Theorem 9 Suppose that 0 < p < z and {(tn, Zn)} be a solution of the system (2) such that lim t, = i and lim z, = 7. Then the error vector n-00 en tn - In-1 -i Zn - 2 En en n-1 Zn-1 - of every solution of system (2) satisfies both of the following asymptotic relations: lim V| E,| = |A1,2,3,4F)(ī, 2)|, || En+1|| n-00 lim no || En|| = |21,2.3,4 F,(7, 2)|. where 11,2,3,4 FJ (i, 2) are the characteristic roots of the Jacobian matrix F,(i, 7). Proof To find the error terms, we set 1 1 In+1 - i = a; (fn-i - i) +EBi (zn-i – 2), | i=0 i=0 Zn+l – 7 = En (in-i – i) +di (zn-i – 2), i=0 i=0 and e, = t, – i, e, = zn – 7. Thus we have 1 n+1 i=0 i=0
Theorem 9 Suppose that 0 < p < z and {(tn, Zn)} be a solution of the system (2) such that lim t, = i and lim z, = 7. Then the error vector n-00 en tn - In-1 -i Zn - 2 En en n-1 Zn-1 - of every solution of system (2) satisfies both of the following asymptotic relations: lim V| E,| = |A1,2,3,4F)(ī, 2)|, || En+1|| n-00 lim no || En|| = |21,2.3,4 F,(7, 2)|. where 11,2,3,4 FJ (i, 2) are the characteristic roots of the Jacobian matrix F,(i, 7). Proof To find the error terms, we set 1 1 In+1 - i = a; (fn-i - i) +EBi (zn-i – 2), | i=0 i=0 Zn+l – 7 = En (in-i – i) +di (zn-i – 2), i=0 i=0 and e, = t, – i, e, = zn – 7. Thus we have 1 n+1 i=0 i=0
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
Show me the steps of deremine yellow and inf is here i need evey I need all the details step by step and inf is here
Transcribed Image Text:Motivated by difference equations and their systems, we consider the following
system of difference equations
Уп
Xn+1 = A + B-
Yn-1
Xn
, Yn+1 = A+B-
2
(1)
Xn-1
where A and B are positive numbers and the initial values are positive numbers. In
From this, system (1) transform into following system:
tn
Zn
1+ PT
Zn-1
tn+1 =
Zn+1 = 1+ p
,2
(2)
'n-1
where p =
> 0. From now on, we study the system (2).
Now we study the rate of convergence of system (2). Hence, we consider the following
system:
En+1
(А + B (п)) Еп,
(14)
where En is a k-dimensional vector, A e Ck*k is a constant matrix, and B : Z+ →
Ckxk is a matrix function satisfying
|| B (n)|| –
→ 0,
(15)
Theorem 9 Suppose that 0 < p < ; and {(tn, Zn)} be a solution of the system (2)
such that lim tn
t and lim zn = 7. Then the error vector
%3D
n-00
-1
tn-1 -
E, =
Zn - 2
Zn-1
п-1
of every solution of system (2) satisfies both of the following asymptotic relations:
lim |E,|| = |21,2.3,4F1(ī, 7)|,
|| En+1||
lim
= |A1,2.3,4 F1(ī, 2)| ,
%3|
n→00 || En||
where A1,2,3,4 F,(i, 7) are the characteristic roots of the Jacobian matrix F, (i, 7).
Proof To find the error terms, we set
1
1
In+1 -i =a; (fn-i - i) +Bi (zn-i – 2),
i=0
i=0
1
Zn+1
= =Evi (in-i –i) +8; (zn-i – E),
|
i=0
i=0
and e, = tn – i, e, = zn – z. Thus we have
1
1
n+1
n-i
i=0
i=0
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 5 steps
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
