Modelli Clamfim Equazione del Calore Lezione ottobre 2014
|
|
- Marcus Lynch
- 5 years ago
- Views:
Transcription
1 CLAMFIM Bologna Modell Clamfm Equazone del Calore Lezone ottobre 2014 professor Danele Rtell danele.rtell@unbo.t 1/24?
2 Convoluton The convoluton of two functons g(t) and f(t) s the functon (g f)(t) = + g(t x)f(x)dx 2/24?
3 Eserczo Se Dmostrare che 0 se t 0 f(t) = e t se t > 0 f f(t) = te t 3/24?
4 Eserczo Se Dmostrare che Eserczo Se a > 0 Dmostrare che 0 se t 0 f(t) = e t se t > 0 f f(t) = te t e ta se t 0 f(t) = 0 se t < 0 f f(t) = te at 3/24?
5 Eserczo Se a > 0, b > 0, a b e ta se t 0 f(t) = 0 se t < 0 g(t) = e tb se t 0 0 se t < 0 Dmostrare che f g(t) = e at e bt b a 4/24?
6 Eserczo Calcolare, se a > 0 la trasformata d Fourer ˆf(s) d 1 t < a f(t) = 0 t > a 5/24?
7 Eserczo Calcolare, se a > 0 la trasformata d Fourer ˆf(s) d 1 t < a f(t) = 0 t > a Calcolare la trasformata d Fourer ˆf(s) d 1 t 2 t < 1 f(t) = 0 t > 1 5/24?
8 artal dfferental equatons A partal dfferental equaton (DE) s a relaton that nvolves partal dervatves of an unknown functon. Let the unknown functon be u, and x, y, z,... be ndependent varables,.e., u = u(x, y, z,...). Often, one of these varables represents the tme. F (x, y, z, u, u x, u y, u z, u xx, u xy,, u xxx, ) = 0. (pde) 6/24?
9 artal dfferental equatons A partal dfferental equaton (DE) s a relaton that nvolves partal dervatves of an unknown functon. Let the unknown functon be u, and x, y, z,... be ndependent varables,.e., u = u(x, y, z,...). Often, one of these varables represents the tme. F (x, y, z, u, u x, u y, u z, u xx, u xy,, u xxx, ) = 0. (pde) We have used the subscrpt notaton for the partal dfferentaton u x = u x, u xy = 2 u x y, We wll always assume that the unknown functon u s suffcently well behaved so that all necessary partal dervatves exst and correspondng mxed partal dervatves are equal 6/24?
10 As n the case of ordnary dfferental equatons, we defne the order of (pde) to be the hghest order partal dervatve appearng n the equaton. Furthermore, we say that (pde) s lnear f F s lnear as a functon of the varables u, u x, u y, u z, u xx,,.e., F s a lnear combnaton of the unknown functon and ts dervatves. 7/24?
11 As n the case of ordnary dfferental equatons, we defne the order of (pde) to be the hghest order partal dervatve appearng n the equaton. Furthermore, we say that (pde) s lnear f F s lnear as a functon of the varables u, u x, u y, u z, u xx,,.e., F s a lnear combnaton of the unknown functon and ts dervatves. The followng are examples of partal DEs: u x + u y = 3u z 2x 2 5z u xx + u y = x 2 frst order lnear second order lnear 7/24?
12 By a soluton of (pde) we mean a contnuous functon u = u(x, y, z, ), wth contnuous partal dervatves, whch, when substtuted n (pde), reduces equaton (pde) to an dentty. For nstance u(x, t) = xe x t solves u t = u xx 2u x 8/24?
13 By a soluton of (pde) we mean a contnuous functon u = u(x, y, z, ), wth contnuous partal dervatves, whch, when substtuted n (pde), reduces equaton (pde) to an dentty. For nstance u(x, t) = xe x t solves u t = u xx 2u x Example For the frst-order partal DE for the unknown u = u(x, y) u x + u y = 0 t s possble to show that u = φ(x y) where φ s any functon havng contnuous frst-order partal dervatves s a soluton. Indeed, snce u x = φ (x y) and u y = φ (x y) t mmedately follows that u x + u y = φ (x y) φ (x y) = 0 8/24?
14 The heat equaton Besdes ths equaton arses n Mathematcal hyscs t s of nterest n Mathematcal Fnance also. We deal wth the partal dfferental equaton u t (x, t) = u xx (x, t) for t > 0, x R (H) u(x, 0) = f(x) for t = 0, x R we assume u and u x fnte as x, t > 0 and both f(x) and u(x, t) are defned for < x < +. 9/24?
15 The heat equaton Besdes ths equaton arses n Mathematcal hyscs t s of nterest n Mathematcal Fnance also. We deal wth the partal dfferental equaton u t (x, t) = u xx (x, t) for t > 0, x R (H) u(x, 0) = f(x) for t = 0, x R we assume u and u x fnte as x, t > 0 and both f(x) and u(x, t) are defned for < x < +. Knowng the Fourer transform of the Gaussan s essental for the treatment we are about to gve. The dea s to take the Fourer transform of both sdes of the heat equaton, wth respect to x thnkng t as a parameter 9/24?
16 Theorem Soluton to (H) s gven by the formula u(x, t) = 1 + ( (x y) 2) exp f(y)dy 4π t 4t 10/24?
17 Theorem Soluton to (H) s gven by the formula u(x, t) = 1 + ( (x y) 2) exp 4π t 4t or, ntroducng the heat kernel H(x, t) = 1 4π t e x2 /4t f(y)dy by the formula u(x, t) = H(x y, t) f(y)dy 10/24?
18 The heat equaton u t (x, t) = u xx (x, t) for t > 0, x R u(x, 0) = f(x) for t = 0, x R (H) Theorem Soluton to (H) s gven by the formula u(x, t) = 1 + ( (x y) 2) exp f(y)dy 4π t 4t 11/24?
19 or, ntroducng the heat kernel H(x, t) = 1 4π t e x2 /4t by the formula u(x, t) = H(x y, t) f(y)dy 12/24?
20 The Fourer transform of the rght hand sde of the equaton (H), u xx (x, t), s Fu xx (s, t) = (2π s) 2 Fu(s, t) = 4π 2 s 2 Fu(s, t) 13/24?
