Bazı mühendislik problemlerinin çözümünde Lie simetrilerinin kullanılması
Use of Lie symmetries in the solution of boundary -value problems
- Tez No: 39510
- Danışmanlar: PROF.DR. VURAL ÇİNEMRE
- Tez Türü: Yüksek Lisans
- Konular: İnşaat Mühendisliği, Civil Engineering
- Anahtar Kelimeler: Belirtilmemiş.
- Yıl: 1994
- Dil: Türkçe
- Üniversite: İstanbul Teknik Üniversitesi
- Enstitü: Fen Bilimleri Enstitüsü
- Ana Bilim Dalı: Belirtilmemiş.
- Bilim Dalı: Belirtilmemiş.
- Sayfa Sayısı: 82
Özet
ÖZET Bu çalışmada, Lie grupları teorisinin katı cisim mekaniğinde rastlanan bazı sınır değer problemlerinin çözümüne uygulanması ele alınmıştır. Bu çerçeve içinde adi diferansiyel denklemle ifade edilen lineer olmayan zemine oturmuş elastik kiriş problemine ait dördüncü basamaktan olan lineer olmayan diferansiyel denklemin basamağının Lie grubu simetrileri göz önüne alınarak uygun dönüşümlerle nasıl iki defa düşürülebileceği ortaya konmuştur. Yine lineer olmayan bir elastik ortamın zorlanmış kayma titreşimleri ile ilgili sınır değer probleminin similarite çözümleri Lie grupları simetrileri kullanılarak elde edilmiştir. Klasik Elastisite Teorisinin yerdeğiştirme bileşenleri yardımı ile formüle edilmesi sonucu bulunan Navier denklemlerinin silindirik simetri özelliği gösteren temel çözümü Lie grupları kullanılarak elde edilmiştir.
Özet (Çeviri)
USE OF LIE SYMMETRIES IN THE SOLUTION OF BOUNDARY-VALUE PROBLEMS SUMMARY In this work, the theory of Lie groups is applied to the solution of some boundary value problems of the mechanics of deformable bodies. At the start, fundamentals of the Lie groups have been recollected. As is well known, a group is a set of elements on which a certain binary composition law is defined. This composition law is not an arbitrary one, instead it must satisfy four properties. The law assigns an element to an ordered pair of elements. The operation is closed (that is the designated element is again an element of the set) and is associative. The composition law determines a single neutral element in the set and every element of the set has an inverse defined by the operation. Now, we try to explain what a transformation group is. Let D be a manifold in a Euclidean space. Consider a set of transformations depending on a parameter e : x* = X (x, e) Here the transformation X maps point x of the manifold to an another point x* of the same manifold. X is a homemorphism for every value of e. The set of the parameters e form a group with respect to a certain composition law, say cp (£,5), and if x* = X (x, e), x** = X (x*, e) we have x** = X (x, (p (e,8)) Not every group of transformations is a Lie group, but some of them are. To be qualified as a Lie group the following three additional properties must be satisfied: 1 ) The parameter e should be a continuous one freely selectable from a given interval S of the real axis. 2) The transformations should be diffeomorphisms from D to D VI3) Transformations must be analytical functions of the parameter and the composition law for the parameters must as well be analytical with respect to both of its arguments. By an appropriate transformation, the parameter can be redefined so that the neutral element becomes zero and the composition law becomes the ordinary operation of addition cp (e, 8) = e + 8 The vector - (x, e)lE=0 = Ux) is called the infinitesimal of the Lie group. On the other hand, the operator X=ÎU(x)J- i=1 oXj is called the infinitesimal generator of the group. If XF(x)lRX) = o =0 then the function F(x) = 0 is called an invariant function of the Lie group. The aim of our work is the application of the theory of Lie groups to the boundary value problems. The boundary value problems are frequently given by differential equations. We say that a differential equation admits a Lie group if the application of the proper prolongation of the infinitesimal generator of the Lie group on the differential equation gives zero. XF(x, u, u,..., u)lF=0 = 0 Where k denotes the order of the differential equation F(x,u>u,...,u) =0 On the other hand x and u denote independent and dependent variables respectively. The subscripts u, u>-->ü are inserted to indicate the derivatives of the dependent variable u with respect to the independent VIIvariable x as well as their order. But the subscript under X, X is the symbol of the k th prolongation. Here for an infinitesimal generator k n j 9 m 9 X=I £(x,u) - + I ti«(x,u) ° =1 dx a=1 dua the prolongations are given by X = X+ II Diif-I uP'Di^l 1 i=1a=1 I j=1 Iduf- where u?- = 3xJ Di = -^+S I...I Ui. Ul1...ln 3xi « i, i,,“”r-aufjv (dx'f (dx2f (dxnf x - x =.i nf k k-i i,...in ?