The general solution of the nonhomogeneous equation 17.2.3.5 is given by the sum yx = ux + 2yx, where ux is the general solution of the corresponding homogeneous equation with f≡ 0, and2
Trang 117.2.3-2 Linear nonhomogeneous difference equations with constant coefficients.
An mth-order linear nonhomogeneous integer-difference equation with constant
coeffi-cients has the form
a m y(x + m) + a m–1y(x + m –1) +· · · + a1y(x +1) + a0y(x) = f (x), (17.2.3.5)
where a0a m0
The general solution of the nonhomogeneous equation (17.2.3.5) is given by the sum
y(x) = u(x) + 2y(x), where u(x) is the general solution of the corresponding homogeneous equation (with f≡ 0), and2y(x) is a particular solution of the nonhomogeneous equation (17.2.3.5) Regarding the
solution of the homogeneous equation see Paragraph 17.2.3-1
Below, we consider some methods for the construction of a particular solution of the nonhomogeneous equation (17.2.3.5)
1◦ Table 17.1 lists the forms of particular solutions corresponding to some special cases
of the function on the right-hand side of the linear nonhomogeneous difference equation (17.2.3.5)
2◦ Let
P (λ) = a m λ m + a
m–1λ m–1+· · · + a1λ + a0
be the characteristic polynomial of equation (17.2.3.5) Consider the function
g(λ) = 1
P (λ) =
∞
k=0
g k λ k, |λ|<|λ1|, (17.2.3.6)
where λ1 is the root of the equation P (λ) = 0 with the smallest absolute value Then lim
k→∞|g k|1/k =|λ1|– 1.
From (17.2.3.6), it follows that
a0g0 =1, a1g0+ a0g1 =0, .,
s–1
k=0
a s–k–1g k=0, s≥ 2, a ν =0 if ν > m.
If
lim
k→∞|f (x + k)|1/k = σ
f <|λ1|, then the series
y(x) = g0f (x) + g1f (x +1) +· · · + g k f (x + k) + · · ·
is convergent and its sum gives a solution of equation (17.2.3.5)
3◦ Let the right-hand side of equation (17.2.3.5) can be represented by the integral
f (x) =
L F (λ)λ
where the line of integration L does not cross the roots of the characteristic polynomial
P (λ) Direct verification shows that the integral
y(x) =
L
F (λ)
P (λ) λ
represents a particular solution of equation (17.2.3.5)
Trang 2TABLE 17.1 Forms of particular solutions of the linear nonhomogeneous difference equation with constant
coefficients a m y (x + m) + a m–1y (x + m –1 ) +· · · + a1 y (x +1) + a0y (x) = f (x)
in some special cases of the function f (x); a0a m 0
Form of the
function f (x)
Roots of the characteristic equation
a m λ m + a m–1λ m–1+· · · + a1 λ + a0= 0 Form of a particularsolution y = 2y(x)
λ= 1 is not a root of the characteristic equation
(i.e., a m + a m–1 +· · · + a1 + a0 ≠ 0 )
n
k=0b x
k
x n
λ= 1 is a root of the
characteristic equation (multiplicity r)
n+r
k=0b x
k
λ = e βis not a root of the
e βx
(β is a real constant) λ = e βis a root of the
characteristic equation (multiplicity r) e
βxr
k=0b x
k
βis not a root of the
βxn
k=0b x
k
x n e βx
(β is a real constant) βis a root of the
characteristic equation (multiplicity r) e βx
n+r
k=0b x
k
iβis not a root of the characteristic equation
2
P ν (x) cos βx
+ 2Q ν (x) sin βx
P m (x) cos βx
+ Q n (x) sin βx
iβis a root of the
characteristic equation (multiplicity r)
2
P ν+r (x) cos βx
+ 2Q ν+r (x) sin βx
α + iβ is not a root of the
characteristic equation
[ 2P ν (x) cos βx
+ 2Q ν (x) sin βx]e αx [P m (x) cos βx
+ Q n (x) sin βx]e αx
α + iβ is a root of the characteristic equation (multiplicity r)
[ 2P ν+r (x) cos βx
+ 2Q ν+r (x) sin βx]e αx
Notation: P m (x) and Q n (x) are polynomials of degrees m and n with given coefficients; 2 P m (x), 2 P ν (x),
and 2Q ν (x) are polynomials of degrees m and ν whose coefficients are determined by substituting the particular solution into the basic equation; ν = max(m, n); and α and β are real numbers, i2= – 1
Example 1 Taking F (λ) = aλ b , b≥ 0, and L ={0 ≤x≤ 1} in (17.2.3.7), we have
f (x) =
1 0
aλ b λ x–1dλ= a
x + b. Therefore, if the characteristic polynomial P (λ) has no roots on the segment [0 , 1 ], then a particular solution
of equation (17.2.3.5) with the right-hand side
f (x) = a
can be obtained in the form
y (x) = a
1 0
λ b
P (λ) λ
Remark. For L ={0 ≤x<∞}, the representation (17.2.3.7) may be regarded as the Mellin transformation
that maps F (λ) into f (x).
