دستگاه بهینه جبر تقارنی و طبقه بندی جوابهای دقیق معادلۀ نیروی جریان

نویسنده
مهدی جعفری
دانشگاه
چکیده
در این مقاله گروه لی تقارنهای معادلۀ نیروی جریان را تحلیل و بررسی می‌کنیم. معادلۀ نیروی جریان یکی از معادلات مهم و کاربردی در مکانیک سیالات است که میزان انرژی گرمایی حاصل از جریان را بررسی می‌کند. به کمک گروه لی تقارنی معادلۀ نیروی جریان، می‌توانیم جوابهای جدیدی را با استفاده از جوابهای قبلی این معادله بدست آوریم. این جوابها در مطالعه وابستگی جوابها به پارامترها و حل کردن مسائل مقدار مرزی کاربرد فراوان دارد. در ادامه، به نوعی طبقه بندی زیرجبرهای لی تقارنی بر اساس نگاشت الحاقی پرداخته و به کمک این طبقه بندی، دستگاه بهینه زیرجبرهای یک پارامتری جبر لی تقارنی معادله را به دست می‌آوریم. همچنین با روش کاهش مرتبه و استفاده از ناورداهای دیفرانسیلی، جوابهای دقیقی را برای معادلۀ نیروی جریان بدست آورده و طبقه بندی می‌کنیم. در پایان با استفاده از گروه لی تقارنی به محاسبه قوانین پایستگی این معادله می‌پردازیم.





کلیدواژه‌ها
گروه لی تقارنی معادله
معادله نیروی جریان
قوانین پایستگی

عنوان مقاله English

Optimum system of symmetry algebra and classification of exact solutions of the flow energy equation.

نویسنده English

Mehdi Jafari
university
چکیده English

The Lie symmetry method, first developed by Sophus Lie, is a fundamental tool in the analysis of differential equations, particularly for generating exact solutions and reducing equation order. One of its key applications lies in identifying invariant solutions and simplifying complex nonlinear systems through symmetry reductions. In recent decades, this method has been extended and applied to a wide range of physical models. The present study focuses on the flow energy equation, which arises in fluid mechanics and describes thermal energy distribution in incompressible Newtonian pipe flow. Using classical Lie point symmetries, we obtain the symmetry generators of the equation and classify its one-parameter subalgebras through the adjoint representation. Based on this classification, an optimal system is constructed to systematically generate non-equivalent invariant solutions. In addition to symmetry reductions, we compute several exact solutions of the reduced equations and analyze the structure of conservation laws associated with the flow energy equation. These conservation laws are derived using both direct and variational approaches. The results highlight the effectiveness of Lie symmetry analysis in understanding and solving nonlinear PDEs in applied mathematics and physics.

کلیدواژه‌ها English

Symmetry Lie group of equations
flow energy equation
conservation laws
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