Yerçekimi Akımlarının Simetri Grup Analizi Ve Benzerlik Çözümleri

Konferans Bildirisi -- Teorik ve Uygulamalı Mekanik Türk Milli Komitesi, 2008Conference Paper -- Theoretical and Applied Mechanical Turkish National Committee, 2008Yoğunlukları birbirinden farklı iki akışkandan birinin diğerinin içine doğru yerçekiminin etkisiyle akması ile oluşan ve bazen yoğunluk akımları diye de adlandırılan yerçekimi akımları, birçok doğal ve insan yapımı durumda ortaya çıkmaktadır. Literatürde sabit hacimli bir sıvı tarafından üretilen yerçekimi akımlarıyla ilgili çalışmalara bakıldığında kullanılan en genel yaklaşımın boyut analiz yaklaşımı olduğu görülür. Boyut analizi sığ su denklemlerinin benzerlik analizin kullanılan tek yöntemdir ve tek bir benzerlik değişkeninin elde edilmesini mümkün kılmaktadır. Bu çalışma içerisinde boyut analizine alternatif bir yaklaşım olarak Lie grup teorisi kullanılmıştır ve Lie grup teorisinin, Boyut analizinin bir genelleştirilmesi olduğu gösterilmiştir. Bu çalışma içerisinde kullanılan bu yeni yaklaşım, mevcut ve benzeri problemlerin incelenmesinde, Lie grup teorisinin çok daha genel ve trivial olmayan benzerlik yapıları ve benzerlik çözümlerinin elde edilebilmesinde kullanılabileceğini göstermektedir.Gravity currents, (sometimes called density currents or buoyancy currents), which consist of fluid of one density flowing under the influence of gravity into fluid of another density, occur in many natural and man-made situations. In the literature, dimensional analysis is the most common method for the gravity currents which are occured by the fixed volume fluids. All of the studies in the current literature are related to the self-similarity analysis of the shallowwater related problems based on the dimensional analysis. In fact, the dimensional analysis enables to find only one particular type of similarity variable. In this study, Lie group theory was used as an alternative approach of dimensional analysis approximations and Lie group theory was shown as the generalization of the dimensional analysis. This new approach mentioned in this study presents the opportunity to obtain more general and nontrivial similarity forms and similarity solutions for the problem

On New Conservation Laws of Fin Equation

We study the new conservation forms of the nonlinear fin equation in mathematical physics. In this study, first, Lie point symmetries of the fin equation are identified and classified. Then by using the relationship of Lie symmetry and λ-symmetry, new λ-functions are investigated. In addition, the Jacobi Last Multiplier method and the approach, which is based on the fact λ-functions are assumed to be of linear form, are considered as different procedures for lambda symmetry analysis. Finally, the corresponding new conservation laws and invariant solutions of the equation are presented

On symmetries, conservation laws and invariant solutions of the foam-drainage equation

This study deals with symmetry group properties and conservation laws of the foam-drainage equation. Firstly, we study the classical Lie symmetries, optimal systems, similarity reductions and similarity solutions of the foam-drainage equation which are obtained through the Lie group method of infinitesimal transformations. Secondly, using the new general theorem on non-local conservation laws and partial Lagrangian approach, local and non-local conservation laws are also studied and, finally, non-classical symmetries are derive

Gradient elasticity solutions of 2D nano-beams

In this study, the exact analytical solutions of a two-dimensional linear homogeneous isotropic nano-beam in gradient elasticity are studied. Four different types of two-dimensional cantilever beams and related boundary conditions are considered. The cases are a cantilever beam under a concentrated force at the end, a cantilever beam under a uniform load, a propped cantilever beam under a uniform load, and a fixed-end beam under a uniform load. The two-dimensional stress gradient fields are investigated and obtained from the analytical solutions of a linear second-order partial differential equation written in terms of the classical and the gradient Airy stress functions. Additionally, the micro-size effects in the displacement components for different loads and support conditions for the two-dimensional cantilever beams by using strain gradient elasticity theory are investigated. Furthermore, for one-dimensional Euler–Bernoulli beam model, the associated stress and strain elasticity solutions are obtained from two-dimensional analytical solutions. The graphical presentations of the exact closed-form solutions are provided and discussed