4◦ Let y(x) be a solution of equation (17.2.3.5) Then u(x) = βy
x (x) is a solution of the
nonhomogeneous equation
a m u(x + m) + a m–1u(x + m –1) +· · · + a1u(x +1) + a0u(x) = βf x (x).
Trang 3Example 2 In order to find a solution of equation (17.2.3.5) with the right-hand side
f (x) = a (x + b)2 = –
d dx
a
x + b, (17.2 3 11 )
let us multiply (17.2.3.9)–(17.2.3.10) by β = –1 and then differentiate the resulting expressions Thus, we obtain the following particular solution of the nonhomogeneous equation (17.2.3.5) with the right-hand side (17.2.3.11):
y (x) = –a
1 0
ln λ
P (λ) λ
Consecutively differentiating expressions (17.2.3.11) and (17.2.3.12), we obtain a solution of equation (17.2.3.5) with the right-hand side
f (x) = a (x + b) n = (–1 )n–1
(n –1 )!
d n–1
dx n–1
a
x + b
This solution has the form
y (x) = a(–1 )n–1
(n –1 )!
1 0
(ln λ) n–1
P (λ) λ
x+b–1dλ.
5◦ In Paragraph 17.2.3-4, Item3◦, there is a formula that allows us to obtain a particular
solution of the nonhomogeneous equation (17.2.3.5) with an arbitrary right-hand side
17.2.3-3 Linear homogeneous difference equations with variable coefficients
1◦ An mth-order linear homogeneous integer-difference equation with variable
coeffi-cients has the form
a m (x)y(x + m) + a m–1(x)y(x + m –1) +· · · + a1(x)y(x +1) + a0(x)y(x) =0, (17.2.3.13)
where a0(x)a m (x)0
This equation admits the trivial solution y(x)≡ 0
The set E of all singular points of equation (17.2.3.13) consists of points of three classes: 1) zeroes of the function a0(x), denoted by μ1, μ2, ;
2) zeroes of the function a m (x – m), denoted by ν1, ν2, ;
3) singular points of the coefficients of the equation, denoted by η1, η2,
The points of the set
S(E) ={μ s – n, ν s + n, η s – n, η s + m + n; n =0, 1, 2, , s =1, 2,3, }
are called comparable with singular points of equation (17.2.3.13)
Let
y1= y1(x), y2= y2(x), , y m = y m (x) (17.2.3.14)
be particular solutions of equation (17.2.3.13) Then the function
y =Θ1(x)y1(x) +Θ2(x)y2(x) + · · · + Θ m (x)y m (x) (17.2.3.15) with arbitrary1-periodic functions Θ1(x),Θ2(x), ,Θm (x) is also a solution of equation
(17.2.3.13)
The Casoratti determinant is the function defined as
D(x) =
y1(x) y2(x) · · · y m (x)
y1(x +1) y2(x +1) · · · y m (x +1)
y1(x + m –1) y2(x + m –1) · · · y m (x + m –1)
(17.2.3.16)
Trang 4THEOREM(CASORATTI) Formula (17.2.3.15) gives the general solution of the linear
homogeneous difference equation (17.2.3.13) if and only if for any point xS(E) such
that x + kS(E) for k =0,1, , m, the condition D(x)≠ 0holds
The Casoratti determinant (17.2.3.16) satisfies the first-order difference equation
D(x +1) = (–1)m a0(x)
a m (x) D(x). (17.2.3.17)
2◦ Let y0= y0(x) be a nontrivial particular solution of equation (17.2.3.13) Then the order
of equation (17.2.3.13) can be reduced by unity Indeed, making the replacement
y(x) = y0(x)u(x) (17.2.3.18)
in equation (17.2.3.13), we get
m
k=0
a k (x)y0(x + k)u(x + k) =0 (17.2.3.19) Let us transform this relation with the help of the Abel identity
m
k=0
F k G k= –
m–1
k=0
(G k+1– G k)
k
s=0
F s + G m
m
k=0
F k,
in which we take F k = a k (x)y0(x + k) and G k = u(x + k) As a result, we get
–
m–1
k=0
r k (x)[u(x + k +1) – u(x + k)] + u(x + m)
m
k=0
a k (x)y0(x + k) =0, (17.2.3.20) where
r k (x) =
k
s=0
a s (x)y0(x + s), k=0, 1, , m –1
