Handbook of mathematics for engineers and scienteists part 80 ppt

Asymptotic Solutions of Linear Equations This subsection presents asymptotic solutions, as ε → 0 ε > 0, of some higher-order linear ordinary differential equations containing arbitrary f

Example 1 Consider a special case of equation (12.4.2.7):

xy xx  + y  x + axy =0 (12 4 2 8 )

Denote y(0) = y0and y  x( 0) = y1 Let us apply the Laplace transform to this equation using formulas (12.4.2.6).

On rearrangement, we obtain a linear first-order equation for2y(p):

–(p22y – y0p – y1) p + (p 2y – y0) – a 2y 

p= 0 =⇒ (p2+ a) 2y 

p + p 2y =0 Its general solution is expressed as

2y = C

p2+ a, (12.4 2 9 )

where C is an arbitrary constant Applying the inverse Laplace transform to (12.4.2.9) and taking into account

formulas 19 and 20 from Subsection T3.2.3, we find a solution to the original equation (12.4.2.8):

y(x) =



CJ0(x √

a) if a >0 ,

CI0(x √ –a ) if a <0 , (12.4 2 10 )

where J0(x) is the Bessel function of the first kind and I0(x) is the modified Bessel function of the first kind.

In this case, only one solution (12.4.2.10) has been obtained This is due to the fact that the other solution

goes to infinity as x →0 , and hence formula (12.4.2.6) cannot be applied to it; this formula is only valid for finite initial values of the function and its derivatives.

12.4.2-7 Solution of equations using the Laplace integral

Solutions to linear differential equations with polynomial coefficients can sometimes be

represented as a Laplace integral in the form

y (x) =



K e

px u (p) dp. (12.4.2.11) For now, no assumptions are made about the domain of integrationK; it could be a segment

of the real axis or a curve in the complex plane

Let us exemplify the usage of the Laplace integral (12.4.2.11) by considering equation (12.4.2.7) It follows from (12.4.2.11) that

y(k)

x (x) =



K e

px p k u (p) dp,

xy(k)

x (x) =



K xe

px p k u (p) dp =*

e px p k u (p)+

K–



K e

px d

dp

*

p k u (p)+

dp

Substituting these expressions into (12.4.2.7) yields



K e

pxn

k=0

a k p k u (p) –

n



k=0

b k dp d

*

p k u (p)+

dp+

n



k=0

b k

*

e px p k u (p)+

K=0 (12.4.2.12) This equation is satisfied if the expression in braces vanishes, thus resulting in a linear

first-order ordinary differential equation for u(p):

u (p)

n



k=0

a k p k– dp d

*

u (p)

n



k=0

b k p k

+

=0 (12.4.2.13)

The remaining term in (12.4.2.12) must also vanish:

*n k=0

b k e px p k u (p)

+

K=0 (12.4.2.14) This condition can be met by appropriately selecting the path of integrationK Consider

the example below to illustrate the aforesaid

522 ORDINARYDIFFERENTIALEQUATIONS

Example 2 The linear variable-coefficient second-order equation

xy  xx + (x + a + b)y  x + ay =0 (a >0, b >0 ) (12 4 2 15 )

is a special case of equation (12.4.2.7) with n =2, a2 = 0, a1 = a + b, a0 = a, b2 = b1 = 1, and b0 = 0 On

substituting these values into (12.4.2.13), we arrive at an equation for u(p):

p(p +1)u  p – [(a + b –2)p + a –1]u =0 Its solution is given by

u(p) = p a–1(p +1 )b–1 (12 4 2 16 )

It follows from condition (12.4.2.14), in view of formula (12.4.2.16), that

*

e px (p + p2)u(p)+β

α= *

e px p a (p +1 )b+β

α= 0 , (12 4 2 17 ) where a segment of the real axis,K = [α, β], has been chosen to be the path of integration Condition (12.4.2.17)

is satisfied if we set α = –1and β =0 Consequently, one of the solutions to equation (12.4.2.15) has the form

y(x) =

 0 – 1

e px p a–1(p +1 )b–1dp. (12 4 2 18 ) Remark 1. If a is noninteger, it is necessary to separate the real and imaginary parts in (12.4.2.18) to

obtain real solutions.