21 The Fourer transform of the rght hand sde of the equaton (H), u xx (x, t), s Fu xx (s, t) = (2π s) 2 Fu(s, t) = 4π 2 s 2 Fu(s, t) For the left hand sde, u t (x, t), we do somethng dfferent. We have 13/24?
22 The Fourer transform of the rght hand sde of the equaton (H), u xx (x, t), s Fu xx (s, t) = (2π s) 2 Fu(s, t) = 4π 2 s 2 Fu(s, t) For the left hand sde, u t (x, t), we do somethng dfferent. We have Fu t (s, t) = + u t (x, t)e 2π sx dx 13/24?
23 The Fourer transform of the rght hand sde of the equaton (H), u xx (x, t), s Fu xx (s, t) = (2π s) 2 Fu(s, t) = 4π 2 s 2 Fu(s, t) For the left hand sde, u t (x, t), we do somethng dfferent. We have Fu t (s, t) = = t + + u t (x, t)e 2π sx dx u(x, t)e 2π sx dx 13/24?
24 The Fourer transform of the rght hand sde of the equaton (H), u xx (x, t), s Fu xx (s, t) = (2π s) 2 Fu(s, t) = 4π 2 s 2 Fu(s, t) For the left hand sde, u t (x, t), we do somethng dfferent. We have Fu t (s, t) = = t + + = Fu(s, t) t u t (x, t)e 2π sx dx u(x, t)e 2π sx dx 13/24?
25 Thus takng the Fourer transform (wth respect to x) of both sdes of the equaton u t (x, t) = u xx (x, t) leads to t Fu(s, t) = 4π2 s 2 Fu(s, t) 14/24?
26 Thus takng the Fourer transform (wth respect to x) of both sdes of the equaton u t (x, t) = u xx (x, t) leads to t Fu(s, t) = 4π2 s 2 Fu(s, t) Ths s a dfferental equaton n t, an ordnary dfferental equaton, despte the partal dervatve symbol, and we can solve t: Fu(s, t) = Fu(s, 0) e 4π2 s 2 t 14/24?
27 Thus takng the Fourer transform (wth respect to x) of both sdes of the equaton u t (x, t) = u xx (x, t) leads to t Fu(s, t) = 4π2 s 2 Fu(s, t) Ths s a dfferental equaton n t, an ordnary dfferental equaton, despte the partal dervatve symbol, and we can solve t: Fu(s, t) = Fu(s, 0) e 4π2 s 2 t What s the ntal condton Fu(s, 0)? Fu(s, 0) = + u(x, 0)e 2π sx dx = + f(x)e 2π sx dx = Ff(s) 14/24?
28 uttng t all together Fu(s, t) = Ff(s) e 4π2 s 2 t 15/24?
29 uttng t all together Fu(s, t) = Ff(s) e 4π2 s 2 t We recognze that the exponental factor on the rght hand sde s the Fourer transform of the heat (Gaussan) hernel H(x, t) = 1 ( ) x 2 exp 4π t 4t snce for α > 0 f(x) = e αx2 ˆf(s) = π α e π2 α s2 15/24?
30 We then have a product of two Fourer transforms Fu(s, t) = Ff(s) FH(s, t) and we nvert ths to obtan a convoluton n the x doman u(x, t) = H(x, t) f(x) or, wrtten out u(x, t) = 1 4π t + exp ( (x y) 2) f(y)dy 4t 16/24?
31 We then have a product of two Fourer transforms Fu(s, t) = Ff(s) FH(s, t) and we nvert ths to obtan a convoluton n the x doman u(x, t) = H(x, t) f(x) or, wrtten out u(x, t) = 1 4π t + ( (x y) 2) exp f(y)dy 4t The functon H(x, t) s also called Green s functon, or fundamental soluton for the heat equaton. 16/24?
32 Ths technque apples also to the problem u t (x, t) = cu xx (x, t) for t > 0, x R u(x, 0) = f(x) for t = 0, x R (H c ) where u and u x fnte as x, t > 0 17/24?
33 Ths technque apples also to the problem u t (x, t) = cu xx (x, t) for t > 0, x R u(x, 0) = f(x) for t = 0, x R (H c ) where u and u x fnte as x, t > 0 whose soluton s 1 + ( (x y) 2) u(x, t) = exp f(y)dy 4π ct 4ct 17/24?
34 Ths s not surprsng snce, f we assume that u(x, t) solves u t = cu xx, f we defne w(x, t) = u(x, t/c) then w s soluton of w t = w xx. In fact 18/24?
35 Ths s not surprsng snce, f we assume that u(x, t) solves u t = cu xx, f we defne w(x, t) = u(x, t/c) then w s soluton of w t = w xx. In fact w t (x, t) = 1 c u t(x, t/c) 18/24?
36 Ths s not surprsng snce, f we assume that u(x, t) solves u t = cu xx, f we defne w(x, t) = u(x, t/c) then w s soluton of w t = w xx. In fact w t (x, t) = 1 c u t(x, t/c) = 1 c c u xx(x, t/c) 18/24?
37 Ths s not surprsng snce, f we assume that u(x, t) solves u t = cu xx, f we defne w(x, t) = u(x, t/c) then w s soluton of w t = w xx. In fact w t (x, t) = 1 c u t(x, t/c) = 1 c c u xx(x, t/c) = u xx (x, t/c) 18/24?
38 Ths s not surprsng snce, f we assume that u(x, t) solves u t = cu xx, f we defne w(x, t) = u(x, t/c) then w s soluton of w t = w xx. In fact w t (x, t) = 1 c u t(x, t/c) = 1 c c u xx(x, t/c) = u xx (x, t/c) = w xx (x, t) 18/24?
39 Ths s not surprsng snce, f we assume that u(x, t) solves u t = cu xx, f we defne w(x, t) = u(x, t/c) then w s soluton of w t = w xx. In fact w t (x, t) = 1 c u t(x, t/c) = 1 c c u xx(x, t/c) = u xx (x, t/c) = w xx (x, t) That s we can assume wthout loss of generalty c = 1 and concern only wth u t = u xx 18/24?
40 Remark The heat kernel H(x, t) s defned for t > 0 only and s an odd functon of x Fgure 1: Graph of the heat kernel for dfferent values of t 19/24?
