Handbook of mathematics for engineers and scienteists part 131 ppsx

Nonhomogeneous linear equations.A second-order nonhomogeneous linear difference equation with constant coefficients has the form ay n+2+ by n+1+ cy n = f n.. Second-order homogeneous lin

O

O

O

y

y

y y

ξ

ξ

ξ

ξ

ξ

ξ

ξ

ξ

z

z

z

z

z=y

z=y

z = y

( )c

( )a

( )d

( )b

z=y

z=f y( )

z=f y( )

z=f y( )

z=f y( )

- <1 f ( ) < 0ξ

0<f ξ( )<1 1<f ξ( )

f ( ) ξ < -1

Figure 17.2 Iterative sequences y n+1 = f (y n ) in a neighborhood of a fixed point ξ = f (ξ) Qualitative analysis

of different cases using the graphs of the function z = f (y).

2◦ Let b2–4ac=0 Then the quadratic equation (17.1.3.2) has one double real root

λ= – b

2a, and the general solution of the difference equation (17.1.3.1) has the form

y n = C1(1– n)λ n + C2nλ n–1.

This formula can be obtained from (17.1.3.3) by taking λ1 = λ, λ2 = λ(1– ε) and passing

to the limit as ε →0

3◦ Let b2–4ac< 0 Then the quadratic equation (17.1.3.2) has two complex conjugate roots

λ1 = ρ(cos ϕ + i sin ϕ), λ2 = ρ(cos ϕ – i sin ϕ),

a, tan ϕ = –1

b

√

4ac – b2, and the general solution of the difference equation (17.1.3.1) has the form

y n = C1ρ n sin[(n –1)ϕ]

sin ϕ + C2ρ

n–1sin(nϕ)

sin ϕ . This formula can also be obtained from (17.1.3.3) by expressing λ1and λ2in terms of ρ and ϕ.

17.1.3-2 Nonhomogeneous linear equations.

A second-order nonhomogeneous linear difference equation with constant coefficients has the form

ay n+2+ by n+1+ cy n = f n. (17.1.3.4) The general solution of this equation is given by

y n = C1λ1λ

n

2 – λ n1λ2

λ1– λ2 + C2

λ n

1 – λ n2

λ1– λ2 +2y n, where

2y0=2y1=0, 2y n= 1

a

n–2



k=0

λ n–k–1

1 – λ n–k–2 1

λ1– λ2 f k,

C1 and C2 are arbitrary constants, and λ1, λ2 are the roots of the quadratic equation

(17.1.3.2)

17.1.3-3 Boundary value problem

The solution of the boundary value problem for equation*

ay n+1+ by n + cy n–1= f n, n=1, 2, , N –1, with the boundary conditions

y0= μ1, y N–1= μ2

is given by the formula

y n= (λ1λ2)

n (λ N–n

1 – λ N–n2 )

λ N

1 – λ N2

μ1+ λ

n

1 – λ n2

λ N

1 – λ N2

μ2

–1

a

n–1



k=1

(λ1λ2)n–k (λ N–n1 – λ N–n2 )(λ k1– λ k2)

(λ1– λ2)(λ N1 – λ N2 ) f k–

1

a

N–1



k=n

(λ N–k1 – λ N–k2 )(λ n1 – λ n2)

(λ1– λ2)(λ N1 – λ N2 ) f k.

17.1.4 Second-Order Linear Difference Equations with Variable

Coefficients

17.1.4-1 Second-order homogeneous linear difference equations General solution

1◦ A second-order homogeneous linear difference equation with variable coefficients has

the form

a n y n+2+ b n y n+1+ c n y n=0 (17.1.4.1)

The trivial solution y n=0is a particular solution of the homogeneous linear equation

Let y n(1), y(n2)be particular solutions of equation (17.1.4.1) satisfying the condition

y(1 )

0 y(12)– y0(2)y(11)≠ 0 (17.1.4.2) Then the general solution of equation (17.1.4.1) is given by

y n = C1y(1 )

n + C2y(2 )

where C1and C2are arbitrary constants

* This equation is obtained from (17.1.3.4) by shifting the subscript of the sought function by unity.

