دفاعیه دکتری در دانشکده ریاضی و علوم کامپیوتر
سمر النصار (دانشجوی دوره دکتری دانشکده ریاضی و علوم کامپیوتر- گرایش هندسه)، ۲۹ شهریورماه ۱۴۰۲ از رساله دکتری خود با عنوان «ساختارهای هندسی و جواب‌های دقیق معادلات فوکر - پلانک» دفاع کرد.
چکیده این رساله که به راهنمایی دکتر مهدی نجفی‌خواه انجام شده، به شرح زیر است:
چکیده:
In this thesis we establish the use of the classical and nonclassical symmetry methods on secondorder linear Fokker-Planck equations (FPEs) of the form ut = [f′(x)u + f(x)ux] + ۱۲ g(t)uxx. This includes first finding the symmetries and then using them to reduce and solve equations; determining the optimal system of lie symmetry subalgebras; formulating general initial value problems (IVPs) solvable via the lie symmetries (including those with initial conditions (ICs) that
are not left invariant under the symmetries); and establishing how symmetries can be used to find functionally separable solutions. In our nonclassical analysis: firstly, establish the relationship between the arbitrary functions in the governing equation and the infinitesimals; then, based on the link, the system of the determining equations of the nonclassical symmetry is extended.
New strictly nonclassical symmetries (NCLS) of the Fokker-Planck equation (FPE) in three main cases result, determined by assisting the Computer Algebra System Reduce. Furthermore, we explored using symmetries to solve initial value problems (IVPs). In general, it is thought that to be able to solve IVPs, the given condition of the form u(x; ۰) = F(x) needs to be left invariant under the one-parameter Lie group of transformations that leaves the partial differential equation (PDE) invariant.
In accordance with this procedure, using the symmetries found for our governing FPE, general ICs solvable with our FPE were instituted. Several examples are provided to explain the method. Further, we investigated a paper [۱] on symmetries and ICs and expanded the main result of this paper, allowing the most general form for a first-order IC to be conceded by a given classical symmetry (CLS). The result of their paper supposed that the generator required to leave the initial condition invariant.
In this thesis, their result is extended to the case where the first-order IC need not be left-invariant, and therefore many more IVP could be solved with the symmetries. Thus, we set the result to our governing FPE with first-order ICs and provided some examples with classical symmetries(CLS). In addition, by considering the form of the invariant surface condition (ISC), we have shown the application of classical symmetries of FPE to finding functionally separable
solutions of different forms.
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