Handbook of mathematics for engineers and scienteists part 133 pps

Consider equation 17.2.1.18 whose right-hand side is an analytic function that admits the expansion in power series: g x = ∞ k=0a k x which converges on the entire complex plane it follo

3◦ For equations with rational right-hand side

y (x +1) – ay(x) = b

x + c, where c≥ 0and a[0,1], a particular solution is written in the form of the integral

2y(x) = b

 1

0

λ x+c–1

λ – a dλ.

4◦ For the equation with exponential right-hand side

y (x +1) – ay(x) =

n



k=0

b k e λkx,

there is a particular solution of the form

2y(x) =

⎧

⎪

⎪

⎪

⎪

n



k=0

b k

e λk – a e λkx if a≠e λm,

b m xe λm(x–1)+

n



k=0 ,k m

b k

e λk – a e λkx if a = e λm, where m =0,1, , n.

5◦ For the equation with sinusoidal right-hand side

y (x +1) – ay(x) = b sin(βx),

there is a particular solution of the form

y (x) = b

a2+1–2a cos β



(cos β – a) sin(βx) – sin β cos(βx)

6◦ For the equation with cosine in the right-hand side

y (x +1) – ay(x) = b cos(βx),

there is a particular solution of the form

a2+1–2a cos β



(cos β – a) cos(βx) + sin β sin(βx)

17.2.1-6 Nonhomogeneous linear equations with the right-hand side of general form

1◦ Consider equation (17.2.1.18) whose right-hand side is an analytic function that admits

the expansion in power series:

g (x) = ∞

k=0a k x

which converges on the entire complex plane (it follows that lim

k→∞|a k|1/k=0) If lim

k→∞ k!|a k|1/k<2π, (17.2.1.21) then the solution of equation (17.2.1.18) can be represented as a convergent series

2y(x) =∞

k=0

a k

k+1B k+1(x),

where B k (x) are Bernoulli polynomials (see Paragraph 17.2.1-5, Item1◦).

Below we state a more general result that requires no conditions of the type (17.2.1.21)

2◦ Suppose that the right-hand side of equation (17.2.1.18) is an analytic function

admit-ting expansion by power series (17.2.1.20) convergent on the entire complex plane (such functions are called entire functions) In this case, the solution of equation (17.2.1.18) can

be represented in the form

2y(x) =∞

k=0

a k β k (x),

where β k (x) are Hurwitz functions, defined by

β k (x) = 2k πi!



|t| = 1

(e tx–1) dt (e t–1)t k+1 + k!

k



s=–k, s≠0

e2isπx–1

(2isπ)s+1 = P k+1(x) +Θk (x),

P n+1(x) is a polynomial of degree n +1,Θk (x) is an entire1-periodic function,|t|=1is the

unit circle, and i2= –1

17.2.1-7 Nonhomogeneous linear difference equations Cauchy’s problem

1◦ Consider the Cauchy problem for a nonhomogeneous linear difference equation of special form (17.2.1.18) with the initial condition (17.2.1.10), where g(x) is a continuous function defined for x≥ 0, and ϕ(x) is a given continuous function defined on the segment

0 ≤x≤ 1

In order to find a solution of this Cauchy problem, the step method can be used: on the interval 1 ≤x < 2 one constructs a solution from equation (17.2.1.18) and the boundary condition (17.2.1.10); on the interval 2 ≤ x < 3, one uses equation (17.2.1.18) and the solution obtained for1 ≤x<2; on the interval3 ≤x<4, one uses equation (17.2.1.18) and the solution obtained for2 ≤x<3, etc As a result, we have

y (x) = ϕ(x – n) + g(x – n) + g(x – n +1) +· · · + g(x –1), n≤x < n +1, (17.2.1.22)

where n =1, 2,

Solution (17.2.1.22) is continuous if it is continuous at integer points x =1,2, , and

this leads us to the condition

ϕ(1) = ϕ(0) + g(0) (17.2.1.23)

The solution is continuously differentiable if the functions g(x) and ϕ(x) are continuously

differentiable and, together with (17.2.1.23), the additional condition

ϕ (1) = ϕ (0) + g (0) for the corresponding one-sided derivatives is satisfied

2◦ In a similar way, one considers the Cauchy problem for a nonhomogeneous linear

difference equation of general form (17.2.1.15) with the initial condition (17.2.1.10) As a result, we obtain a solution of the form

y (x) = ϕ({x})

