Definition 4. A symmetry generator of (4) is denoted by U and is given by + SQ- +...+ (9) dun+k-1 une z =n [+une. Q is the characteristic of the group of transformations. 27 In the above definition, SİQ(n,un) = Q(n+j,u,n+i). The operator S is known as the shift operator. Consider the system of ordinary difference equations of the form 28 29 Un+k+2 = w1(un+k, Vn, Vn+2), Vn+k+2 = w2 2(un, Un+2, Vn+k), (10) where the independent variable here is denoted by n and the dependent variables are denoted by un, Vn and their shifts. Consider the group of transformations (n, un, Vn) → (n, ũn = Un + ɛQ1(n, un) +0(?), 0n (11) = Un + EQ2(n,vn)+O(ɛ²)). In (11), the characteristic of the group of transformations is Q = (Q1, Q2). The infinitesimal generator corresponds to U = Q1du, + Q2do, (12) where d, = . In this work, we will need the kth extension dx uk = Qidu, + Q2do, + s²Q1ðun+2+ S?Q2dv+2+ S*Qiðun+* +s*Q2dv+* (13) Un+k Un+k of (12). For set solutions of (10) to be mapped to itself, the following linearised symmetry conditions s(k+2)Q1 - ukwi = 0 and s*+2) Q2 – uklw2 = = 0, (14) whenever (10) is true, must be satisfied. If the conditions given in(14), that is, Q;(n+k+ 2,0;) – uk (n;) = 0, j = 1,2, are satisfied, then the group of transformations (11) is a 32 symmetry group. 30 31 3. Symmetries and Solutions of the System of Difference Equations (1) 33 Equivalently, equation (3) can be written as Un+kVn Un+k+2 =W1 = Vn+2(an + bnUn+kOn)' UnVn+k (15) Un+k+2 =W2 = Un+2(Cn+ dnUnVn+k)' where an and bn are real sequences. Applying (14) onto (15) yields,
Definition 4. A symmetry generator of (4) is denoted by U and is given by + SQ- +...+ (9) dun+k-1 une z =n [+une. Q is the characteristic of the group of transformations. 27 In the above definition, SİQ(n,un) = Q(n+j,u,n+i). The operator S is known as the shift operator. Consider the system of ordinary difference equations of the form 28 29 Un+k+2 = w1(un+k, Vn, Vn+2), Vn+k+2 = w2 2(un, Un+2, Vn+k), (10) where the independent variable here is denoted by n and the dependent variables are denoted by un, Vn and their shifts. Consider the group of transformations (n, un, Vn) → (n, ũn = Un + ɛQ1(n, un) +0(?), 0n (11) = Un + EQ2(n,vn)+O(ɛ²)). In (11), the characteristic of the group of transformations is Q = (Q1, Q2). The infinitesimal generator corresponds to U = Q1du, + Q2do, (12) where d, = . In this work, we will need the kth extension dx uk = Qidu, + Q2do, + s²Q1ðun+2+ S?Q2dv+2+ S*Qiðun+* +s*Q2dv+* (13) Un+k Un+k of (12). For set solutions of (10) to be mapped to itself, the following linearised symmetry conditions s(k+2)Q1 - ukwi = 0 and s*+2) Q2 – uklw2 = = 0, (14) whenever (10) is true, must be satisfied. If the conditions given in(14), that is, Q;(n+k+ 2,0;) – uk (n;) = 0, j = 1,2, are satisfied, then the group of transformations (11) is a 32 symmetry group. 30 31 3. Symmetries and Solutions of the System of Difference Equations (1) 33 Equivalently, equation (3) can be written as Un+kVn Un+k+2 =W1 = Vn+2(an + bnUn+kOn)' UnVn+k (15) Un+k+2 =W2 = Un+2(Cn+ dnUnVn+k)' where an and bn are real sequences. Applying (14) onto (15) yields,
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
![In [3], the author determined and formulated the analytical solutions of the rational
recursive equations:
Xn-1Yn-3
Yn-1Xn-3
Хn+1
Уп+1
Yn-1(±1+xp-1Yn–3)
Xn-1(+1+ yn-1Xn–3)'
here x-3, x_2, x_1, X0, y-3, y-2, y1 and yo are the initial conditions which are arbi-
10 trary non-zero real numbers.