41 ropertes of H ) H t (x, t) = H xx (x, t) for each t > 0, x R ) ) lm t 0 + H(x, t)dx = 1 for each t > 0 H(x, t) = 0 x 0 + x = 0 20/24?
42 ropertes of H ) H t (x, t) = H xx (x, t) for each t > 0, x R ) ) lm t 0 + H(x, t)dx = 1 for each t > 0 H(x, t) = 0 x 0 + x = 0 ) stems from a drect calculaton. ) follows from the change of varable q = x = dx = 4t dq whch mples 4t + H(x, t) dx = 1 π + e q2 dq = 1 20/24?
43 Eventually to obtan ) when x 0 we change varable puttng s = 1 t and we use Hosptal rule 21/24?
44 Eventually to obtan ) when x 0 we change varable puttng s = 1 t and we use Hosptal rule 1 lm e x2 /4t t 0 + 4π t 21/24?
45 Eventually to obtan ) when x 0 we change varable puttng s = 1 t and we use Hosptal rule 1 s lm e x2 /4t = lm t 0 + 4π t s + 4π e sx 2 /4 21/24?
46 Eventually to obtan ) when x 0 we change varable puttng s = 1 t and we use Hosptal rule 1 s lm e x2 /4t = lm = lm 1 t 0 + 4π t s + 4π e sx 2 /4 s + x 2 sπ e = 0 sx2 /4 21/24?
47 Eventually to obtan ) when x 0 we change varable puttng s = 1 t and we use Hosptal rule 1 s lm e x2 /4t = lm = lm 1 t 0 + 4π t s + 4π e sx 2 /4 s + x 2 sπ e = 0 sx2 /4 whle for x = 0 ) s obvous 21/24?
48 Remark Usng the change of varable y = x+2s t = dy = 2 t ds we can wrte soluton of (H) as u(x, t) = 1 π e s2 f(x + 2s t)ds (H s ) 22/24?
49 Exercse rove, usng (H s ) that u(x, t) = x 2 + 2t solves u t = u xx x R, t > 0 u(x, 0) = x 2 x R (pb1) 23/24?
50 Exercse rove, usng (H s ) that u(x, t) = x 2 + 2t solves u t = u xx x R, t > 0 u(x, 0) = x 2 x R (pb1) From (H s ) we can wrte u(x, t) = 1 π e s2 ( x + 2s t) 2 ds 23/24?
51 Exercse rove, usng (H s ) that u(x, t) = x 2 + 2t solves u t = u xx x R, t > 0 u(x, 0) = x 2 x R (pb1) From (H s ) we can wrte Now u(x, t) = 1 π e s2 ( x + 2s t) 2 ds ( e s2 x + 2s 2 t) ( = e s 2 x 2 + 4xs ) t + 4s 2 t 23/24?
52 Exercse rove, usng (H s ) that u(x, t) = x 2 + 2t solves u t = u xx x R, t > 0 u(x, 0) = x 2 x R (pb1) From (H s ) we can wrte Now u(x, t) = 1 π e s2 ( x + 2s t) 2 ds ( e s2 x + 2s 2 t) ( = e s 2 x 2 + 4xs ) t + 4s 2 t Observe that s 4xs t s an odd functon of s, so that 23/24?
53 u(x, t) = 1 π e s2 ( x 2 + 4s 2 t ) ds 24/24?
54 u(x, t) = 1 π = x2 π e s2 ( x 2 + 4s 2 t ) ds e s2 ds + 4t π s 2 e s2 ds 24/24?
55 u(x, t) = 1 π = x2 π = x 2 + 4t π c e s2 ( x 2 + 4s 2 t ) ds e s2 ds + 4t π s 2 e s2 ds 24/24?
56 u(x, t) = 1 π = x2 π = x 2 + 4t π c e s2 ( x 2 + 4s 2 t ) ds e s2 ds + 4t π s 2 e s2 ds To fnd the value of the constant c we can mpose that the found functons solves (pb1) 24/24?
57 u(x, t) = 1 π = x2 π = x 2 + 4t π c e s2 ( x 2 + 4s 2 t ) ds e s2 ds + 4t π s 2 e s2 ds To fnd the value of the constant c we can mpose that the found functons solves (pb1) It s u t = 4 π c, u xx = 2 24/24?
58 u(x, t) = 1 π = x2 π = x 2 + 4t π c e s2 ( x 2 + 4s 2 t ) ds e s2 ds + 4t π s 2 e s2 ds To fnd the value of the constant c we can mpose that the found functons solves (pb1) It s u t = 4 π c, u xx = 2 then c = π 2 = s 2 e s2 ds 24/24?