[x]



j=1

f (x – j) +

[x]



i=1

g (x – i)

i–1



j=1

f (x – j),

where [x] and {x} denote, respectively, the integer and the fractional parts of x (x = [x] +{x}); the product and the sum over the empty index set (for [x] =0) are assumed equal

to1and0, respectively

If a nontrivial particular solution y1(x) of the homogeneous equation (17.2.1.1) is known,

then the solution of the Cauchy problem for the nonhomogeneous equation (17.2.1.15) with the initial condition (17.2.1.10) is given by

y (x) = y1(x)



ϕ({x})

y1(0) +

[x]



i=1

g (x – i)

y1(x – i +1)



17.2.2 Second-Order Linear Difference Equations with Integer

Differences

17.2.2-1 Linear homogeneous difference equations with constant coefficients

A second-order homogeneous linear integer-difference equation with constant coefficients

has the form

ay (x +2) + by(x +1) + cy(x) =0, ac≠ 0 (17.2.2.1)

This equation has the trivial solution y(x)≡ 0

The general solution of the difference equation (17.2.2.1) is determined by the roots of the characteristic equation

aλ2+ bλ + c =0 (17.2.2.2)

1◦ For b2–4ac>0, the quadratic equation (17.2.2.2) has two distinct roots:

λ1= –b +

√

b2–4ac

2a , λ2 = –b –

√

b2–4ac

The general solution of the difference equation (17.2.2.1) is given by

y (x) =Θ1(x)λ x1+Θ2(x)λ x2 if ab <0, ac >0;

y (x) =Θ1(x)λ x1+Θ2(x)|λ2|x cos(πx) if ac <0;

y (x) =Θ1(x)|λ1|x cos(πx) +Θ2(x)|λ2|x cos(πx) if ab >0, ac >0,

(17.2.2.3)

whereΘ1(x) andΘ2(x) are arbitrary1-periodic functions,Θk (x) =Θk (x +1), k =1, 2

2◦ For b2–4ac=0, the quadratic equation (17.2.2.2) has one double real root

λ= – b

2a, and the general solution of the difference equation (17.2.2.1) is given by

y=

Θ1(x) + xΘ2(x)

λ x if ab<0,

y=

Θ1(x) + xΘ2(x)

|λ|x cos(πx) if ab>0 (17.2.2.4)

3◦ For b2–4ac<0, the quadratic equation (17.2.2.2) has two complex conjugate roots

λ1 , 2 = ρ(cos β i sin β), ρ= c

a, β = arccos



– b

2√ ac



, and the general solution of the difference equation (17.2.2.1) has the form

y=Θ1(x)ρ x cos(βx) +Θ2(x)ρ x sin(βx), (17.2.2.5) whereΘ1(x) andΘ2(x) are arbitrary1-periodic functions

17.2.2-2 Linear nonhomogeneous difference equations with constant coefficients.

1◦ A second-order nonhomogeneous linear integer-difference equation with constant

co-efficients has the form

ay (x +2) + by(x +1) + cy(x) = f (x), ac≠ 0 (17.2.2.6)

Let y1(x) and y2(x) be two particular solutions of the homogeneous equation (17.2.2.1).

According to the roots of the characteristic equation (17.2.2.2), these solutions are defined

by (17.2.2.3)–(17.2.2.5), respectively, withΘ1(x)≡ 1,Θ2(x)≡ 0andΘ1(x)≡ 0,Θ2(x)≡ 1

A particular solution of the nonhomogeneous equation (17.2.2.6) satisfying zero initial conditions

y (x) =0, y (x +1) =0 for 0 ≤x<1

has the form

2y(x) = 1

a

[x]–1

j=1

Δ(x – j, x) Δ(x – j, x – j +1)f (x – j), Δ(x, z) =

y1(x) y2(x)

y1(z) y2(z)



 (17.2.2.7)

2◦ The solution of the Cauchy problem for the nonhomogeneous equation (17.2.2.6) with

arbitrary initial conditions

y (x) = ϕ(x), y (x +1) = ψ(x) for 0 ≤x<1 (17.2.2.8)

is the sum of the particular solution (17.2.2.7) and the function

u (x) = – 1

Δ({x},{x}+1)







