Show me the steps of deremine blue and inf is here i need evey I need all the details step by step and inf is her Theorem 9 Suppose that 0 < p < and {(tn, Zn)} be a solution of the system (2)such that lim tn = i and lim zn = 7. Then the error vectorn-00n-00In -iIn-1 -iZn - 7En =e?´n-1Zn-1 – 7)of every solution of system (2) satisfies both of the following asymptotic relations:lim En| = |21,2,3,4F, (7, 2)|,n-00||En+1|||21,2.3,4F1(, 2)|,limn→∞ || En ||where 11,2,3,4 FJ(ī, z) are the characteristic roots of the Jacobian matrix F1(ī, z).Proof To find the error terms, we set11In+1 –i =Eai (In-i – i) + E Bi (zn-i – 2),i=0i=011Zn+1 – 7 =ri (tn-i – i) + ôi (zn-i – 2),i=0i=0and e = tn - i, e, = zn – 7. Thus we have11= I+"ai=0i=011ent!i=0i=0whereao = aj = 0,-p (7 + Zn-1)PBo =2Zn-1, B1 =n-1-p (ī + tn-1)iYo =, Yi =12'n-1n-180 = 81 = 0.Now we take the limitslim ¤on→00lim aj = 0,n-00-2plim Bo =lim B1n-00-2plim yo =n-00limYIn-00lim 8o= lim 81 = 0.n-00n-00Hence-2p+ an, B1 =+ bn,Bo =-2pYO =+ Cn, Y1 =+ dn,where an → 0, b, → 0, cn → 0, dn → 0 as n → ∞. Therefore, we obtain thesystem of the form (14)En+1(A + B (n)) En%3Dwhere0 01A =0 010 0 an bn1000B(n) =Cn dn 0 0 Motivated by difference equations and their systems, we consider the followingsystem of difference equationsXnУп, Yn+1 = A + B-Уп-1Xn+1 = A + B-(1)Xn-1where A and B are positive numbers and the initial values are positive numbers. InFrom this, system (1) transform into following system:Zn, Zn+1 = 1 +p,n-1tn+1 = 1+P7(2)%3Dn-1where p = > 0. From now on, we study the system (2).Now we study the rate of convergence of system (2). Hence, we consider the followingsystem:En+1 = (A + B (n)) En,(14)where En is a k-dimensional vector, A e Ckxk is a constant matrix, and B : Z+ →Ckxk is a matrix function satisfying||B(n)|| → 0,(15)

Given: limn→∞tn=t¯ limn→∞zn=z¯ To do: Determine how the following equations arrived in the proof…

Answered: Theorem 9 Suppose that 0 < p < and…

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Show me the steps of deremine blue and inf is here i need evey I need all the details step by step and inf is her

Expert Solution

Step 1

Given:

To do:

Determine how the following equations arrived in the proof of Theorem 9.

$t_{n + 1} - \bar{t} = \sum_{i = 0}^{1} α_{i} (t_{n - i} - \bar{t}) + \sum_{i = 0}^{1} β_{i} (z_{n - i} - \bar{z})$ .......... $(1)$
$z_{n + 1} - \bar{z} = \sum_{i = 0}^{1} γ_{i} (t_{n - i} - \bar{t}) + \sum_{i = 0}^{1} δ_{i} (z_{n - i} - \bar{z})$ .......... $(2)$
$e_{n + 1}^{1} = \sum_{i = 0}^{1} α_{i} e_{n - i}^{1} + \sum_{i = 0}^{1} β_{i} e_{n - i}^{2}$ .......... $(3)$
$e_{n + 1}^{2} = \sum_{i = 0}^{1} γ_{i} e_{n - i}^{1} + \sum_{i = 0}^{1} δ_{i} e_{n - i}^{2}$ .......... $(4)$