Remark. If condition (17.1.4.2) holds, then the solutions y(n1), y(n2)are linearly independent and for all

nthe following inequality holds: Δn = y( 1 )

n y(n+12) – y( 2 )

n y(n+11) ≠ 0 It is convenient to single out two linearly

independent solutions y( 1 )

n , y( 2 )

n with the help of the initial conditions

y0(1)= 1 , y(11)= 0 ; y0(2)= 0 , y(12)= 1

2◦ Let y ∗

nbe a nontrivial particular solution of equation (17.1.4.1) Then the replacement

yields the equation

a n y n+ ∗ 2u n+2+ b n y ∗ n+1u n+1+ c n y ∗ n u n=0 (17.1.4.5)

Taking into account that equation (17.1.4.1) holds for y ∗ n, and substituting

b n y ∗ n+1= –a n y ∗

n+2– c n y ∗

n

into (17.1.4.5), we find, after simple transformations, that

a n y ∗ n+2(u n+2– u n+1) – c n y n ∗ (u n+1– u n) =0 Introducing a new variable by

we come to the homogeneous first-order difference equation

a n y ∗ n+2w n+1– c n y ∗

n w n=0 Solving this equation (see Paragraph 17.1.1-1), one finds a solution of the nonhomogeneous first-order equation with constant coefficients (17.1.4.6) (see Paragraph 17.1.1-3), and then, using (17.1.4.4), one finds a solution of the original equation

17.1.4-2 Second-order nonhomogeneous linear equations General solution

1◦ A second-order nonhomogeneous linear difference equation with variable coefficients

has the form

a n y n+2+ b n y n+1+ c n y n = f n. (17.1.4.7) The general solution of the nonhomogeneous linear equation (17.1.4.7) can be rep-resented as a sum of the general solution (17.1.4.3) of the corresponding homogeneous equation (17.1.4.1) and a particular solution2y nof the nonhomogeneous equation (17.1.4.7):

y n = C1y(1 )

n + C2y n(2)+2y n, where

2y0=2y1=0, 2y n=

n–2



j=0

y(1 )

j+1y(n2)– y(n1)y(j+2)1

y(1 )

j+1y(j+2)2– y(j+1)2y j+(2)1

f j

a j, n=2, 3,

2◦ Let y ∗

nbe a nontrivial particular solution of the homogeneous equation (17.1.4.1) Then

the substitutions (17.1.4.4) and (17.1.4.6) yield a nonhomogeneous first-order difference equation

a n y n+ ∗ 2w n+1– c n y n ∗ w n = f n.

With regard to the solution of this equation see Paragraph 17.1.1-2

3◦ Superposition principle Let y( 1 )

n and y(n2)be solutions of two nonhomogeneous linear

difference equations with the same left-hand sides and different right-hand sides:

a n y n+2+ b n y n+1+ c n y n = f n,

a n y n+2+ b n y n+1+ c n y n = g n.

Then αy( 1 )

n + βy n(2)is a solution of the equation

a n y n+2+ b n y n+1+ c n y n = αf n + βg n,

where α and β are arbitrary constants.

17.1.5 Linear Difference Equations of Arbitrary Order with Constant

Coefficients

17.1.5-1 Homogeneous linear equations

An mth-order homogeneous linear difference equation with constant coefficients has the

general form

a m y n+m + a m–1y n+m–1+· · · + a1y n+1+ a0y n=0 (17.1.5.1) The general solution of this equation is determined by the roots of the characteristic equation

a m λ m + a m–1λ m–1+· · · + a1λ + a0 =0 (17.1.5.2) The following cases are possible:

1◦ All roots λ1, λ2, , λ

mof the characteristic equation (17.1.5.2) are real and distinct.

Then the general solution of the homogeneous linear differential equation (17.1.5.1) has the form

y n = C1λ n

1 + C2λ n

2 +· · · + C m λ n

m,

where C1, C2, , C mare arbitrary constants

2◦ There are k equal real roots λ1 = λ2 =· · · = λ k (k≤m), and the other roots are real

and distinct In this case, the general solution is given by

y n = (C1+ C2n+· · · + C k n k–1)λ n

1 + C k+1λ n

k+1+· · · + C m λ n

m.