Taking into account that the second sum in (17.2.3.20) is equal to zero (since y0 is a particular solution of the equation under consideration) and setting
w(x) = u(x +1) – u(x) (17.2.3.21)
in (17.2.3.20), we come to an (m –1)th-order difference equation
m–1
k=0
r k (x)w(x + k) =0
17.2.3-4 Linear nonhomogeneous difference equations with variable coefficients
1◦ An mth-order linear nonhomogeneous integer-difference equation with variable
coef-ficients has the form
a m (x)y(x + m) + a m–1(x)y(x + m –1) +· · · +a1(x)y(x +1) + a0(x)y(x) = f (x), (17.2.3.22)
where a0(x)a m (x)0
The general solution of the nonhomogeneous equation (17.2.3.22) is given by the sum
y(x) = u(x) + 2y(x), (17.2.3.23)
where u(x) is the general solution of the corresponding homogeneous equation (with f≡ 0) and2y(x) is a particular solution of the nonhomogeneous equation (17.2.3.22) The general
solution of the homogeneous equation is defined by the right-hand side of (17.2.3.15) Every solution of equation (17.2.3.22) is uniquely determined by prescribing the values
of the sought function on the interval [0, m).
Trang 52◦ A particular solution2y(x) of the linear nonhomogeneous difference equation
a m (x)y(x + m) + a m–1(x)y(x + m –1) +· · · + a1(x)y(x +1) + a0(x)y(x) =
n
k=1
f k (x)
can be represented by the sum
2y(x) =n
k=1
2y k (x),
where2y k (x) are particular solutions of the linear nonhomogeneous difference equations
a m (x)y(x + m) + a m–1(x)y(x + m –1) +· · · + a1(x)y(x +1) + a0(x)y(x) = f k (x).
3◦ The solution of the Cauchy problem for the nonhomogeneous equation (17.2.3.22) with
arbitrary initial conditions
y(x + j) = ϕ j (x) for 0 ≤x<1, j =0, 1, , m –1, (17.2.3.24)
is given by the sum (17.2.3.23), where
u(x) = – 1
D({x})
ϕ0({x}) y1({x}) · · · y m({x})
ϕ m–1({x}) y1({x}+ m –1) · · · y m({x}+ m –1)
is a solution of the homogeneous equation (17.2.3.13) with the boundary conditions (17.2.3.24), and
2y(x) =
[x]
j=m
D ∗ (x – j +1)
D(x – j +1)
f (x – j)
a m (x – j) (17.2.3.25)
is a particular evolution of the nonhomogeneous equation (17.2.3.22) with zero initial conditions2y(x) =0for0 ≤x < m Formula (17.2.3.25) contains the determinant
D ∗ (t +1) =
y1(t +1) · · · y m (t +1)
· · · ·
y1(t + m –1) · · · y m (t + m –1)
y1(x) · · · y m (x)
,
which is obtained from the determinant D(t +1) by replacing the last row [y1(t + m), , y m (t+m)] with the row [y1(x), , y m (x)] Note that D ∗ (t+1) =0for x–m+1≤t≤x–1
17.2.3-5 Equations reducible to equations with constant coefficients
1◦ The difference equation with variable coefficients
a m f(x + m)y(x + m) + a m–1f (x + m –1)y(x + m –1) +· · ·
+ a1f (x +1)y(x +1) + a0f (x)y(x) = g(x)
can be reduced, with the help of the replacement
y(x) = f (x)u(x),
Trang 6to the equation with constant coefficients
a m u(x + m) + a m–1u(x + m –1) +· · · + a1u(x +1) + a0u(x) = g(x).
2◦ The difference equation with variable coefficients
a m y(x + m) + a m–1f (x)y(x + m –1) + a m–2f (x)f (x –1)y(x + m –2) +· · ·
+ a0f (x)f (x –1) f (x – m +1)y(x) = g(x)
can be reduced to a nonhomogeneous equation with constant coefficients with the help of the replacement
y(x) = u(x) exp
ϕ(x – m)
,
with ϕ(x) satisfying the auxiliary first-order difference equation
ϕ(x +1) – ϕ(x) = ln f (x).
The resulting equation with constant coefficients has the form
a m u(x + m) + a m–1u(x – m –1) +· · · + a1u(x +1) + a0u(x) = g(x) exp[–ϕ(x)].