0 y1(x) y2(x)

ϕ({x}) y1({x}) y2({x})

ψ({x}) y1({x}+1) y2({x}+1)





,

which is a solution of the homogeneous equation (17.2.2.1) with the initial conditions (17.2.2.8)

17.2.2-3 Linear homogeneous difference equations with variable coefficients

1◦ A second-order linear homogeneous integer-difference equation has the form

a (x)y(x +2) + b(x)y(x +1) + c(x)y(x) =0, a (x)c(x)0 (17.2.2.9)

The trivial solution, y(x) =0, is a particular solution of the homogeneous linear equation

Let y1(x), y2(x) be two particular solutions of equation (17.2.2.9) with the condition*

D (x)≡y1(x)y2(x +1) – y2(x)y1(x +1)≠ 0 (17.2.2.10) Then the general solution of equation (17.2.2.9) is given by

y (x) =Θ1(x)y1(x) +Θ2(x)y2(x), (17.2.2.11) where Θ1(x) andΘ2(x) are arbitrary1-periodic functions,Θ1 , 2(x) =Θ1 , 2(x +1)

* Condition (17.2.2.10) may be violated at singular points of equation (17.2.2.9); for details see Para-graph 17.2.3-3.

2◦ Let y0(x) be a nontrivial particular solution of equation (17.2.2.9) Then the substitution

y (x) = y0(x)u(x) (17.2.2.12) results in the equation

a (x)y0(x +2)u(x +2) + b(x)y0(x +1)u(x +1) + c(x)y0(x)u(x) =0 (17.2.2.13)

Taking into account that y0(x) satisfies equation (17.2.2.9), let us substitute the expression

b (x)y0(x +1) = –a(x)y0(x +2) – c(x)y0(x)

into (17.2.2.13) Then, after simple transformations, we obtain

a (x)y0(x +2)[u(x +2) – u(x +1)] – c(x)y0(x)[u(x +1) – u(x)] =0

Introducing a new variable by

w (x) = u(x +1) – u(x) (17.2.2.14)

we come to the first-order difference equation

a (x)y0(x +2)w(x +1) – c(x)y0(x)w(x) =0 After solving this equation, one solves the nonhomogeneous first-order equation with constant coefficients (17.2.2.14), and then, using (17.2.2.12), one finds a solution of the original equation

17.2.2-4 Linear nonhomogeneous difference equations with variable coefficients

1◦ A second-order linear nonhomogeneous difference equation with integer differences

has the form

a (x)y(x +2) + b(x)y(x +1) + c(x)y(x) = f (x), a (x)c(x)0 (17.2.2.15) The general solution of the nonhomogeneous equation (17.2.2.15) 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), and 2y(x) is a particular solution of equation (17.2.2.15) The general solution of the

homogeneous equation is defined by the right-hand side of (17.2.2.11)

Every solution of equation (17.2.2.15) is uniquely determined by given values of the sought function on the interval [0,2)

2◦ A particular solution2y(x) of the linear nonhomogeneous difference equation

a (x)y(x +2) + b(x)y(x +1) + c(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 (x)y k (x +2) + b(x)y k (x +1) + c(x)y k (x) = f k (x).

3◦ Let y1(x) and y2(x) be particular solutions of the corresponding linear homogeneous

equation (17.2.2.9) that satisfy the condition (17.2.2.10) A particular solution of the linear nonhomogeneous equation (17.2.2.15) can be sought in the form

2y(x) = ϕ1(x)y1(x) + ϕ2(x)y2(x), (17.2.2.16)

where ϕ1(x) and ϕ2(x) are functions to be determined.

For what follows, we need the identity

Δ[ϕ(x)ψ(x)]≡ψ (x +1)Δϕ(x) + ϕ(x)Δψ(x), (17.2.2.17) whereΔϕ(x) is the standard notation for the difference,

Δϕ(x)≡ϕ (x +1) – ϕ(x).