Symmetry groups of equations of nonlocal elasticity

Tez (Doktora) -- İstanbul Teknik Üniversitesi, Fen Bilimleri Enstitüsü, 1999Thesis (Ph.D.) -- İstanbul Technical University, Institute of Science and Technology, 1999Bu çalışmada amaç, integro-diferansiyel denklem sistemlerine ait Lie nokta simetri gruplarının oluşturulması ve bu simetri gruplarına bağlı olarak bir sınıflandırmaya gidilmesidir. Bir problemin kabul ettiği Lie grupları bakımından sınıflandırılması büyük önem taşımaktadır. Özellikle son yirmi yıl içinde denklem ya da denklem sistemlerine ait Lie simetrilerinin araştırma teknikleri önemli ölçüde geliştirilmiştir. Bununla birlikte integro-diferansiyel denklem sistemleri için genel bir yöntem verilememektedir. Fakat yerel olmayan elastisite ya da visko elastisite problemleri gibi mekanikte karşılaşılan pek çok problemin matematiksel formülasyonu integro diferansiyel denklem yapısaldadırlar. Visko-elastisiteye ait denklemler S.V. Meleshko tarafından incelenmiş ve probleme ait bir sınıflandırmaya gidilmiştir. A.V.Bobylev, S.I. Senashov, V.B. Taranov, V.N.Chetverikov ve A.G. Kudryavtsev integro-diferansiyel denklemler içeren problemler üzerinde çalışmalarda bulunmuş diğer matematikçilerdir. Birinci bölümde Lie gruplarının temel özellikleri hakkında genel bir değerlendirmede bulunulmuş ve integro-diferansiyel denklemlerin simetri grupları hakkında açıklayıcı bilgi verilmiştir. İkinci bölümde, Navier denklemlerine ait Lie grupları belirlenmiş ve Lie gruplarının sınır değer problemlerine bir uygulanması olarak denkleme ait simetriler kullanılarak Boussinesq probleminin çözümü elde edilmiştir. Bilindiği gibi lineer homogen izotrop bir ortamda elasto-statik Navier denklemleri aşağıdaki formda verilebilir: (X, + n.)grad divw+uAH+p/>=p« (1) Burada X,\ı Lam6 sabitlerini, u yer değiştirme vektörünü, p yoğunluğu ve/> kütle kuvvetlerini göstermektedir. (r,(p,z) silindirik koordinatlarda eksenel simetrik bir problem için denklem sistemi eksene paralel ve eksene dik u,w ile gösterilen bileşenlerden oluşan bir denklem sistemine indirgenir. Eğer sonsuz küçük üreticiye ait ifadenin ikinci uzanımı Navier denkleminin homogen olan kısmına uygulanırsa denkleme ait belirleyici denklemlere ulaşılır ve bu denklemlerin çözülmesi ile denkleme ait aşağıdaki simetri grubunun sonsuz küçük üreticilerine ulaşılır: Xx=rdr+zdz, X2=udu+wdw (2) Homogen olmayan denklem sistemi için aşağıdaki şekilde tanımlanan sadece tek bir sonsuz küçük üretici söz konusudur. X = XX-X2 (3) Yukarıda elde edilen simetri grupları kullanılarak Boussinesq problemine ait çözüme ulaşılır: Mx(r2+z2J +Arz(r2+Kz2\l + ^- / 2, v 2\ -M=- " 2V ^ r\»M=A(>% s,t) + 2Vf{r,s,t)}lrds -oo-oo 00 00 (p, (h) = (X + 2ıı)f(x, y, t) + 2\xe{x, y,t)+ j j K(x, y, r, s\{X + 2u)/(r, s, t) + 2ue(r, s, t)}tnts -00-00 00 00 l,(p2,K(x,y,r,s) problemin verileridir. Başlangıç fonksiyonları kullanılarak ve belirleyici denklemler çözülerek ele alınan problem için bir grup sınıflandırması aşağıdaki gibi teşkil edilir: Tablo 2 Simetri Grubu Sınıflandırması burada *ı=ö" Y2=%x{y)dv, Y3=%2(x)dw, X4=8x+dy, Xs=xdx+y8y+tdt Y6=-^tdt+x3(x,y)de+x4(x,y)df+^vdv+^-wdw+(ylk + Sl)dk+(ylm + Pl)dn + (y}h + Hl)dh 77 = -!±td, +(%ı(x,y)-y2e)de +kA{xty)-yj)bf +^-vdv +^-wdw+S,dk + Hxdh+Pxdm Ys=~ (y3 - Y4 K + - (y3 + Y4 )vö, + - (y4 + Y3 V^w + (x3 (*. y) + Y4e& + (X4(x,y)+y4f)df+{y3k + Sl)dk+.(y3m + Pl^m+(y3h + Hl)dh Y9 = vdv +wdw +ede +fdf +kdk +hdh +mdm cp| (m) = 8: + Kİn(y,m + p} ) k * 0, q>2 (m) = e + exp(y2m) y2 * 0 (Pı(m) = a + (/?! +my3)Y3/74 Y4*0 dır ve y,,y2,y3,y4,5'1,//'1,P],51,E,a keyfi sabitlerdir.In this study, the aim is to investigate the Lie point symmetries of integro-differential equations and get a classification. The classification of a problem with respect to Lie symmetries it accepts for different data is very important. The research techniques of Lie symmetries of an equation or a system of equations were improved in last twenty years. However the general methods cannot be given for a system of integro- differential equations. Nevertheless some mathematical formulations in mechanics produce integro- differential equations such as problems of visco-elasticity and nonlocal mechanics. The equations of visco-elasticity are handled and their classification with respect to Lie symmetries are done by S. V. Meleshko. A. V. Bobylev, S. I. Senashov, V. B Taranov,V. N. Chetverikov, A. G. Kudryavtsev are some other mathematicians who have studied problems with integro-differential equations. In part 1, we introduce the general properties of Lie groups and give information about the calculation of the Lie point symmetries of system of integro-differential equations. In part 2, Lie groups of Navier equations are determined and the solution of Boussinesq problem is obtained by using its symmetries as an application of Lie groups to the boundary value problems. As is known the Navier equations for the elasto-statics of linear homogeneous isotropic media is given in the form: (A, + u)grad divM+uA«+p/>=p« (1) where X,\i are Lame constants, u is displacement vector, p is mass density and/; is body force for unit mass. For an axially symmetric problem in cylindrical coordinate system (r, y>t)+e(x>y>t))+ J JK(x,y,r,s)ıı{f(r,s,t)+e(r,s,t))drds -00-00 (11) where v,w,e,f,m,k,h are dependent variables, x,y are independent variables, t is the time variable and q>x,DoktoraPh.D