59 u(x, t) = 1 π = x2 π = x 2 + 4t π c e s2 ( x 2 + 4s 2 t ) ds e s2 ds + 4t π 24/24? s 2 e s2 ds To fnd the value of the constant c we can mpose that the found functons solves (pb1) It s u t = 4 π c, u xx = 2 then c = π 2 = s 2 e s2 ds Concluson: soluton to (pb1) s u(x, t) = x 2 + 4t π π 2 = x2 + 2t
Modelli Clamfim Equazioni differenziali 7 ottobre 2013
CLAMFIM Bologna Modell 1 @ Clamfm Equazon dfferenzal 7 ottobre 2013 professor Danele Rtell danele.rtell@unbo.t 1/18? Ordnary Dfferental Equatons A dfferental equaton s an equaton that defnes a relatonshp
More informationModelli Clamfim Equazioni differenziali 22 settembre 2016
CLAMFIM Bologna Modell 1 @ Clamfm Equazon dfferenzal 22 settembre 2016 professor Danele Rtell danele.rtell@unbo.t 1/22? Ordnary Dfferental Equatons A dfferental equaton s an equaton that defnes a relatonshp
More informationCALCULUS CLASSROOM CAPSULES
CALCULUS CLASSROOM CAPSULES SESSION S86 Dr. Sham Alfred Rartan Valley Communty College salfred@rartanval.edu 38th AMATYC Annual Conference Jacksonvlle, Florda November 8-, 202 2 Calculus Classroom Capsules
More informationGroup Analysis of Ordinary Differential Equations of the Order n>2
Symmetry n Nonlnear Mathematcal Physcs 997, V., 64 7. Group Analyss of Ordnary Dfferental Equatons of the Order n> L.M. BERKOVICH and S.Y. POPOV Samara State Unversty, 4430, Samara, Russa E-mal: berk@nfo.ssu.samara.ru
More informationThe Feynman path integral
The Feynman path ntegral Aprl 3, 205 Hesenberg and Schrödnger pctures The Schrödnger wave functon places the tme dependence of a physcal system n the state, ψ, t, where the state s a vector n Hlbert space
More informationCalculus of Variations Basics
Chapter 1 Calculus of Varatons Bascs 1.1 Varaton of a General Functonal In ths chapter, we derve the general formula for the varaton of a functonal of the form J [y 1,y 2,,y n ] F x,y 1,y 2,,y n,y 1,y
More informationAPPENDIX A Some Linear Algebra
APPENDIX A Some Lnear Algebra The collecton of m, n matrces A.1 Matrces a 1,1,..., a 1,n A = a m,1,..., a m,n wth real elements a,j s denoted by R m,n. If n = 1 then A s called a column vector. Smlarly,
More informationNUMERICAL DIFFERENTIATION
NUMERICAL DIFFERENTIATION 1 Introducton Dfferentaton s a method to compute the rate at whch a dependent output y changes wth respect to the change n the ndependent nput x. Ths rate of change s called the
More informationn α j x j = 0 j=1 has a nontrivial solution. Here A is the n k matrix whose jth column is the vector for all t j=0
MODULE 2 Topcs: Lnear ndependence, bass and dmenson We have seen that f n a set of vectors one vector s a lnear combnaton of the remanng vectors n the set then the span of the set s unchanged f that vector
More informationLecture 21: Numerical methods for pricing American type derivatives
Lecture 21: Numercal methods for prcng Amercan type dervatves Xaoguang Wang STAT 598W Aprl 10th, 2014 (STAT 598W) Lecture 21 1 / 26 Outlne 1 Fnte Dfference Method Explct Method Penalty Method (STAT 598W)
More informationInner Product. Euclidean Space. Orthonormal Basis. Orthogonal
Inner Product Defnton 1 () A Eucldean space s a fnte-dmensonal vector space over the reals R, wth an nner product,. Defnton 2 (Inner Product) An nner product, on a real vector space X s a symmetrc, blnear,
More informationModelli Clamfim Integrali Multipli 7 ottobre 2015
CLAMFIM Bologna Modell 1 @ Clamfm Integral Multpl 7 ottobre 2015 professor Danele Rtell danele.rtell@unbo.t 1/30? roduct of σ-algebras Let (Ω 1, A 1, µ 1 ), (Ω 2, A 2, µ 2 ) two measure spaces. ut Ω :=
More information% & 5.3 PRACTICAL APPLICATIONS. Given system, (49) , determine the Boolean Function, , in such a way that we always have expression: " Y1 = Y2
5.3 PRACTICAL APPLICATIONS st EXAMPLE: Gven system, (49) & K K Y XvX 3 ( 2 & X ), determne the Boolean Functon, Y2 X2 & X 3 v X " X3 (X2,X)", n such a way that we always have expresson: " Y Y2 " (50).
More informationIntroduction to Vapor/Liquid Equilibrium, part 2. Raoult s Law:
CE304, Sprng 2004 Lecture 4 Introducton to Vapor/Lqud Equlbrum, part 2 Raoult s Law: The smplest model that allows us do VLE calculatons s obtaned when we assume that the vapor phase s an deal gas, and
More informationAnother converse of Jensen s inequality
Another converse of Jensen s nequalty Slavko Smc Abstract. We gve the best possble global bounds for a form of dscrete Jensen s nequalty. By some examples ts frutfulness s shown. 1. Introducton Throughout
More informationLectures - Week 4 Matrix norms, Conditioning, Vector Spaces, Linear Independence, Spanning sets and Basis, Null space and Range of a Matrix
Lectures - Week 4 Matrx norms, Condtonng, Vector Spaces, Lnear Independence, Spannng sets and Bass, Null space and Range of a Matrx Matrx Norms Now we turn to assocatng a number to each matrx. We could
More informationAffine and Riemannian Connections
Affne and Remannan Connectons Semnar Remannan Geometry Summer Term 2015 Prof Dr Anna Wenhard and Dr Gye-Seon Lee Jakob Ullmann Notaton: X(M) space of smooth vector felds on M D(M) space of smooth functons
More information9 Characteristic classes
THEODORE VORONOV DIFFERENTIAL GEOMETRY. Sprng 2009 [under constructon] 9 Characterstc classes 9.1 The frst Chern class of a lne bundle Consder a complex vector bundle E B of rank p. We shall construct
More informationA new Approach for Solving Linear Ordinary Differential Equations
, ISSN 974-57X (Onlne), ISSN 974-5718 (Prnt), Vol. ; Issue No. 1; Year 14, Copyrght 13-14 by CESER PUBLICATIONS A new Approach for Solvng Lnear Ordnary Dfferental Equatons Fawz Abdelwahd Department of
More informationDigital Signal Processing
Dgtal Sgnal Processng Dscrete-tme System Analyss Manar Mohasen Offce: F8 Emal: manar.subh@ut.ac.r School of IT Engneerng Revew of Precedent Class Contnuous Sgnal The value of the sgnal s avalable over
More informationErrors for Linear Systems
Errors for Lnear Systems When we solve a lnear system Ax b we often do not know A and b exactly, but have only approxmatons  and ˆb avalable. Then the best thng we can do s to solve ˆx ˆb exactly whch
More informationProf. Dr. I. Nasser Phys 630, T Aug-15 One_dimensional_Ising_Model
