Construction of Pseudospectral Meshless Radial Point Interpolation for Sobolev Equation with Error Analysis
محورهای موضوعی : مجله بین المللی ریاضیات صنعتی
S. Abbasbandy
1
(Department of Applied Mathematics, Imam Khomeini International University, Qazvin, Iran.)
E. Shivanian
2
(Department of Applied Mathematics, Imam Khomeini International University, Qazvin, Iran.)
کلید واژه: meshless technique, Pseudospectral method, Sobolev equation, Radial point interpolation (RPI), Radial basis function,
چکیده مقاله :
In this study, we develop an approximate formulation for two-dimensional (2D) Sobolev equations based on pseudospectral meshless radial point interpolation (PSMRPI). The Sobolev equations which are arisen in the fluid flow penetrating rocks, soils, or different viscous media do not have an exact solution except in some special cases. The problem can be rigorously solved particularly when the geometry of the domain is more complex. In the PSMRPI method, the nodal points do not need to be regularly distributed and can even be quite arbitrary. It is easy to have high order derivatives of unknowns in terms of the values at nodal points by constructing operational matrices. It is proved that the method is convergent and unconditionally stable in some sense with respect to the time. The main results of the Sobolev equation are demonstrated by some examples to show the validity and trustworthiness of the PSMRPI technique.
در این مطالعه، یک روش تقریبی برای معادلات دو بعدی (2D) Sobolev بر اساس درون یابی نقطه شعاعی بدون شبکه طیفی (PSMRPI) ارایه می شود. معادلات سوبولف که در جریان نفوذ جریان سیال در سنگها، خاکها یا محیطهای مختلف چسبناک مشاهده می شود، به جز در برخی موارد خاص، جواب دقیق ندارند. این مسئله را به سختی می توان حل کرد، به ویژه هنگامی که هندسه دامنه پیچیده تر باشد. در روش PSMRPI، نقاط گره ای نیازی به توزیع منظم ندارند و حتی می توانند کاملاً دلخواه باشند. بوسیله ساخت ماتریس عملیاتی تقریب، مشتقات مرتبه بالا برحسب مقادیر تابع مجهول در نقاط گره ای آسان است. ثابت می کنیم که این روش نسبت به زمان پایداری بی قید و شرط دارد. نتایج اصلی روی معادله سوبولف با چند مثال نشان می دهد که روش PSMRPI قابل اعتماد بوده و خوب عمل می کند.
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