Ordinary Differential Equations
Ordinary Differential Equations MATH 123
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NE 255 LECTURE 2 Relevant Courses Math 123 Ordinary Differential Equations Math 204 Ordinary Differential Equations Math 128A Numerical Analysis solution of ordinary differential equations Math 128B Numerical Analysis evaluations of eigenvalues and eigenvectors solution of simple partial differential equations Math 222A 222B Partial Differential Equations The theory of boundary value and initial value problems Laplace s equation heat equation wave equation Fourier transformations elliptic parabolic and hyperbolic equations Math 228A 228B Numerical Solutions of Partial Differential Equations Runge Kutta and predictorcorrector methods boundary value problems finite differences and finite element solutions of elliptic equations 3 Types of differential and integral equations in reactor analysis In nuclear engineering and reactor analysis we encounter a wide range of mathematical physics equations In today s lecture we will introduce some of them A Ordinary Differential Equations ODEs A1 Linear 15I order ODE s The most general form of an nth order linear ordinary differential equation is n quot71 1 ammy anxy a2Jcy a1Jcy 006 fx 2a 1 Boundary conditions i if y and derivatives are given at one end of domain interval Initial Value Problem IVP ii if y and or derivatives are given at each end of the interval Boundary Value Problem BVP Example y 3ysinx 2a2 IVP if boundary conditions are y01 y 02 BVP if boundary conditions are y0 1 y13 General solution obtained through the use of an integrating factor Example a1 Point kinetics analysis of a nuclear reactor is an IVP linear 1st order ODE dnm pB N TI A nt Expo 2a3 dC z TEint Aicit11N 2a4 Note IX r t Do rnt n s 5 2 fraction of delayed neutrons Xi 2 effective decay constant of the i 11 pre cursor Ci 2 delayed neutron concentration for the ith pre cursor E mean neutron lifetime p k l k reactivity A E k BC s nt0 2 I10 and Cit0 Ci0 for i1 2 N Example 212 Depletion calculation radioactive decay is an IVP linear 15I order ODE dN t TIE Oi Oai Ni 0 Li1Ni1t XiFi il2 l 2a5 BC Nit0Niv0 A2 2nd order ODE s d d 7pxi x qx x 506 2a6 dx dx defined for a s x s 3 d BC d7 y 0 at xoc and x5 BVP X Mixed BC if y 0 Neuman BC if Y 0 Dirichlet BC if y gt 000y const it is reduced to x const If Sx is nonzero at least at a point in a s x s 3 a unique solution exists The homogeneous equation ipxi x qx x MOCMJOC la7 dx dx where px gt 0 fx 2 0 and A eigenvalue and 7 yL a at xoc dx yR c7 at x5 with yL 2 0 and yR 2 0 is known as the STURM LIOUVILLE eigenvalue problem If M yR 0 the BC become a 3 0 The solution has an infinite number of eigenfunctions pi with corresponding REAL and DISTINCT eigenvalues Ki If we number them in the sequence k0 lt1lt2 ltlttw then K0 is the lowest eigenvalue and p0 is the fundamental mode eigenfunction If qx 2 0 then all Ki are positive Example 213 1 dimension 1 group energy time independent neutron diffusion equation 1Dxi x 2a x x SX 2a8 dX dx 1 d 1 7DXi x 2a x x i uzf x x 2a9 dX dX k BC BVP vacuum 50 Equation 2 a 8 is a fixed source problem and Eq 2a9 is an eigenfunction problem B Partial Differential Equations PDEs A partial differential equation is an equation containing an unknown function of two or more variables and its derivatives with respect to those variables 2 6 U Example 6 a 4X 3y is a 2nd order PDE in two variables X y If the partial differential equation is linear in U and all derivatives of U then we say that the PDE is linear 2 2 2 A Ba UC DEEEFUG 2a10 3X2 6X6y ayz 6X 6y Equation 2 a 10 is a 2 1d order PDE in two variables It is linear if A G depend on Xy but not on U Classification of 2 1d order PDEs a ELLIPTIC if B2 4AC lt 0 b HYPERBOLIC if B2 4AC gt 0 c PARABOLIC if B2 4AC 0 B 1 Elliptic PDE s a One group 2 dimensional steady state neutron diffusion equation 1 2a 0 2a11 9x 1 D6X Dgy g 932 k b Stationary heat conduction with a volumetric source Poisson s equation V39kVTqquot39 0 2a12 a T is temperature k is thermal conductivity and q is the volumetric heat generation rate recall that the gradient is VT E 6X 6y az aLer L and that the divergence is V 39 V 6X 6y az BC s Dirichlet if fab const 6f 6f Neumannifn39VfnXinyi Z7 6X 6y az B2 Parabolic PDEs a Transient heat conduction in a slab 6T t 6 k 6 777TXtq Xt 2a13 at 6x pc 6X where p 2 mass density and c 2 specific heat capacity b Time dependent one group neutron diffusion equation 1 3D iIX t UEf Ea X t SX t 2a 14 V at 6X 6X B3 Hyperbolic PDEs a The wave equation i ci0 2a15 where fxt is the displacement field and c is the speed of wave propagation b The basic equations of 1 D unsteady flow of compressible fluids b 1 1 D flow of compressible fluid in a uniform conduit 6p 6p 6V 7 7 70 2a16 6t Yax pax where p is the fluid density and V is the fluid velocity b2 The equation of motion under the in uence of gravity and flow