Remark 2. By setting α = – ∞ and β =0 in (12.4.2.17), one can find a second solution to equation

(12.4.2.15) (at least for x >0 ).

12.4.3 Asymptotic Solutions of Linear Equations

This subsection presents asymptotic solutions, as ε → 0 (ε > 0), of some higher-order linear ordinary differential equations containing arbitrary functions (sufficiently smooth), with the independent variable being real

12.4.3-1 Fourth-order linear equations

1◦ Consider the equation

ε4y 

xxxx – f (x)y =0

on a closed interval a≤x≤b With the condition f >0, the leading terms of the asymptotic

expansions of the fundamental system of solutions, as ε →0, are given by the formulas

y1 = [f (x)]–3 8exp

 –1

ε



[f (x)]1 4dx

 , y2= [f (x)]–3 8exp

ε



[f (x)]1 4dx

 ,

y3 = [f (x)]–3 8cos

ε



[f (x)]1 4dx

 , y4= [f (x)]–3 8sin

ε



[f (x)]1 4dx

2◦ Now consider the “biquadratic” equation

ε4y 

xxxx–2ε2g (x)y 

xx – f (x)y =0 (12.4.3.1) Introduce the notation

D (x) = [g(x)]2+ f (x).

In the range where the conditions f (x)≠ 0 and D(x)≠ 0 are satisfied, the leading terms

of the asymptotic expansions of the fundamental system of solutions of equation (12.4.3.1) are described by the formulas

y k = [λ k (x)]–1 2[D(x)]–1 4exp



1

ε



λ k (x) dx – 1

2



[λ k (x)]  x

√

D (x) dx

; k=1, 2, 3, 4, where

λ1(x) =

g (x) + √

D (x), λ2(x) = –

g (x) + √

D (x),

λ3(x) =

g (x) – √

D (x), λ4(x) = –

g (x) – √

D (x).

12.4.3-2 Higher-order linear equations.

1◦ Consider an equation of the form

ε n y(n)

x – f (x)y =0

on a closed interval a≤x≤b Assume that f≠ 0 Then the leading terms of the asymptotic

expansions of the fundamental system of solutions, as ε →0, are given by

y m=

f (x)–1

2 + 21nexp



ω m

ε

 

f (x)1

n dx

1+ O(ε)

,

where ω1, ω2, , ω n are roots of the equation ω n=1:

ω m= cos

n



+ i sin2πm

n

 , m=1, 2, , n.

2◦ Now consider an equation of the form

ε n y(n)

x + ε n–1f n–1(x)y(x n–1)+· · · + εf1 (x)y x  + f0(x)y =0 (12.4.3.2)

on a closed interval a ≤ x ≤ b Let λ m = λ m (x) (m = 1,2, , n) be the roots of the

characteristic equation

P (x, λ)≡λ n + f

n–1(x)λ n–1+· · · + f1 (x)λ + f0(x) =0

Let all the roots of the characteristic equation be different on the interval a≤x≤b, i.e., the

conditions λ m (x) ≠λ k (x), m ≠k, are satisfied, which is equivalent to the fulfillment of

the conditions P λ (x, λ m)≠ 0 Then the leading terms of the asymptotic expansions of the

fundamental system of solutions of equation (12.4.3.2), as ε →0, are given by

y m= exp

ε



λ m (x) dx – 1

2



[λ m (x)]  x P λλ x , λ m (x)

P λ x , λ m (x) dx

 , where

P λ (x, λ)≡ ∂P

∂λ = nλ n–1+ (n –1)f n–1(x)λ n–2+· · · +2λf2(x) + f1(x),

P λλ (x, λ)≡ ∂2P

∂λ2 = n(n –1)λ n–2+ (n –1)(n –2)f n–1(x)λ n–3+· · · +6λf3(x) +2f2(x).

