یک طرح تفاضل متناهی برای معادلات انتشار مرتبه‌ی توزیعی زمانی-مکانی با جواب‌های منفرد ضعیف

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، معادلات انتشار مرتبه‌ی توزیعی زمانی-مکانی با جواب‌های منفردی ضعیف در زمان اولیه در نظر گرفته شده است. طرح‌ تفاضلی جدید با استفاده از یک شبکه غیریکنواخت برای حل این معادلات ارائه می‌شود. پایداری و همگرایی طرح‌ تفاضلی پیشنهادی مورد بحث قرار می‌گیرد. نشان می‌دهیم که طرح تفاضلی بدون قید و شرط پایدار است و ثابت می‌کنیم که طرح‌ تفاضلی همگرا است. همچنین، نشان می‌دهیم که مرتبه همگرایی زمانی روی شبکه غیریکنواخت بیش از شبکه یکنواخت است. در پایان‏، چندین مثال‌ عددی برای تایید نتایج تئوری ارائه می‌شود.

کلیدواژه‌ها

موضوعات


عنوان مقاله [English]

A finite difference scheme for time-space fractional distributed-order diffusion eqations with weakly ‎singular ‎solutions

نویسندگان [English]

  • Mojtaba Fardi 1
  • Ebrahim Amini 2
  • Soheila Mohammadi 1
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, ‎the time-space distributed-order fractional diffusion equations with weakly singular solutions are considered‎. We provide the difference scheme using a nonuniform mesh to solve ‎equations‎‎. ‎The stability and convergence of the difference scheme are discussed‎, ‎We prove that the difference scheme is unconditionally stable‎. ‎We find that the difference scheme is convergent‎. ‎We also show that the temporal convergence order on the nonuniform mesh is higher than on the uniform mesh‎. Finally, some numerical examples are given to verify the theoretical analysis‏.

کلیدواژه‌ها [English]

  • Distributed-order
  • Time-‎space
  • Nonuniform mesh‎
  • Nonsmooth solutions
  • Stability
  • Error estimate
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