-? auP ;a _ r^k-l...a i,...h k, nîLin=Dr'nLin-J ujt,nD^ An equivalent definition of the admittance of a Lie group by a differential equation is the following: If y = f(x) is a solution of the differential equation and x* = X (x, y, e) y* = Y (x, y, e) is the Lie group admitted by the differential equation, the function g(x, u) = 0 obtained by g(x, u) = x* = X [x, f(x), e] y* = Y [x, f(x), e] is another solution of the same equation. For a specific differential equation it is possible to find all Lie groups admitted. For this purpose an overdetermined system of partial differential equations can be written in terms of infinitesimals, Q (x,u), r|a (x,u) which is called the determining equation. As an illustrative first example, the Lie groups admitted by the VIIIeight-parameters (aside from a trivial infinite-parameter group) and it is determined by the following infinitesimals £, = ax2 + bxy + ex + dy + f r| = -ay2 + ayk + gx + hy + k As the second example the non-linear ordinary differential equation d4y dx< + byJ = 0 is considered. The aim is to reduce the order of the equation using the Lie group symmetries. Unfortunately the equation is found to have only a two-parameter symmetry group whose infinitesimal generators are w d x, d r. d Xi = 5-, X2 = x- - 2y - dX dx dy where X^ is easily seen to generate a normal subgroup. In concordance wfth the appropriate order indicated by the theory, first, the invariants associated with X-| are found. Choosing these invariants as coordinates the degree has been reduced by one. Using the invariants of X2 the order of the equation has been reduced to final two. But the resulting equation is so complex and highly non-linear that no practical gain has been obtained toward the solution. The Lie group of the partial differential equation Uxx - A.Utt + m Uxx = 0 has been found to be of five parameters: Ç1 = ax + b ¥ = at + c r| = au + dt + e Here a great deal of effort has been spent for the aim of finding a solution to a decent boundary value problem. But not a single meaningful example could have been constructed. Later the symmetry group of IXUxx - ?lUyy + PU^ UXX+ YUxxy = 0 has been found to be of four parameters. of the Xi = -, X2 = -, X3 = --, X4 = y - dx 9y du du And lastly a boudary value problem, namely“Kelvin problem”linear elasticity theory has been attempted to be solved using implications of the Lie groups admitted. The attempt was quite successful and the well-known exact solution of the problem has been obtained. As is known the Navier equations for the elastostatics of linear homogeneous isotropic media is given in the form (A,+2uJ V div u - |J, curl curl u + pf = 0 where X, u. are Lame constants, u displacement vector, p mass density and f body force for unit mass. For an axially symmetric problem the system of equations reduces to the equations for parallel and perpendicular components shown by w, u respectively. They are Urr + T Ur - J-U + Wrz + ±^L (uZ2-Wrz) + - £- fr = 0 r r2 2(1 -v) ^+2u. Uzr + Wzz + 1 Uz - -^2\L (urz-Wrz + JrUz - 1 Wr) + - ^- fz = 0 r 2(1 -v)* r r I X+2^ where parallel coordi nate is z, and perpen dicular coordinate is r. The problem is formulat ed as follows: The infi nite medium is subjected to a single force F acting along z axis at the origin of coordinates. Then we have iiZ w (r, z) P I ^ u(r,z) r^-4 p Figure 1X+2\l X+2\l X+2\i 2kt The sought - for (so-called) singular solution is required to comply with the following three conditions: 1) It should satisfy the system of homogeneous equations in its classical sense everywhere except at the origin, so that it is a classical solution if the origin is excepted 2) As a whole it should satisfy the non - homogeneous system in the sense of distributions 3) It should approach zero with all of its derivatives whenever the point approaches zero and it should be of the class C°° everywhere except at the origin. To be able to find such a solution we intend to use the direction indicated by the Lie group symmetries of the system. After some long and tedious computations the non-trivial Lie group of the system is found to be of two parameters whose infinitesimal generators are Xi =-, X2 = u- + w- - dz du dW which shows that the solution is in the form u (r, z) = LeTZcp (r, y), w (r, z) = S eVy (r, y) not disregarding the fact that our system is linear. But here y is completely arbitrary and can be obviously considered as a continuous parameter. But then the integral sign replaces the sum sign. Further the third condition imposed on the looked-for solution brings the necessary restriction of a completely imaginary y from which we infer that u and w are (the inverse) Fourier transforms of certain functions Y (r, X), V (r,X,) respectfully: ±-\ e^Y(r,X)dX, w(r,z)=^ J-oo J--a u(r,z)=-L| e^ Y (r, X) dX, w (r, z) = J- | e*zV (r, X) dX Then the system of equations is converted into a system of ordinary differential equations in terms of r depending on X as well but parametrically: XIYrr + 1 Yr - J- Y - iA.Vr + k[-?i2 Y + avr] = 0. r-2 -ir^Yr - rA,2V - iAV - k [-iA,rYr - r Vrr - \XY - Vr] + - E- 5{r) = 0 2îtc§p where 2(1 -v) v being Poisson's ratio. To have a simple enough system which can be separated into two independent equations without raising the order another integral transform proves to be useful, namely, the Laplace transform: 00 00 A (u, X) = I e-n* Y (r, x) dr, B (^ x) = I e-m V (r, x) dr 0 0 Then we have a quite simple system of equations in terms of the variable ji: (|i2 - kX2) A“ + 3pA' = - (1-k) İA. (|iB”+ 2B') (1-k) i^uA' = {X2 - k\i2) B' - ku B + f where f= F 271 (?i+2jl) An independent equation for B can be easily obtained from the system: BII[5n2 + ^2B, ( 3n2 B= fKX2+2n2 ^-x2) (^.x2)2 n(n2-^2) of which the general solution is XIIBM=xj2ln 1+7J_U(i-k) A,2ln((i + y \i2-X2) \i \ (^T ^ 2^3/2 where C(>.) and D(^) are arbitrary functions of their arguments. Then we have A( ^.^L\^y^]._A + V X (\i2-X2) 2\l/2 cW + 2\i? - 3X~\jl K\X (1-k) x3 (\i2-x2f2 (1-k) x t2-r) i\3/2 D (X) \ + E (A.) Considering the third condition C (X), D(X), E{X) = 0 can be determined and by inverse transformation u and w can be found: rz,fO-K) 4K (r2+z2) i3/2 W = _j 1 -fil^l rl_ 2k (r2+z2)1/2 4k (r2+z2); xm
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