12.4.4 Collocation Method and Its Convergence

12.4.4-1 Statement of the problem Approximate solution

1◦ Consider the linear boundary value problem defined by the equation

Ly≡y(n)

x + f n–1(x)y x(n–1)+· · · + f1 (x)y  x + f0(x)y = g(x), –1< x <1, (12.4.4.1) and the boundary conditions

n–1



j=0



α ij y(j)x (–1) + β ij y(j)

x (1)

=0, i=1, , n. (12.4.4.2)

2◦ We seek an approximate solution to problem (12.4.4.1)–(12.4.4.2) in the form

y m (x) = A1ϕ1(x) + A2ϕ2(x) + · · · + A m ϕ m (x),

where ϕ k (x) is a polynomial of degree n + k –1 that satisfies the boundary conditions

(12.4.4.2) The coefficients A kare determined by the linear system of algebraic equations

Ly m – g(x)

x=x i =0, i=1, , m, (12.4.4.3)

with Chebyshev nodes x i = cos

2i–1

2m π



, i =1, , m.

524 ORDINARYDIFFERENTIALEQUATIONS

12.4.4-2 Convergence theorem for the collocation method

THEOREM Let the functions f j (x) (j = 0, , n –1) and g(x) be continuous on the

interval [–1,1] and let the boundary value problem (12.4.4.1)–(12.4.4.2) have a unique

solution, y(x) Then there exists an m0such that system (12.4.4.3) is uniquely solvable for

m≥m0; and the limit relations

max

– 1≤x≤1y(k)

m (x) – y(k) (x)  →0, k=0,1, , n –1;

– 1

y(n)

m (x) – y(n) (x)2

√

1– x2 dx

1 2

hold for m → ∞.

Remark A similar result holds true if the nodes are roots of some orthogonal polynomials with some weight function If the nodes are equidistant, the method diverges.

12.5 Nonlinear Equations of Arbitrary Order

12.5.1 Structure of the General Solution Cauchy Problem

12.5.1-1 Equations solved for the highest derivative General solution

An nth-order differential equation solved for the highest derivative has the form

y(n)

x = f (x, y, y x  , , y x(n–1)). (12.5.1.1)

The general solution of this equation depends on n arbitrary constants C1, , C n In some cases, the general solution can be written in explicit form as

y = ϕ(x, C1, , C n) (12.5.1.2)

12.5.1-2 Cauchy problem The existence and uniqueness theorem

The Cauchy problem: find a solution of equation (12.5.1.1) with the initial conditions

y (x0) = y0, y 

x (x0) = y(01), ., y(n–1 )

x (x0) = y(0n–1). (12.5.1.3)

(At a point x0, the values of the unknown function y(x) and all its derivatives of orders

≤n–1are prescribed.)

EXISTENCE AND UNIQUENESS THEOREM Suppose the function f (x, y, z1, , z n–1) is

continuous in all its arguments in a neighborhood of the point (x0, y0, y(01), , y(0n–1))

and has bounded derivatives with respect to y, z1, , z n–1 in this neighborhood Then

a solution of equation (12.5.1.1) satisfying the initial conditions (12.5.1.3) exists and is unique

12.5.1-3 Construction of a differential equation by a given general solution.

Suppose a general solution (12.5.1.2) of an unknown nth-order ordinary differential equation

is given The equation corresponding to the general solution can be obtained by eliminating

the arbitrary constants C1, , C nfrom the identities

y = ϕ(x, C1, , C n),

y 

x = ϕ  x (x, C1, , C n),

y(n)

x = ϕ(x n) (x, C1, , C n),

obtained by differentiation from formula (12.5.1.2)

12.5.1-4 Reduction of an nth-order equation to a system of n first-order equation The differential equation (12.5.1.1) is equivalent to the following system of n first-order

equations:

y 

0= y1, y 

1= y2, ., y 

n–2= y n–1, y 

n–1 = f (x, y0, y1, , y n–1),

where the notation y0≡yis adopted

12.5.2 Equations Admitting Reduction of Order

12.5.2-1 Equations not containing y, y  x , , y(x k) explicitly

An equation that does not explicitly contain the unknown function and its derivatives up to

order k inclusive can generally be written as

F x , y x(k+1), , y(x n)

=0 (1 ≤k+1< n). (12.5.2.1)

Such equations are invariant under arbitrary translations of the unknown function, y →

y + const (the form of such equations is also preserved under the transformation u(x) =

y +a k x k+· · ·+a1 x +a0, where the a m are arbitrary constants) The substitution z(x) = y(k+1)

x

reduces (12.5.2.1) to an equation whose order is by k +1smaller than that of the original

equation, F x , z, z x  , , z(n–k–1)

=0

12.5.2-2 Equations not containing x explicitly (autonomous equations).