resistance pgval fpgcos6 2a17 at 6x 6X where f 2 flow resistance P 2 pressure gcos e is the component of gravity in the axial direction b3 The energy equation in terms of specific enthalpy VE1 V7PqW 2a18 X plt 6t 6X J at where H 2 specific enthalpy q is the volumetric heat generation rate and J is a conversion factor C IntegroDifferential Equations One group time independent neutron transport equation in slab geometry 6 X 2 1 r I HM2twXH7S fdu wow SXu 2a19 6X 2 1 where the angular neutron flux is a function of space and angle The general enerr39f J J J J J thr J39 39 neutron transport equation is presented in Eq 3a 20 lgw E 2 t 20 E1p7 E 52 t 2 V1p E 2 t V de fd zgaE gt E 2quot 2Ip7E 2 t 2a20 0 4n my M 7 dE d9 E 2 E EQt EX EQt 4 v fr Mr Qr where 1110 E t is the angular neutron flux as a function of three special variables x y 2 one energy variable two angular variables Q gap and one time variable D Integral equations rE fdf f dEKf Er E f E ffE SrE 2a21 v where e TGE E KfE E 223iE gt E 4313 fl R tr39E de39Zt in E optical length 0 e crf E fE fdi 2Frquot E fission source V 4 f fl 6 1633 SE fdiquot 2JrquotE surface source ie S 275 quotrquot incoming current Over the course of the semester we will further define these types of differential integral and integral differential equations specifically applied to neutron transport and diffusion And we will utilize deterministic discrete ordinates finite difference finite elements and stochastic Monte Carlo methods to solve them THE NONCOMMUTATIVE HAHNBANACH THEOREMS WILLIAM ARVESON The Hahn Banach theorem in its simplest form asserts that a bounded linear functional de ned on a subspace of a Banach space can be extended to a linear functional de ned everywhere without increasing its norm There is an order theoretic version of this extension theorem Theorem 01 below that is often more useful in context The purpose of these lecture notes is to discuss the noncommutative generalizations of these two results and their relation to each other We make use of several standard terms such as op erator space operator system n positiue n contractiue completely positiue completely contractiue and refer the reader to Pau02 for de nitions The original proof of the extension theorem for completely positive maps is found in Arv69 I will sketch a somewhat simpli ed proof that is based on the following theorem of M G Krein see page 63 of Nai70 Theorem 01 Krein Let P be a cone in a real topological uector space X such that the interior of P is nonempty Let M be a linear subspace of X and let f M gt R be a linear functional such that fM P Q 0 Then f can be extended to a linear functional f on X satisfying Q 0 A straightforward application of Krein s theorem to the cone P of all positive elements of a C algebra leads to the following extension theorem for complex linear functionals de ned on operator systems Corollary 02 Let S be an operator system in a unital C algebra A and let f S gt C be a complex linear functional such that fS Q 000 S denoting the set of positiue elements of S Then f can be extended to a positiue linear functional on A Aside from a compactness argument that will be described in the lecture the key assertion of the completely positive extension theorem is the follow ing result about extending completely positive maps into matrix algebras Theorem 03 Let S be an operator system in a unital C algebra A and let H be a nite dimensional Hilbert space Then euery completely positiue linear map f S H 3H can be extended to a completely positiue map ofA into Sketch of proof lt suf ces to show that there is a Hilbert space K a rep resentation 7r A a 3K and a linear operator V H a K such that Date 8 September 2003 2 WILLIAM ARVESON s V7rsV s E S that is because the map a E A H V7raV is rather obviously completely positive To that end let n dim H let 51 n be a linearly independent set that spans H and consider the linear functional 1 de ned on MAS by 71 01 f8ij Z lt gtSij j7 igt7 ij1 3H denoting the n x 71 matrix with entries SH 1 is a positive linear functional on MAS because 1 is n positive Corollary 02 implies that there is a positive linear functional 9 on MnA that extends 1 By the GNS construction we obtain a Hilbert space K a representation 7 of A on K and avector 77 E K such that gz ltfrz7777gt for all x E A bit of re ection and a straightforward computation shows that we can realize K as a direct sum of 71 copies of a single Hilbert space K 77 as a column vector with 71 components m E K and a single representation 7139 A a BK such that is given by an n x n operator matrix as follows 9W ltWij77777gt ZltWijmvmgt ii If we let V be the unique linear map of H to K that satis es V56 77k 1 S k S n and choose mij 37 6 S then the above formula implies n 71 n 2 lt lt8ij5j7 igt 9Sij Z lt7r8ijVEj7V5igt Z ltVWSijV5j75igt ijl ij1 ij1 Since the 31739 can be chosen arbitrarily in S the latter formula implies that