Introduction to Applied Mathematics
Introduction to Applied Mathematics MA 325
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This 6 page Class Notes was uploaded by Braeden Lind on Thursday October 15, 2015. The Class Notes belongs to MA 325 at North Carolina State University taught by Robert White in Fall. Since its upload, it has received 9 views. For similar materials see /class/223679/ma-325-north-carolina-state-university in Mathematics (M) at North Carolina State University.
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Date Created: 10/15/15
Lecture 9 Analysis of von Neumann Series Introduction In the applications to heat and mass transfer the discrete time dependent models have the form yk1 A yk b Under stability conditions on the time step the solution would converge to the solution of the steady state problem y A y b One condition that ensured this was when A converged to the zero matrix the yk converges to y We would like to be more precise about the term converge De nition The in nity norm of the vector x is a real number E maxl x l where 1 S 139 S n 1 The in nity norm of the square matrix A aij is IIAIIEmIaleawl J Example 1 1 3 4 LetX 6 andA13 1 9 3 0 5 x max169 9 and IIAII maX858 8 Basic Propelties Z 0 and 0 if and only if 0 2 llx Y S quotXquot M 3 llaxax where a is real 4 llell 3 quotAll llxll 5 quotABquot s llAllllBll The proofs of the first three basic properties are a consequence of the analogous properties of the absolute value Real valued functions that satisfy 13 are called norms Proof of 4 In order to prove the fourth property apply the de nition of in nity norm to the matrixvector product AX Z all x J Ax maX S mlaxzjzlay I 39le S mlaxzjllav I mjaxlxj I Proof of 5 nmwmz J 2k alkka mwgymw wwymaw s max gnaw m x 21ka llAllllBll De nition Let yk and y be vectors yk converges to y if and only if each component of yik converge to yi This is equivalent to k k quoty y maxlyl y1 converges to zero 1 Like the geometric series of single numbers the iterative scheme A y b can be expressed as a summation via recursion yk1 Ayk b AAyk391bb Azyk391 Ab b A2A yk392 b Ab b A3yk392 A2 A1 Ib Ak1y0 Ak Ib 1 De nition The summation I Ak and the series I Ak are generalizations of the geometric series and they are often referred to as the von Neumann series Previously we showed if Ak converges to the zero matrix then yk1 A yk b must converge to the solution of y A y b which is also a solution of I 7 Ay b If I 7 A has an inverse this and line 1 suggest that the von Neumann series must converge to the inverse ofI 7 A Ifthe norm of A is less than 1 then all this is true 1AykerandletysatisfyyAyb Ifthenorm Theorem Consider the scheme yk ofA is less than one then 1 yk1 A yk b converges to y A y b 2 I 7 A has an inverse and 3 I Ak converges to the inverse ofI 7 A Proof of 1 yk y Ayk 7 y Apply property 4 of the norm as follows lly yll W ygt S Ayk yll WVW d IIAIIZ lly yll k1 s quotA y Because the norm of A is less than one the right side must go to zero This forces the norm of the error to go to zero Proof of 2 This requires some additional information from matrix algebra a square matrix B has an inverse matrix if and only if Bx 0 implies x 0 Let B I 7A and suppose Bx 0 Then 0 Bx I 7 Ax Ix 7 Ax so that x Ax If x 7 0 then by property 1 0 Apply the norm to both sides ofx Ax and use property 4 to get MdeMWl Now divide by the norm of x to get a contradiction to the assumption that the norm of A is strictly less than one Proof of 3 By the associative and distributive properties of matrices we have I AI AAk IIAAk AIAAk I A Multiply both sides by the inverse of I 7 A to get I AAk I A391I Ak1 1 A 1 I I Arm 1 AAk I A 1 I A 1Ak1 Apply property 5 of the norm to get quotI A Ak I A 1quot1 AVA s quot 0 Ar llllA k1 SllUAVIIIIA Since the norm is less than one the right side must go to zero Thus the series must converge to the inverse of I 7 A Application to the Cooling Wire Consider a cooling wire with the heat loss through the lateral surface It will be assumed to be directly proportional to the change in time the lateral surface area and to the difference in the surrounding temperature and the temperature in the wire Let csur be the proportionality constant which measures insulation Let r be the radius of the wire so that the lateral surface area of a small wire segment is 21trh If usur is the surrounding temperature of the wire then the heat loss through the small lateral area is csur At 21trh usur ui Heat loss or gain from a source such as electrical current and from left and right diffusion will remain the same as in the previous lecture Let er Kpc AIAXZ and AX L n1 and combine the above Explicit Finite Difference Model of Heat Transfer uik1 Atpc f csur 2r usur 0cui1k ui1k l 20c Atpc csur2r uik where 2 i lnl k 0man l uio 0 fori lnl and 3 uok u k 0 for k lman 4 Equation 3 is the initial temperature set equal to zero and 4 is the temperature at the left and right ends set equal to zero Equation 2 may be put into the matrix version of the first order nite difference method For example if the wire is divided into four equal parts then n 4 and 1 may be written as a 3D vector equation uk1 A uk b where uf 1 uk u bAtpcFl and u l l 20c d at A 0L l 20c d at with ac l 20c d F f csm2 rusm and d At pccsm2 r Stability Condition for 2 l 20c Atpc csur2r gt 0 and at gt 0 The norm of the above 3x3 matrix under the stability condition is maxl 2a dla0 al 2a da 0l 2a da maxl 2a daal 2a dal 2a da maxl a dl dl a d 1 dlt1 5 Application to Pollutant in a Stream The model will have the general form change in amount m amount entering from upstream amount leaving to downstream amount decaying in a time interval In lecture 7 we formulated the following discrete model for the approximation of the pollutant Explicit Finite Difference Model of Flow and Decay of a Pollutant uik vel AtAx ui1k 1 vel AtAx At dec uik where 5 i lnl k 0maxk l uio given for i lnl and 6 uok given for k lmaxk 7 Equation 6 is the initial concentration and 7 is the concentration far upstream Equation 5 may be put into the matrix version of the rst order nite difference method For example if the stream is divided into four equal parts then n 4 and 1 may be written as 3D vector equation uk1 A uk b where uf uf dug c ukl1 d ulz 0 wheredvelAtAxandcl d decAt 2 c k1 k u3 d c u3 0 Stability Condition for 5 l vel AtAx dec At and vel dec gt 0 When the stability condition holds the norm of the 3x3 matrix is given by maxc00 dc0 0dc maxl d dec Atdl d dec Atdl d dec At l dec At lt l Homework 1 Find the norms of the following 1 4 5 3 7 x 0 andA 0 10 l 11 2 4 3 2 Verify properties 13 of the norm 3 Consider the application to cooling Let n 5 Find the matrix and determine when its norm will be less than one 4 Consider the application to pollution of stream Let n 5 Find the matrix and determine when its norm will be less than one