EXACT OE-DIMESIOAL ISIG MODEL The one-dmensonal Isng model conssts of a chan of spns, each spn nteractng only wth ts two nearest neghbors. The smple Isng problem n one dmenson can be solved drectly n several
More informationModule 3 LOSSY IMAGE COMPRESSION SYSTEMS. Version 2 ECE IIT, Kharagpur
Module 3 LOSSY IMAGE COMPRESSION SYSTEMS Verson ECE IIT, Kharagpur Lesson 6 Theory of Quantzaton Verson ECE IIT, Kharagpur Instructonal Objectves At the end of ths lesson, the students should be able to:
More informationBernoulli Numbers and Polynomials
Bernoull Numbers and Polynomals T. Muthukumar tmk@tk.ac.n 17 Jun 2014 The sum of frst n natural numbers 1, 2, 3,..., n s n n(n + 1 S 1 (n := m = = n2 2 2 + n 2. Ths formula can be derved by notng that
More informationLecture 10 Support Vector Machines II
Lecture 10 Support Vector Machnes II 22 February 2016 Taylor B. Arnold Yale Statstcs STAT 365/665 1/28 Notes: Problem 3 s posted and due ths upcomng Frday There was an early bug n the fake-test data; fxed
More informationON A DETERMINATION OF THE INITIAL FUNCTIONS FROM THE OBSERVED VALUES OF THE BOUNDARY FUNCTIONS FOR THE SECOND-ORDER HYPERBOLIC EQUATION
Advanced Mathematcal Models & Applcatons Vol.3, No.3, 2018, pp.215-222 ON A DETERMINATION OF THE INITIAL FUNCTIONS FROM THE OBSERVED VALUES OF THE BOUNDARY FUNCTIONS FOR THE SECOND-ORDER HYPERBOLIC EUATION
More informationNumerical Solution of Ordinary Differential Equations
Numercal Methods (CENG 00) CHAPTER-VI Numercal Soluton of Ordnar Dfferental Equatons 6 Introducton Dfferental equatons are equatons composed of an unknown functon and ts dervatves The followng are examples
More informationTHE VIBRATIONS OF MOLECULES II THE CARBON DIOXIDE MOLECULE Student Instructions
THE VIBRATIONS OF MOLECULES II THE CARBON DIOXIDE MOLECULE Student Instructons by George Hardgrove Chemstry Department St. Olaf College Northfeld, MN 55057 hardgrov@lars.acc.stolaf.edu Copyrght George
More informationTransfer Functions. Convenient representation of a linear, dynamic model. A transfer function (TF) relates one input and one output: ( ) system
Transfer Functons Convenent representaton of a lnear, dynamc model. A transfer functon (TF) relates one nput and one output: x t X s y t system Y s The followng termnology s used: x y nput output forcng
More informationImplicit Integration Henyey Method
Implct Integraton Henyey Method In realstc stellar evoluton codes nstead of a drect ntegraton usng for example the Runge-Kutta method one employs an teratve mplct technque. Ths s because the structure
More informationUsing T.O.M to Estimate Parameter of distributions that have not Single Exponential Family
IOSR Journal of Mathematcs IOSR-JM) ISSN: 2278-5728. Volume 3, Issue 3 Sep-Oct. 202), PP 44-48 www.osrjournals.org Usng T.O.M to Estmate Parameter of dstrbutons that have not Sngle Exponental Famly Jubran
More informationCHAPTER 5 NUMERICAL EVALUATION OF DYNAMIC RESPONSE
CHAPTER 5 NUMERICAL EVALUATION OF DYNAMIC RESPONSE Analytcal soluton s usually not possble when exctaton vares arbtrarly wth tme or f the system s nonlnear. Such problems can be solved by numercal tmesteppng
More informationTHE CHINESE REMAINDER THEOREM. We should thank the Chinese for their wonderful remainder theorem. Glenn Stevens
THE CHINESE REMAINDER THEOREM KEITH CONRAD We should thank the Chnese for ther wonderful remander theorem. Glenn Stevens 1. Introducton The Chnese remander theorem says we can unquely solve any par of
More informationC/CS/Phy191 Problem Set 3 Solutions Out: Oct 1, 2008., where ( 00. ), so the overall state of the system is ) ( ( ( ( 00 ± 11 ), Φ ± = 1
C/CS/Phy9 Problem Set 3 Solutons Out: Oct, 8 Suppose you have two qubts n some arbtrary entangled state ψ You apply the teleportaton protocol to each of the qubts separately What s the resultng state obtaned
More informationMath1110 (Spring 2009) Prelim 3 - Solutions
Math 1110 (Sprng 2009) Solutons to Prelm 3 (04/21/2009) 1 Queston 1. (16 ponts) Short answer. Math1110 (Sprng 2009) Prelm 3 - Solutons x a 1 (a) (4 ponts) Please evaluate lm, where a and b are postve numbers.
More informationChapter 4 The Wave Equation
Chapter 4 The Wave Equaton Another classcal example of a hyperbolc PDE s a wave equaton. The wave equaton s a second-order lnear hyperbolc PDE that descrbes the propagaton of a varety of waves, such as
More informationChapter 12. Ordinary Differential Equation Boundary Value (BV) Problems
Chapter. Ordnar Dfferental Equaton Boundar Value (BV) Problems In ths chapter we wll learn how to solve ODE boundar value problem. BV ODE s usuall gven wth x beng the ndependent space varable. p( x) q(
More information3 Basic boundary value problems for analytic function in the upper half plane
3 Basc boundary value problems for analytc functon n the upper half plane 3. Posson representaton formulas for the half plane Let f be an analytc functon of z throughout the half plane Imz > 0, contnuous
More informationprinceton univ. F 17 cos 521: Advanced Algorithm Design Lecture 7: LP Duality Lecturer: Matt Weinberg
prnceton unv. F 17 cos 521: Advanced Algorthm Desgn Lecture 7: LP Dualty Lecturer: Matt Wenberg Scrbe: LP Dualty s an extremely useful tool for analyzng structural propertes of lnear programs. Whle there
More information2 Finite difference basics
Numersche Methoden 1, WS 11/12 B.J.P. Kaus 2 Fnte dfference bascs Consder the one- The bascs of the fnte dfference method are best understood wth an example. dmensonal transent heat conducton equaton T
More informationDifference Equations
Dfference Equatons c Jan Vrbk 1 Bascs Suppose a sequence of numbers, say a 0,a 1,a,a 3,... s defned by a certan general relatonshp between, say, three consecutve values of the sequence, e.g. a + +3a +1
More informationSolution of Linear System of Equations and Matrix Inversion Gauss Seidel Iteration Method
Soluton of Lnear System of Equatons and Matr Inverson Gauss Sedel Iteraton Method It s another well-known teratve method for solvng a system of lnear equatons of the form a + a22 + + ann = b a2 + a222
More informationOpen Systems: Chemical Potential and Partial Molar Quantities Chemical Potential
Open Systems: Chemcal Potental and Partal Molar Quanttes Chemcal Potental For closed systems, we have derved the followng relatonshps: du = TdS pdv dh = TdS + Vdp da = SdT pdv dg = VdP SdT For open systems,