3◦ There are k pairs of distinct complex conjugate roots λ j = ρ j (cos ϕ j i sin ϕ j

(j =1, , k; 2k≤m), and the other roots are real and distinct In this case, the general

solution is

y n = ρ n1[A1cos(nϕ1) + B1sin(nϕ1)] +· · · + ρ n k [A k cos(nϕ k ) + B k sin(nϕ k)]

+ C2k+1λ n

2k+1+· · · + C m λ n

m,

where A1, , A k , B1, , B k , C2k+1, , C m are arbitrary constants

4◦ In the general case, if there are r different roots λ1, λ2, , λ

r of multiplicities

k1, k2, , k r, respectively, the left-hand side of the characteristic equation (17.1.5.2) can

be represented as the product

P(λ) = a m (λ – λ1)k1 (λ – λ2)k2 (λ – λ r)kr,

where k1+ k2+· · · + k r = m The general solution of the original equation is given by the

formula

y n=

r



s=1

(C s,1+ C s,2n+· · · + C s,ks n ks–1)λ n

s,

where C s,kare arbitrary constants

If the characteristic equation (17.1.5.2) has complex conjugate roots of the form

λ = ρeiϕ = ρ(cos ϕ i sin ϕ) , then in the above solution, one should extract the real part

on the basis of the relation λ n = ρ n e inϕ = ρ n [cos(nϕ) i sin(nϕ)].

17.1.5-2 Nonhomogeneous linear equations.

1◦ An mth-order nonhomogeneous linear difference equation with constant coefficients

has the general form

a m y n+m + a m–1y n+m–1+· · · + a1y n+1+ a0y n = f n. (17.1.5.3)

The general solution of this equation can be represented as the sum y n = Y n+2y n, where

Y n is the general solution of the corresponding homogeneous equation (for f n≡ 0) and2y n

is any particular solution of the nonhomogeneous equation (17.1.5.3) The general solution

of the corresponding homogeneous equation is constructed with the help of the formulas from Paragraph 17.1.5-1, and a particular solution of the nonhomogeneous equation for an

arbitrary function f nis constructed with the help of the formulas from Paragraph 17.1.6-2

2◦ If the roots λ1, λ2, , λ

mof the characteristic equation (17.1.5.2) are mutually

dis-tinct, the particular solution of the nonhomogeneous difference equation (17.1.5.3) has the form

2y n=

n



ν=m

f n–ν

m



k=1

λ ν–1

k

where P (λ) is the characteristic polynomial [coinciding with the left-hand side of equation

(17.1.5.2)], and the prime indicates its derivative

P  (λ)≡a m mλ m–1+ a m–1(m –1)λ m–2+· · · +2a2λ + a1

In the case of complex conjugate roots, solution (17.1.5.4) should be split into the real and the imaginary parts

17.1.6 Linear Difference Equations of Arbitrary Order with Variable

Coefficients

17.1.6-1 Homogeneous linear difference equations

An mth-order homogeneous linear difference equation with variable coefficients has the

form

a m (n)y n+m + a m–1(n)y n+m–1+· · · + a1(n)y n+1+ a0(n)y n=0 (17.1.6.1)

Functions u( 1 )

n , u(n2), , u(m)n are called linearly independent solutions of equation

(17.1.6.1) if

1) they take finite values and satisfy equation (17.1.6.1),

2) the relation

C1u(1 )

n + C2u(2 )

n +· · · + C m u(m)

n =0, for all n=1,2, , with constants C1, C2, , C m implies that C1=· · · = C m=0

If u(n1), u(n2), , u(n m)are linearly independent solutions of equation (17.1.6.1), then the determinant









u(1 )

n u(n+1)1 · · · u( 1 )

n+m–1

u(2 )

n u(n+2)1 · · · u( 2 )

n+m–1

· · · ·

u(m)

n u(m)n+1 · · · u(m)n+m–1









(17.1.6.2)

differs from zero for all admissible n Conversely, if the determinant (17.1.6.2) for solutions

u(1 )

n , u(n2), , u(n m) of equation (17.1.6.1) differs from zero for some n, then the solutions

are linearly independent

If u(n1), u(n2), , u(n m)are linearly independent solutions of equation (17.1.6.1), then the general solution of this equation has the form

y n = C1u(1 )

n + C2u(2 )

n +· · · + C m u(m)