Invariant solutions and conservation laws to nonconservative FP equation

We generate conservation laws for the one dimensional nonconservative Fokker-Planck (FP) equation, also known as the Kolmogorov forward equation, which describes the time evolution of the probability density function of position and velocity of a particle, and associate these, where possible, with Lie symmetry group generators. We determine the conserved vectors by a composite variational principle and then check if the condition for which symmetries associate with the conservation law is satisfied. As the Fokker-Planck equation is evolution type, no recourse to a Lagrangian formulation is made. Moreover, we obtain invariant solutions for the FP equation via potential symmetries

Solution of Boussinesq Problem using lie Symmetries

Firstly the Lie point symmetries of cylindrically symmetric homogeneous Nawier equations are obtained. Using the symmetries the general class of similarity solutions are found. The subclass that also satisfies the non-homogeneous system of the medium subject to a singular force is determined. Substituting the subclass into the non-homogeneous system, a system of ordinary differential equations is obtained. The solution of the system satisfying the boundary conditions of Bousssinesq problem gives the exact solution

Conservation laws for one-layer shallow water wave systems

The problem of correspondence between symmetries and conservation laws for one-layer shallow water wave systems in the plane flow, axisymmetric flow and dispersive waves is investigated from the composite variational principle of view in the development of the study [N.H. lbragimov, A new conservation theorem, journal of Mathematical Analysis and Applications, 333(1) (2007) 311-328]. This method is devoted to construction of conservation laws of non-Lagrangian systems. Composite principle means that in addition to original variables of a given system, one should introduce a set of adjoint variables in order to obtain a system of Euler-Lagrange equations for some variational functional. After studying Lie point and Lie-Backlund symmetries, we obtain new local and nonlocal conservation laws. Nonlocal conservation laws comprise nonlocal variables defined by the adjoint equations to shallow water wave systems. In particular, we obtain infinite local conservation laws and potential symmetries for the plane flow case

Group properties and conservation laws for nonlocal shallow water wave equation

Symmetry groups, symmetry reductions, optimal system, conservation laws and invariant solutions of the shallow water wave equation with nonlocal term are studied. First, Lie symmetries based on the invariance criterion for nonlocal equations and the solution approach for nonlocal determining equations are found and then the reduced equations and optimal system are obtained. Finally, new conservation laws are generated and some similarity solutions for symmetry reduction forms are discussed