More informationConvexity preserving interpolation by splines of arbitrary degree
Computer Scence Journal of Moldova, vol.18, no.1(52), 2010 Convexty preservng nterpolaton by splnes of arbtrary degree Igor Verlan Abstract In the present paper an algorthm of C 2 nterpolaton of dscrete
More informationBezier curves. Michael S. Floater. August 25, These notes provide an introduction to Bezier curves. i=0
Bezer curves Mchael S. Floater August 25, 211 These notes provde an ntroducton to Bezer curves. 1 Bernsten polynomals Recall that a real polynomal of a real varable x R, wth degree n, s a functon of the
More informationThe Multiple Classical Linear Regression Model (CLRM): Specification and Assumptions. 1. Introduction
ECONOMICS 5* -- NOTE (Summary) ECON 5* -- NOTE The Multple Classcal Lnear Regresson Model (CLRM): Specfcaton and Assumptons. Introducton CLRM stands for the Classcal Lnear Regresson Model. The CLRM s also
More information3.1 Expectation of Functions of Several Random Variables. )' be a k-dimensional discrete or continuous random vector, with joint PMF p (, E X E X1 E X
Statstcs 1: Probablty Theory II 37 3 EPECTATION OF SEVERAL RANDOM VARIABLES As n Probablty Theory I, the nterest n most stuatons les not on the actual dstrbuton of a random vector, but rather on a number
More informationMath 217 Fall 2013 Homework 2 Solutions
Math 17 Fall 013 Homework Solutons Due Thursday Sept. 6, 013 5pm Ths homework conssts of 6 problems of 5 ponts each. The total s 30. You need to fully justfy your answer prove that your functon ndeed has
More informationDifferentiating Gaussian Processes
Dfferentatng Gaussan Processes Andrew McHutchon Aprl 17, 013 1 Frst Order Dervatve of the Posteror Mean The posteror mean of a GP s gven by, f = x, X KX, X 1 y x, X α 1 Only the x, X term depends on the
More informationSELECTED PROOFS. DeMorgan s formulas: The first one is clear from Venn diagram, or the following truth table:
SELECTED PROOFS DeMorgan s formulas: The frst one s clear from Venn dagram, or the followng truth table: A B A B A B Ā B Ā B T T T F F F F T F T F F T F F T T F T F F F F F T T T T The second one can be
More informationSalmon: Lectures on partial differential equations. Consider the general linear, second-order PDE in the form. ,x 2
Salmon: Lectures on partal dfferental equatons 5. Classfcaton of second-order equatons There are general methods for classfyng hgher-order partal dfferental equatons. One s very general (applyng even to
More informationChapter 5. Solution of System of Linear Equations. Module No. 6. Solution of Inconsistent and Ill Conditioned Systems
Numercal Analyss by Dr. Anta Pal Assstant Professor Department of Mathematcs Natonal Insttute of Technology Durgapur Durgapur-713209 emal: anta.bue@gmal.com 1 . Chapter 5 Soluton of System of Lnear Equatons
More informationALGORITHM FOR THE CALCULATION OF THE TWO VARIABLES CUBIC SPLINE FUNCTION
ANALELE ŞTIINŢIFICE ALE UNIVERSITĂŢII AL.I. CUZA DIN IAŞI (S.N.) MATEMATICĂ, Tomul LIX, 013, f.1 DOI: 10.478/v10157-01-00-y ALGORITHM FOR THE CALCULATION OF THE TWO VARIABLES CUBIC SPLINE FUNCTION BY ION
More informationFACTORIZATION IN KRULL MONOIDS WITH INFINITE CLASS GROUP
C O L L O Q U I U M M A T H E M A T I C U M VOL. 80 1999 NO. 1 FACTORIZATION IN KRULL MONOIDS WITH INFINITE CLASS GROUP BY FLORIAN K A I N R A T H (GRAZ) Abstract. Let H be a Krull monod wth nfnte class
More informationEEE 241: Linear Systems
EEE : Lnear Systems Summary #: Backpropagaton BACKPROPAGATION The perceptron rule as well as the Wdrow Hoff learnng were desgned to tran sngle layer networks. They suffer from the same dsadvantage: they
More informationPHYS 705: Classical Mechanics. Hamilton-Jacobi Equation
1 PHYS 705: Classcal Mechancs Hamlton-Jacob Equaton Hamlton-Jacob Equaton There s also a very elegant relaton between the Hamltonan Formulaton of Mechancs and Quantum Mechancs. To do that, we need to derve
More informationPHYS 705: Classical Mechanics. Canonical Transformation II
1 PHYS 705: Classcal Mechancs Canoncal Transformaton II Example: Harmonc Oscllator f ( x) x m 0 x U( x) x mx x LT U m Defne or L p p mx x x m mx x H px L px p m p x m m H p 1 x m p m 1 m H x p m x m m
More informationELASTIC WAVE PROPAGATION IN A CONTINUOUS MEDIUM
ELASTIC WAVE PROPAGATION IN A CONTINUOUS MEDIUM An elastc wave s a deformaton of the body that travels throughout the body n all drectons. We can examne the deformaton over a perod of tme by fxng our look
More informationChapter 9: Statistical Inference and the Relationship between Two Variables
Chapter 9: Statstcal Inference and the Relatonshp between Two Varables Key Words The Regresson Model The Sample Regresson Equaton The Pearson Correlaton Coeffcent Learnng Outcomes After studyng ths chapter,
More informationBézier curves. Michael S. Floater. September 10, These notes provide an introduction to Bézier curves. i=0
Bézer curves Mchael S. Floater September 1, 215 These notes provde an ntroducton to Bézer curves. 1 Bernsten polynomals Recall that a real polynomal of a real varable x R, wth degree n, s a functon of
More informationLINEAR REGRESSION ANALYSIS. MODULE IX Lecture Multicollinearity
LINEAR REGRESSION ANALYSIS MODULE IX Lecture - 30 Multcollnearty Dr. Shalabh Department of Mathematcs and Statstcs Indan Insttute of Technology Kanpur 2 Remedes for multcollnearty Varous technques have
More informationFundamental loop-current method using virtual voltage sources technique for special cases
Fundamental loop-current method usng vrtual voltage sources technque for specal cases George E. Chatzaraks, 1 Marna D. Tortorel 1 and Anastasos D. Tzolas 1 Electrcal and Electroncs Engneerng Departments,
More informationSolutions to exam in SF1811 Optimization, Jan 14, 2015
Solutons to exam n SF8 Optmzaton, Jan 4, 25 3 3 O------O -4 \ / \ / The network: \/ where all lnks go from left to rght. /\ / \ / \ 6 O------O -5 2 4.(a) Let x = ( x 3, x 4, x 23, x 24 ) T, where the varable
More informationLinear Approximation with Regularization and Moving Least Squares
Lnear Approxmaton wth Regularzaton and Movng Least Squares Igor Grešovn May 007 Revson 4.6 (Revson : March 004). 5 4 3 0.5 3 3.5 4 Contents: Lnear Fttng...4. Weghted Least Squares n Functon Approxmaton...
