# بررسی عددی یک طرح تفاضلی جدید روی یک شبکه مدرج برای حل معادلات زیر-انتشار کسری زمانی-مکانی با جواب‌های ناهموار

نوع مقاله : مقاله پژوهشی

نویسندگان

1 گروه ریاضی کاربردی، دانشکده علوم ریاضی، دانشگاه شهرکرد، شهرکرد، ایران

2 گروه ریاضی دانشگاه پیام نور، صندوق پستی 19395-4697، تهران، ایران

چکیده

در این مقاله، یک طرح تفاضلی جدید روی یک شبکه مدرج برای حل مسائل زیر-انتشار کسری زمانی-مکانی ارائه می‌دهیم. در معادلات مذکور مشتقات زمانی از نوع کپوتو با مرتبه‌ی ‎\$gammain (0,1)\$‎ و مشتقات مکانی از نوع ریس با مرتبه‌ی ‎\$alpha in (1,2]\$‎ هستند. پایداری و همگرایی طرح تفاضلی را مورد بررسی قرار می‌دهیم که اساس تئوری روش پیشنهادی است. نشان می‌دهیم که طرح تفاضلی جدید بدون قید و شرط پایدار است. همچنین، اثبات می‌کنیم که این طرح تفاضلی با مرتبه‌ی ‎\$min{2-gamma,rgamma}\$‎ در زمان و مرتبه‌ی دو در مکان برای هر ‎\$gammain (0,1)\$‎ و هر ‎\$alpha in (1,2]\$‎ همگرا است. در پایان، یک مثال عددی برای نشان دادن کارآیی و دقت طرح تفاضلی ارائه می‌شود.

کلیدواژه‌ها

موضوعات

عنوان مقاله [English]

### Numerical investigation of a new difference scheme on a graded mesh for solving the time-space fractional sub-diffusion equations with nonsmooth solutions

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, ‎we provide a new difference scheme on a graded mesh for solving the time-space fractional diffusion problem‎. ‎In ‎this ‎equation ‎the ‎time ‎derivative ‎is ‎the ‎Caputo ‎of ‎order ‎‎\$‎‎gammain(0,1)\$ ‎and ‎the ‎space ‎derivative ‎is ‎the ‎Riesz ‎of ‎order ‎‎\$‎‎alphain(1,2]\$.‎ ‎The stability and convergence of the difference scheme are discussed which provides the theoretical basis of the proposed‎ ‎schemes‎. W‎e prove that the new difference scheme is unconditionally stable‎. Also, ‎we find that the difference scheme is convergent with order \$min{2-gamma,rgamma}\$ in time for all \$gammain (0,1)\$ and \$alpha in (1,2]\$‎. ‎A test example is given to verify the efficiency and accuracy of the difference scheme‎.

کلیدواژه‌ها [English]

• Caputo derivative&lrm
• Riesz derivative&lrm
• Sub-diffusion equation&lrm
• &lrm

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### سابقه مقاله

• تاریخ دریافت: 14 آبان 1400
• تاریخ بازنگری: 20 خرداد 1401
• تاریخ پذیرش: 27 خرداد 1401
• تاریخ اولین انتشار: 01 تیر 1401