cogent Mathematics & Statistics, 5 (2018), 1432521.
[25] D. Baleanu, A. Jajarmi, H. Mohammadi, and S. Rezapour, A new study on the mathematical modelling of human liver with Caputo-Fabrizio fractional derivative, Chaos, Solitons & Fractals, 134 (2020), 109705.
[26] L. Zhengguang, A. Cheng, and L. Xiaoli, A second-order finite difference scheme for quasilinear time fractional parabolic equation based on new fractional derivative, International Journal of Computer Mathematics, 95 (2018), 396-411.
[27] J. Shen, T. Tang, and L. Wang, Spectral Methods: Algorithms, Analysis and Applications, Springer, 2013.
[28] C. Canuto, and A. Quarteroni, Approximation results for orthogonal polynomials in Sobolev spaces, Mathematics of Computation, 38 (1982), 67-86.
[29] A. Quarteroni and A. Valli, Numerical approximation of partial differential equations, Springer Science & Business Media, 2008.
[30] C. Bernardi and Y. Maday, Approximations spectrales de problemes aux limites elliptiques, Springer, 1992.
دوره 13، شماره 1 - شماره پیاپی 29
بهار ۱۴۰۲
تابستان 1402
صفحه 31-51

  • تاریخ دریافت 24 دی 1401
  • تاریخ بازنگری 12 اسفند 1401
  • تاریخ پذیرش 16 اردیبهشت 1402