More informationAssortment Optimization under MNL
Assortment Optmzaton under MNL Haotan Song Aprl 30, 2017 1 Introducton The assortment optmzaton problem ams to fnd the revenue-maxmzng assortment of products to offer when the prces of products are fxed.
More informationAffine transformations and convexity
Affne transformatons and convexty The purpose of ths document s to prove some basc propertes of affne transformatons nvolvng convex sets. Here are a few onlne references for background nformaton: http://math.ucr.edu/
More informationLecture 6/7 (February 10/12, 2014) DIRAC EQUATION. The non-relativistic Schrödinger equation was obtained by noting that the Hamiltonian 2
P470 Lecture 6/7 (February 10/1, 014) DIRAC EQUATION The non-relatvstc Schrödnger equaton was obtaned by notng that the Hamltonan H = P (1) m can be transformed nto an operator form wth the substtutons
More informationComplete subgraphs in multipartite graphs
Complete subgraphs n multpartte graphs FLORIAN PFENDER Unverstät Rostock, Insttut für Mathematk D-18057 Rostock, Germany Floran.Pfender@un-rostock.de Abstract Turán s Theorem states that every graph G
More informationOn the set of natural numbers
On the set of natural numbers by Jalton C. Ferrera Copyrght 2001 Jalton da Costa Ferrera Introducton The natural numbers have been understood as fnte numbers, ths wor tres to show that the natural numbers
More informationNew Method for Solving Poisson Equation. on Irregular Domains
Appled Mathematcal Scences Vol. 6 01 no. 8 369 380 New Method for Solvng Posson Equaton on Irregular Domans J. Izadan and N. Karamooz Department of Mathematcs Facult of Scences Mashhad BranchIslamc Azad
More informationMAE140 - Linear Circuits - Winter 16 Final, March 16, 2016
ME140 - Lnear rcuts - Wnter 16 Fnal, March 16, 2016 Instructons () The exam s open book. You may use your class notes and textbook. You may use a hand calculator wth no communcaton capabltes. () You have
More informationarxiv: v1 [math.ho] 18 May 2008
Recurrence Formulas for Fbonacc Sums Adlson J. V. Brandão, João L. Martns 2 arxv:0805.2707v [math.ho] 8 May 2008 Abstract. In ths artcle we present a new recurrence formula for a fnte sum nvolvng the Fbonacc
More information2.3 Nilpotent endomorphisms
s a block dagonal matrx, wth A Mat dm U (C) In fact, we can assume that B = B 1 B k, wth B an ordered bass of U, and that A = [f U ] B, where f U : U U s the restrcton of f to U 40 23 Nlpotent endomorphsms
More information8.4 COMPLEX VECTOR SPACES AND INNER PRODUCTS
SECTION 8.4 COMPLEX VECTOR SPACES AND INNER PRODUCTS 493 8.4 COMPLEX VECTOR SPACES AND INNER PRODUCTS All the vector spaces you have studed thus far n the text are real vector spaces because the scalars
More informationFixed points of IA-endomorphisms of a free metabelian Lie algebra
Proc. Indan Acad. Sc. (Math. Sc.) Vol. 121, No. 4, November 2011, pp. 405 416. c Indan Academy of Scences Fxed ponts of IA-endomorphsms of a free metabelan Le algebra NAIME EKICI 1 and DEMET PARLAK SÖNMEZ
More informationCanonical transformations
Canoncal transformatons November 23, 2014 Recall that we have defned a symplectc transformaton to be any lnear transformaton M A B leavng the symplectc form nvarant, Ω AB M A CM B DΩ CD Coordnate transformatons,
More informationMATH 5630: Discrete Time-Space Model Hung Phan, UMass Lowell March 1, 2018
MATH 5630: Dscrete Tme-Space Model Hung Phan, UMass Lowell March, 08 Newton s Law of Coolng Consder the coolng of a well strred coffee so that the temperature does not depend on space Newton s law of collng
More informationThe Order Relation and Trace Inequalities for. Hermitian Operators
Internatonal Mathematcal Forum, Vol 3, 08, no, 507-57 HIKARI Ltd, wwwm-hkarcom https://doorg/0988/mf088055 The Order Relaton and Trace Inequaltes for Hermtan Operators Y Huang School of Informaton Scence
More informationLinear, affine, and convex sets and hulls In the sequel, unless otherwise specified, X will denote a real vector space.