3◦ The difference equation with variable coefficients
a m f(x)f (x+1) f (x+m –1)y(x+m)+a m–1f (x)f (x+1) f (x+m –2)y(x+m –1)+· · ·
+ a1f (x)y(x +1) + a0y(x) = g(x)
can be reduced to a nonhomogeneous equation with constant coefficients with the help of the replacement
y(x) = u(x) exp
–ψ(x) ,
where ϕ(x) is a function satisfying the auxiliary first-order difference equation
ψ(x +1) – ψ(x) = ln f (x).
The resulting equation with constant coefficients has the form
a m u(x + m) + a m–1u(x – m –1) +· · · + a1u(x +1) + a0u(x) = g(x) exp[ψ(x)].
Differences
17.2.4-1 Linear homogeneous difference equations
1◦ A linear homogeneous difference equation with constant coefficients, in the case of
arbitrary differences, has the form
a m y(x + h m ) + a m–1y(x + h m–1) +· · · + a1y(x + h1) + a0y(x + h0) =0, (17.2.4.1)
where a0a m ≠ 0, m≥ 1, h i ≠ h j for i≠j; the coefficients a k and the differences h k are
complex numbers and x is a complex variable.
Equation (17.2.4.1) can be reduced to an equation with integer differences if the
quan-tities h k – h0are commensurable in the sense that there is a common constant q such that
h k – h0= qN k with integer N k Indeed, we have
y(x + h k ) = y(x + h0+ qN k ) = y
q
x + h0
q + N k
= w(z + N k),
where the new variables have the form z = (x + h0)/q and w(z) = y(qz) As a result, we obtain an equation with integer differences N k for the function w(z) In particular, this situation takes place if h k – h0 are rational numbers: h k – h0 = p k /r k , where p k and r k
are positive integers (k =1, 2, , m) In this case, one can take q =1/r, where r is the common denominator of the fractions p k /r k.
Trang 72◦ In what follows, we assume that the numbers h
k – h0 are not commensurable
(k =1, , m) We seek particular solutions of equation (17.2.4.1) in the form y = e tx
Substituting this expression into (17.2.4.1) and dividing the result by e tx, we obtain the transcendental equation
A(t)≡a m e h m t + a
m–1e h m–1t+· · · + a1e h1t + a
0e h0t=0, (17.2.4.2)
where A(t) is the characteristic function.
It is known that equation (17.2.4.2) has infinitely many roots
3◦ Let h k be real ordered numbers, h0 < h1 <· · · < h m , and t = t1+ it2, where t1 = Re t and t2 = Im t Then the following statements hold:
(a) There exist constants γ1and γ2such that the following estimates are valid:
|A(t)|> 12|a0|e h0t1 if t1≤γ1,
|A(t)|> 12|a m|e h m t1 if t1≥γ2.
(b) All roots of equation (17.2.4.2) belong to the vertical strip γ1< t1< γ2
(c) Let β1, β2, , β n , be roots of equation (17.2.4.2), and|β1| ≤ |β2| ≤· · ·≤ |β n| ≤· · ·
Then the following limit relation holds:
lim
n→∞
β n
n = 2π
h m – h0.
4◦ A root β
k of multiplicity n k of the characteristic equation (17.2.4.2) corresponds to
exactly n klinearly independent solutions of equation (17.2.4.1):
e β k x, xe β k x, ., x n k 1e β k x (k =1, 2, ). (17.2.4.3) Equation (17.2.4.1) admits infinitely many solutions of the form (17.2.4.3), since the characteristic function (17.2.4.2) has infinitely many roots In order to single out different classes of solutions, it is convenient to use a condition that characterizes the order of their growth for large values of the argument
The class of functions of exponential growth of finite degree σ is denoted by [1, σ] and
is defined as the set of all entire functions* f (x) satisfying the condition
e(σ–ε)|x| <|f (x)|< e(σ+ε)|x| (17.2.4.4)
for any ε > 0, where the right inequality in (17.2.4.4) should hold for all sufficiently
large x: |x| > R(ε), while the left inequality in (17.2.4.4) should hold for some sequence
x = x n = x n (ε) → ∞.
The parameter σ can be found from the relations
σ= lim
r→∞
M (r)
r = lim
k→∞ k!|b k| 1/k, M (r) = max
|x| =r|f (x)| (0 ≤σ<∞), where b k are the coefficients in the power series expansion of the function f (x) (see the
footnote)
* An entire function is a function that is analytic on the entire complex plane (except, possibly, the infinite
point) Any such function can be expanded in power series, f (x) =∞
k=0b x
k, convergent on the entire complex plane, i.e., lim
k→∞|b | 1/k= 0