From (17.2.2.16), using the identity (17.2.2.17), we obtain

Δ2y(x) = y1(x +1)Δϕ1(x) + y2(x +1)Δϕ2(x) + ϕ1(x) Δy1(x) + ϕ2(x) Δy2(x) (17.2.2.18)

As one of the equations to determine the functions ϕ1(x) and ϕ2(x) we take

y1(x +1)Δϕ1(x) + y2(x +1)Δϕ2(x) =0 (17.2.2.19)

In view of the above considerations, from (17.2.2.16) and (17.2.2.18) we find that

2y(x +1) = ϕ1(x)y1(x +1) + ϕ2(x)y2(x +1),

2y(x +2) = ϕ1(x +1)y1(x +2) + ϕ2(x +1)y2(x +2)

= ϕ1(x)y1(x +2) + ϕ2(x)y2(x +2) + y1(x +2)Δϕ1(x) + y2(x +2)Δϕ2(x),

(17.2.2.20)

where the second relation is obtained from the first one by replacing x with x +1

Substituting (17.2.2.16) and (17.2.2.20) into equation (17.2.2.15) and taking into account

that y1(x) and y2(x) are particular solutions of the linear homogeneous equation (17.2.2.9),

we obtain

y1(x +2)Δϕ1(x) + y2(x +2)Δϕ2(x) = f (x)/a(x). (17.2.2.21) Relations (17.2.2.19) and (17.2.2.21) form a system of linear algebraic equations for the differencesΔϕ1(x) and Δϕ2(x) The solution of this system has the form

Δϕ1(x) = – f (x)y2(x +1)

a (x)D(x +1), Δϕ2(x) = – f (x)y1(x +1)

a (x)D(x +1), (17.2.2.22)

where D(x) is the determinant introduced in (17.2.2.10).

Thus, the construction of a particular solution of the second-order nonhomogeneous equation (17.2.2.15) amounts to finding a solution of two independent nonhomogeneous first-order equations (17.2.2.22) considered in detail in Paragraphs 17.2.1-4–17.2.1-7

4◦ The structure of particular solutions of second-order difference equations with

spe-cific (polynomial, exponential, sinusoidal, etc.) right-hand sides is described in Para-graph 17.2.3-2

17.2.3 Linearmth-Order Difference Equations with Integer

Differences

17.2.3-1 Linear homogeneous difference equations with constant coefficients

An mth-order linear integer-difference homogeneous equation with constant coefficients

has the form

a m y (x + m) + a m–1y (x + m –1) +· · · + a1y (x +1) + a0y (x) =0, (17.2.3.1)

where a0a m0

Let us write out its characteristic equation,

a m λ m + a m–1λ m–1+· · · + a1λ + a0 =0 (17.2.3.2) Consider the following cases:

1◦ All roots λ1, λ2, , λ nof equation (17.2.3.2) are real and mutually distinct Then the

general solution of the original equation (17.2.3.1) has the form

y (x) =Θ1(x)λ x1 +Θ2(x)λ x2+· · · + Θ n (x)λ x n, (17.2.3.3) where Θ1(x), Θ2(x), , Θn (x) are arbitrary 1-periodic functions, Θk (x) =Θk (x +1),

k=1, 2, , n.

ForΘk (x)≡C k, formula (17.2.3.3) yields the particular solution

y (x) = C1λ x

1+ C2λ x

2 +· · · + C n λ x

n, where C1, C2, , C nare arbitrary constants

2◦ There are k equal real roots: λ1= λ2=· · · = λ k (k≤m), and the other roots are real and mutually distinct In this case, the solution of the difference equation (17.2.3.1) is defined by

y=

Θ1(x) + xΘ2(x) + · · · + x k–1Θk (x)

λ x

1 +Θk+1(x)λ x k+1+Θk+2(x)λ x k+2+· · · + Θ m (x)λ x m (17.2.3.4)

3◦ There are k equal complex conjugate roots: λ = ρ(cos β i sin β) (2k≤m), and the other roots are real and mutually distinct In this case, forΘn (x)≡constnthe solution of the functional equation has the form

y = ρ x cos(βx)(A1+ A2x+· · · + A k x k–1) +

+ ρ x sin(βx)(B1+ B2x+· · · + B k x k–1) +

+ C2k+1λ x

2k+1+ C2k+2λ x

2k+2+· · · + C m λ x

m, where A1, , A k , B1, , B k , C2k+1, , C mare arbitrary constants In the general case, the arbitrary constants involved in this solution should be replaced by arbitrary1-periodic functions

4◦ In a similar way, one can consider the general situation.