حل تحلیلی معادله انتقال آلاینده در رودخانه با ضرایب متغیر دلخواه با استفاده از تکنیک تبدیل انتگرالی تعمیم‌یافته

نوع مقاله: مقاله پژوهشی

نویسندگان

1 گروه سازه‌های آبی، دانشکده کشاورزی، دانشگاه تربیت مدرس

2 گروه ریاضی، دانشکده ریاضی، دانشگاه صنعتی شریف

چکیده

انتقال آلودگی در رودخانه به‌وسیله معادله دیفرانسیل با مشتق‌های جزئی جابه‌جایی-پراکندگی-واکنش (ADRE) بیان می‌شود. راه‌حل‌های تحلیلی ازجمله تبدیل‌های انتگرالی ابزارهای بسیار قدرتمند و مفیدی در حل معادله ADRE هستند. در پژوهش حاضر، معادله یک‌بعدی انتقال آلودگی در رودخانه با ضرایب وابسته به مکان با استفاده از تکنیک تبدیل انتگرالی تعمیم‌یافته، (GITT)، در دامنه‌ای با طول محدود حل شده است. در تکنیک GITT تبدیل‌های مستقیم و معکوسی تعریف می‌شود که استفاده از آن‌ها در حل مسئله منجر به تولید دستگاهی از معادلات دیفرانسیل وابسته به زمان و بنابراین ساده شدن حل معادله حاکم بر پدیده می‌گردد. صحت‌سنجی پاسخ تحلیلی ارائه شده با استفاده از مقایسه نتایج به‌دست آمده از مدل ریاضی با حل‌های تحلیلی موجود در منابع و نیز روش عددی مبتنی بر تفاضل‌های محدود انجام شد. مقایسه نتایج GITT و حل‌های تحلیلی استفاده شده در صحت‌سنجی و حل عددی به همراه شاخص‌های آماری، نشان از دقت بسیار بالای راه‌حل ارائه شده دارد. همچنین برای نشان دادن اهمیت به‌کارگیری ضرایب متغیر در معادله انتقال آلاینده در رودخانه، نتایج حل معادله با ضرایب ثابت و حل معادله با ضرایب متغیر مقایسه شد. محاسبه شاخص‌های آماری در این حالت بیانگر عدم دقت کافی نتایج معادله انتقال آلودگی با ضرایب ثابت است.

کلیدواژه‌ها

موضوعات


عنوان مقاله [English]

Analytical Solution of Contaminant Transport Equation in River by Arbitrary variable coefficients using Generalized Integral Transform Technique

نویسندگان [English]

  • Nazem Bavandpouri Gilan 1
  • Mehdi Mazaheri 1
  • Morteza Fotouhi Firouzabadi 2
1 Department of Water Structures, Faculty of Agriculture, Tarbiat Modares University, Tehran, Iran
2 Sharif University of Technology
چکیده [English]

Contamination transport in the river is expressed using advection-dispersion-reaction partial differential equation (ADRE). There are a variety of analytical and numerical methods for solving the aforementioned equation. Analytical solutions such integral transforms are very powerful and useful tools in solving ADRE. In the present study, one-dimensional ADRE with space-dependent coefficients in river has been solved using generalized integral transform technique (GITT). Forward and inverse transformations are defined in GITT technique which using them in problem solving leads to generating time-dependent system of ordinary differential equations. Analytical solution verification was accomplished using the comparison of the results of mathematical models with analytical solutions and also numerically model based on finite differences method. To inspect the accuracy of models’ results, statistical indicators were calculated. Comparison of GITTs’ result with analytical solutions that used in verification and numerical solution implied high accuracy of the proposed solution. Also to show the importance of the application of variable coefficients in ADRE in river, the results of solving equation with constant and variable coefficients were compared. 

کلیدواژه‌ها [English]

  • Analytical solution
  • Contaminant transport
  • River
  • Generalized integral transform technique
  • Variable coefficient
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