The aim of this study is to generalize the results in [2,3] by studying the system of
ordinary difference equations
Un-10n–k–1
Un-k-1Vn–1
Иn+1 —
Un+1 =
(3)
Vn–k+1(An+ BnUn–1Vn–k–1)'
Un-k+1(Cn + DnUn-k–1Vn–1)'
where An, Bn, Cn and Dn are real sequences, using a symmetry method. For a similar
11](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9d6994c6-9c0b-4312-919e-29acb733a883%2F422d8972-1a0c-4408-a11e-4d77c8cca778%2Fnkcirt_processed.jpeg&w=3840&q=75)
Transcribed Image Text:In [3], the author determined and formulated the analytical solutions of the rational
recursive equations:
Xn-1Yn-3
Yn-1Xn-3
Хn+1
Уп+1
Yn-1(±1+xp-1Yn–3)
Xn-1(+1+ yn-1Xn–3)'
here x-3, x_2, x_1, X0, y-3, y-2, y1 and yo are the initial conditions which are arbi-
10 trary non-zero real numbers.
The aim of this study is to generalize the results in [2,3] by studying the system of
ordinary difference equations
Un-10n–k–1
Un-k-1Vn–1
Иn+1 —
Un+1 =
(3)
Vn–k+1(An+ BnUn–1Vn–k–1)'
Un-k+1(Cn + DnUn-k–1Vn–1)'
where An, Bn, Cn and Dn are real sequences, using a symmetry method. For a similar
11
Transcribed Image Text:Definition 4. A symmetry generator of (4) is denoted by U and is given by
U = Q;
+ SQ
дип+1
+...+ sk-o
(9)
dun+k–1'
Une.
Q is the characteristic of the group of transformations.
27
In the above definition, SİQ(n, un) = Q(n+j,un+j). The operator S is known as
the shift operator.
Consider the system of ordinary difference equations of the form
28
29
Un+k+2
wi(Un+k, Vn, Vn+2), Vn+k+2
w2(Un, Un+2, Vn+k),
(10)
where the independent variable here is denoted by n and the dependent variables are
denoted by un, Vn and their shifts.
Consider the group of transformations
(п, ип,
tn, Vn)
H (n,ũn
= Un + ɛQ1(n,u,n)
(11)
+0(?), õn
= Vn + EQ2(n, Vn)+0(:²)).
In (11), the characteristic of the group of transformations is
Q = (Q1, Q2). The infinitesimal generator corresponds to
U = Q1du, + Q2dvµ,
(12)
where dx
2. In this work, we will need the kh extension
dx
(13)
On+2
Un+k
of (12).
For the set of solutions of (10) to be mapped to itself, the following linearised symmetry
conditions
s(k+2)Q1 – uklwi
= 0 and S(k+2)O2
uklw2 = 0,
(14)
whenever (10) is true, must be satisfied. If the conditions given in(14), that is, Q;(n+k+
2,N;) – uk (N;) = 0, j = 1,2, are satisfied, then the group of transformations (11) is a
symmetry group.
30
31
32
3. Symmetries and Solutions of the System of Difference Equations (1)
33
Equivalently, equation (3) can be written as
Un+kVn
Un+k+2 =w1 =
Vn+2(an + bnUn+kVn)'
UnVn+k
Un+2(Cn + d,UnVn+k)
(15)
Un+k+2 =w2 =
where an and b, are real sequences. Applying (14) onto (15) yields,
AnUn+kQ2
Vn+2(an + bnU+kUn)²
-sk+2Q1+
(16а)
Un+kVnS²Q2
v+2 (an + bnUn+kVn)
anVnS*Q1
+
Vn+2(an+ bnUn+kVn)² ¯ °r
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