where C1, C2, , C mare arbitrary constants

A solution is determined by prescribing the initial values of the sought function at m

points

17.1.6-2 Nonhomogeneous linear difference equations

1◦ An mth-order nonhomogeneous linear difference equation with variable coefficients

has the form

a m (n)y n+m + a m–1(n)y n+m–1+· · · + a1(n)y n+1+ a0(n)y n = f n. (17.1.6.4) The general solution of this equation can be represented as a sum of the general solution

of the corresponding homogeneous equation (17.1.6.1) and a particular solution2y nof the nonhomogeneous equation (17.1.6.4):

y n = C1u(1 )

n + C2u(2 )

n +· · · + C m u(m)

n +2y n.

A particular solution can be determined by the formula

2y0=2y1=· · · = 2y m–1 =0, 2y n=

n–m



j=0

A m,n,j

B m,j

f j

a m (j), n = m, m +1, ,

where

A m,n,j =









u(1 )

j+1 u(j+2)1 · · · u(m)

j+1

· · · ·

u(1 )

j+m–1 u(j+m–2) 1 · · · u(m)

j+m–1

u(1 )

n u(n2) · · · u(m)

n









, B m,j =









u(1 )

j+1 u(j+1)2 · · · u( 1 )

j+m

· · · ·

u(m–1 )

j+1 u(j+ m–21) · · · u(m–1 )

j+m

u(m) j+1 u(j+ m)2 · · · u(m)

j+m







.

2◦ Superposition principle Let y( 1 )

n and y(n2)be solutions of two nonhomogeneous linear

difference equations with the same left-hand side and different right-hand sides:

a m (n)y n+m + a m–1(n)y n+m–1+· · · + a1(n)y n+1+ a0(n)y n = f n,

a m (n)y n+m + a m–1(n)y n+m–1+· · · + a1(n)y n+1+ a0(n)y n = g n.

Then αy n(1)+ βy n(2)is a solution of the equation

a m (n)y n+m + a m–1(n)y n+m–1+· · · + a1(n)y n+1+ a0(n)y n = αf n + βg n,

where α and β are arbitrary constants.

17.1.7 Nonlinear Difference Equations of Arbitrary Order

17.1.7-1 Difference equations of mth order General solution.

Let y n = y(n) be a function of integer argument n =0, 1, 2, An mth-order difference equation, in the general case, has the form

F (n, y n , y n+1, , y n+m) =0 (17.1.7.1)

A solution of the difference equation (17.1.7.1) is a discrete function y n that, being

substituted into the equation, turns it into identity The general solution of a difference equation is the set of all its solutions The general solution of equation (17.1.7.1) depends

on m arbitrary constants C1, , C m The general solution can be written in explicit form as

y n = ϕ(n, C1, , C m), (17.1.7.2)

or in implicit form as Φ(n, y n , C1, , C m) = 0 Specific values of C1, , C m define

specific solutions of the equation (particular solutions).

Any constant solution y n = ξ of equation (17.1.7.1), where ξ is independent of n, is called an equilibrium solution.

Remark. The term difference equation was introduced in numerical mathematics in connection with the

investigation of equations of the form

with finite differences*

Δy n = y n+1 – y n, Δ 2y = y n+2– 2y n+1 + y n, Δm y = Δm–1 Δy n (17 1 7 4 ) The replacement of the finite differences in (17.1.7.3), by their explicit expressions in terms of the values of the sought function according to (17.1.7.4), brings us to equations of the form (17.1.7.1).

17.1.7-2 Construction of a difference equation by a given general solution

Suppose that the general solution of a difference equation is given in the form (17.1.7.2)

Then the corresponding mth-order difference equation with this solution can be constructed

by eliminating the arbitrary constants C1, , C mfrom the relations

y n = ϕ(n, C1, , C m),

y n+1= ϕ(n +1, C1, , C m),

y n+m = ϕ(n + m, C1, , C m).

17.1.7-3 Cauchy’s problem and its solution The step method

A difference equation resolved with respect to the leading term y n+mhas the form

y n+m = f (n, y n , y n+1, , y n+m–1) (17.1.7.5) The Cauchy problem consists of finding a solution of this equation with given initial values

of y0, y1, , y m–1

* Finite differences are used for the approximation of derivatives in differential equations.