Lnear, affne, and convex sets and hulls In the sequel, unless otherwse specfed, X wll denote a real vector space. Lnes and segments. Gven two ponts x, y X, we defne xy = {x + t(y x) : t R} = {(1 t)x +
More informationHidden Markov Models & The Multivariate Gaussian (10/26/04)
CS281A/Stat241A: Statstcal Learnng Theory Hdden Markov Models & The Multvarate Gaussan (10/26/04) Lecturer: Mchael I. Jordan Scrbes: Jonathan W. Hu 1 Hdden Markov Models As a bref revew, hdden Markov models
More informationHomework & Solution. Contributors. Prof. Lee, Hyun Min. Particle Physics Winter School. Park, Ye
Homework & Soluton Prof. Lee, Hyun Mn Contrbutors Park, Ye J(yej.park@yonse.ac.kr) Lee, Sung Mook(smlngsm0919@gmal.com) Cheong, Dhong Yeon(dhongyeoncheong@gmal.com) Ban, Ka Young(ban94gy@yonse.ac.kr) Ro,
More informationOne-sided finite-difference approximations suitable for use with Richardson extrapolation
Journal of Computatonal Physcs 219 (2006) 13 20 Short note One-sded fnte-dfference approxmatons sutable for use wth Rchardson extrapolaton Kumar Rahul, S.N. Bhattacharyya * Department of Mechancal Engneerng,
More informationThe KMO Method for Solving Non-homogenous, m th Order Differential Equations
The KMO Method for Solvng Non-homogenous, m th Order Dfferental Equatons Davd Krohn Danel Marño-Johnson John Paul Ouyang March 14, 2013 Abstract Ths paper shows a smple tabular procedure for fndng the
More informationMore metrics on cartesian products
More metrcs on cartesan products If (X, d ) are metrc spaces for 1 n, then n Secton II4 of the lecture notes we defned three metrcs on X whose underlyng topologes are the product topology The purpose of
More informationON THE JACOBIAN CONJECTURE
v v v Far East Journal of Mathematcal Scences (FJMS) 17 Pushpa Publshng House, Allahabad, Inda http://www.pphm.com http://dx.do.org/1.17654/ms1111565 Volume 11, Number 11, 17, Pages 565-574 ISSN: 97-871
More informationStrong Markov property: Same assertion holds for stopping times τ.
Brownan moton Let X ={X t : t R + } be a real-valued stochastc process: a famlty of real random varables all defned on the same probablty space. Defne F t = nformaton avalable by observng the process up
More informationLecture 12: Discrete Laplacian
Lecture 12: Dscrete Laplacan Scrbe: Tanye Lu Our goal s to come up wth a dscrete verson of Laplacan operator for trangulated surfaces, so that we can use t n practce to solve related problems We are mostly
More information1 (1 + ( )) = 1 8 ( ) = (c) Carrying out the Taylor expansion, in this case, the series truncates at second order:
68A Solutons to Exercses March 05 (a) Usng a Taylor expanson, and notng that n 0 for all n >, ( + ) ( + ( ) + ) We can t nvert / because there s no Taylor expanson around 0 Lets try to calculate the nverse
More informationAppendix B. The Finite Difference Scheme
140 APPENDIXES Appendx B. The Fnte Dfference Scheme In ths appendx we present numercal technques whch are used to approxmate solutons of system 3.1 3.3. A comprehensve treatment of theoretcal and mplementaton
More informationGlobal Sensitivity. Tuesday 20 th February, 2018
Global Senstvty Tuesday 2 th February, 28 ) Local Senstvty Most senstvty analyses [] are based on local estmates of senstvty, typcally by expandng the response n a Taylor seres about some specfc values
More informationA New Refinement of Jacobi Method for Solution of Linear System Equations AX=b
Int J Contemp Math Scences, Vol 3, 28, no 17, 819-827 A New Refnement of Jacob Method for Soluton of Lnear System Equatons AX=b F Naem Dafchah Department of Mathematcs, Faculty of Scences Unversty of Gulan,
More information5 The Laplace Equation in a convex polygon
5 Te Laplace Equaton n a convex polygon Te most mportant ellptc PDEs are te Laplace, te modfed Helmoltz and te Helmoltz equatons. Te Laplace equaton s u xx + u yy =. (5.) Te real and magnary parts of an
More informationLecture 2: Numerical Methods for Differentiations and Integrations
Numercal Smulaton of Space Plasmas (I [AP-4036] Lecture 2 by Lng-Hsao Lyu March, 2018 Lecture 2: Numercal Methods for Dfferentatons and Integratons As we have dscussed n Lecture 1 that numercal smulaton
More information10-801: Advanced Optimization and Randomized Methods Lecture 2: Convex functions (Jan 15, 2014)
0-80: Advanced Optmzaton and Randomzed Methods Lecture : Convex functons (Jan 5, 04) Lecturer: Suvrt Sra Addr: Carnege Mellon Unversty, Sprng 04 Scrbes: Avnava Dubey, Ahmed Hefny Dsclamer: These notes
More informationPolynomial Regression Models
LINEAR REGRESSION ANALYSIS MODULE XII Lecture - 6 Polynomal Regresson Models Dr. Shalabh Department of Mathematcs and Statstcs Indan Insttute of Technology Kanpur Test of sgnfcance To test the sgnfcance
More informationGeorgia Tech PHYS 6124 Mathematical Methods of Physics I
Georga Tech PHYS 624 Mathematcal Methods of Physcs I Instructor: Predrag Cvtanovć Fall semester 202 Homework Set #7 due October 30 202 == show all your work for maxmum credt == put labels ttle legends
More informationMin Cut, Fast Cut, Polynomial Identities
Randomzed Algorthms, Summer 016 Mn Cut, Fast Cut, Polynomal Identtes Instructor: Thomas Kesselhem and Kurt Mehlhorn 1 Mn Cuts n Graphs Lecture (5 pages) Throughout ths secton, G = (V, E) s a mult-graph.
More informationModule 2. Random Processes. Version 2 ECE IIT, Kharagpur
Module Random Processes Lesson 6 Functons of Random Varables After readng ths lesson, ou wll learn about cdf of functon of a random varable. Formula for determnng the pdf of a random varable. Let, X be
More informationAsymptotics of the Solution of a Boundary Value. Problem for One-Characteristic Differential. Equation Degenerating into a Parabolic Equation
Nonl. Analyss and Dfferental Equatons, ol., 4, no., 5 - HIKARI Ltd, www.m-har.com http://dx.do.org/.988/nade.4.456 Asymptotcs of the Soluton of a Boundary alue Problem for One-Characterstc Dfferental Equaton
More informationAppendix for Causal Interaction in Factorial Experiments: Application to Conjoint Analysis
A Appendx for Causal Interacton n Factoral Experments: Applcaton to Conjont Analyss Mathematcal Appendx: Proofs of Theorems A. Lemmas Below, we descrbe all the lemmas, whch are used to prove the man theorems
More information