for xed 3 E S and all ij between 1 and n we have 57350 ltVWSV5j7 50 from which the required assertion is evident D After a preprint of Arv69 was circulated I received a letter from George Elliott outlining the above argument The original proof of Theorem 03 in Arv69 made no use of Krein s theorem but was somewhat more involved The basic idea of both proofs namely that of using duality to convert state ments about matrix valued maps to statements about functionals can be embellished See chapter 6 of Pau02 for a generalization and a more sys tematic organization of the details As I have already pointed out a compactness argument allows one to generalize Theorem 03 to the case of in nite dimensional Hilbert spaces and the latter result is a noncommutative generalization of the Hahn Banach theorem in its order theoretic form namely Krein s Theorem 01 I will sketch this compactness argument in the lecture it is reproduced in Chapter 7 of Pau02 More than ten years went by before anyone looked seriously for a version of the extension theorem for operator spaces Finally in 1981 Gerd Wittstock proved the following result Wit81 THE NONCOMMUTATIVE HAHNeBANACH THEOREMS 3 Theorem 04 Wittstock Let S be a linear subspace of a unital 0 algebra A and let j S gt 3H be a completely bounded linear map Then j can be extended to a linear map j A gt 3H such that Wittstock s proof of Theorem 04 was somewhat involved by way of gen eralizing the Hahn Banach theorem to set valued maps into Soon afterward Paulsen discovered a simple device that allows one to deduce The orem 04 from the extension theorem for completely positive maps Pau82 We now discuss Paulsen s trick then we sketch the proof of Theorem 04 Lemma 05 Paulsen Let S be an operator space in a unital C algebra A and let j Sgt 3H be an operator ualued linear map Consider the operator system S Q M2A de ned by lt bigtst S wee and the operator ualued linear map I S a M2BH de ned by a1 3 7 a1 a 121 lt t b1 39 If gt is completely contractiue then P is completely positiue Sketch of proof Assuming that j is n contractive for some n 1 2 we will show that E MAS a is positive For this we identify MnM2A with the Ci algebra of all 2 x 2 operator matrices of the form if g ABXY 6 AMA This follows because the natural map of MnM2A onto M2MnA is a isomorphism In this identi cation MAS becomes the set of matrices A X Y B 7 where AB belong to MAC and X Y belong to Let us choose a positive element T of this form in Then A and B are positive n x n matrices and Y X so that for every 6 gt 0 AE A 61 and B6 B 61 are invertible positive elements of MAC and we have T17 AEXiAlZ 0 1Y5 A2 0 E XB5 013 2Y1 0322 where Y5 AzlZXBglZ is an element of Since T 61 is positive it follows that 3 315 Z O which in turn is equivalent to g 1 E To show that EAT is positive it suf ces to show that EAT 61 2 0 for every 6 gt 0 After noting that A6 and B6 are scalar matrices we nd 4 WILLIAM ARVESON that 1415712ltjgtnX3127 and moreover WW ate r1 A2 0 1 MY A2 0 7 0 322 MYEV 1 0 Bel239 Now S 1because j is n contractive7 and g 1 is equivalent to the assertion 1 115 2 0 It follows that the right side of the gtnY5 preceding equation is positive7 hence QnT 61 2 as required D While we will not require the fact7 we remark that the converse of Lemma 05 is true as well indeed7 the reader can easily adapt the above argument to show that for every n 17 2 7 j is n contractive iff Q is n positive We now indicate how one deduces Theorem 04 from the extension theo rem for completely positive linear maps via Lemma 05 Proof of Theorem 04 In order to prove Theorem 04 it is enough to show that that every linear map 1 S a 3H that is completely contractive ie7 H gtch 1 has a completely contractive linear extension to a map of A into By Lemma 057 the associated map Q g a M2BH is completely positive By the extension theorem for completely positive maps7 Q can be extended to a completely positive map Q of M2A into Let 1 be the linear map of A into 3H de ned by 8 3lt 13 956A Obviously7 j is an extension of 1 Now Q may be viewed as a completely pos itive unit preserving operator valued map de ned on a C algebra M2A7 and therefore it has a Stinespring decomposition of the form QX V7rXV X e M2A where 7139 is a representation of M2A and V is a linear operator between appropriate Hilbert spaces satisfying VV Q1 1 Hence V is an isometry and Q is a completely contractive map It is now a routine matter to check that 1 must also be completely contractive see p 100 of Pau02 D REFERENCES AIV69 W Arveson Subalgebras of Calgebras Aota Math 12321417224 1969 Nai70 M A Nailnark Nor ed Rings WoltersiNoordho 7 Groningen 1970 Pau82 V Paulsen Completely bounded maps on Calgebras and invariant opertor ranges Proo AMS 862917967 1982 PauOQ V Paulsen Completely bounded maps and Operator Algebras Cambridge7 Cam bridge UK 2002 Wit81 G Wittstock Ein operatorwertiger HahnBanach satz J Funot Anal 4021277