An equation that does not explicitly contain x has in the general form

F y , y x  , , y(n)x

Such equations are invariant under arbitrary translations of the independent variable, x →

x + const The substitution y x  = w(y) (where y plays the role of the independent variable)

reduces by one the order of an autonomous equation Higher derivatives can be expressed

in terms of w and its derivatives with respect to the new independent variable, y xx  = ww  y,

y 

xxx = w2w  yy + w(w  y)2,

526 ORDINARYDIFFERENTIALEQUATIONS

12.5.2-3 Equations of the form F ax + by, y x  , , y(n)x

=0 Such equations are invariant under simultaneous translations of the independent variable

and the unknown function, x → x + bc and y → y – ac, where c is an arbitrary constant For b =0, see equation (12.5.2.1) For b≠ 0, the substitution w(x) = y + (a/b)x leads

to an autonomous equation of the form (12.5.2.2)

12.5.2-4 Equations of the form F x , xy x  – y, y  xx , , y x(n)

=0

The substitution w(x) = xy  x – y reduces the order of this equation by one.

This equation is a special case of the equation

F x , xy  x – my, y x(m+1), , y(n)x

=0, where m=1, 2, , n –1 (12.5.2.3)

The substitution w(x) = xy  x – my reduces by one the order of equation (12.5.2.3).

12.5.2-5 Homogeneous equations

1◦ Equations homogeneous in the independent variable are invariant under scaling of the

independent variable, x → αx , where α is an arbitrary constant (α≠ 0) In general, such equations can be written in the form

F y , xy x  , x2y 

xx , , x n y x(n)

=0

The substitution z(y) = xy x  reduces by one the order of this equation

2◦ Equations homogeneous in the unknown function are invariant under scaling of the

unknown function, y → αy, where α is an arbitrary constant (α≠ 0) Such equations can

be written in the general form

F x , y  x /y , y xx  /y , , y x(n)/y

=0

The substitution z(x) = y  x /y reduces by one the order of this equation

3◦ Equations homogeneous in both variables are invariant under simultaneous scaling

(dilatation) of the independent and dependent variables, x → αx and y → αy, where α

is an arbitrary constant (α≠ 0) Such equations can be written in the general form

F y/x , y  x , xy  xx , , x n–1y(n)

x

=0

The transformation t = ln|x|, w = y/x leads to an autonomous equation considered in

Paragraph 12.5.2-2

12.5.2-6 Generalized homogeneous equations

1◦ Generalized homogeneous equations (equations homogeneous in the generalized sense)

are invariant under simultaneous scaling of the independent variable and the unknown

function, x → αx and y → α k y , where α≠ 0 is an arbitrary constant and k is a given

number Such equations can be written in the general form

F x–k y , x1 –k y 

x , , x n–k y x(n)

=0

The transformation t = ln x, w = x–ky leads to an autonomous equation considered in Paragraph 12.5.2-2

2◦ The most general form of generalized homogeneous equations is

F x n y m , xy 

x /y , , x n y x(n)/y

=0

The transformation z = x n y m , u = xy 

x /y reduces the order of this equation by one.

12.5.2-7 Equations of the form F e λx y n , y 

x /y , y xx  /y , , y(x n) /y

=0 Such equations are invariant under simultaneous translation and scaling of variables,

x → x + α and y → βy, where β = exp(–αλ/n) and α is an arbitrary constant The transformation z = e λx y n , w = y 

x /y leads to an equation of order n –1

12.5.2-8 Equations of the form F x n e λy , xy 

x , x2y xx  , , x n y(x n)

=0

Such equations are invariant under simultaneous scaling and translation of variables, x →

αx and y →y+β, where α=exp(–βλ/n) and β is an arbitrary constant The transformation

z = x n e λy , w = xy 

x leads to an equation of order n –1 12.5.2-9 Other equations

Consider the nonlinear differential equation

F x, L1[y], , L k [y]

where the Ls [y] are linear homogeneous differential forms,

Ls [y] =

n s



m=0

ϕ(s)

m (x)y(x m), s=1, , k.

Let y0= y0(x) be a common particular solution of the linear equations

Ls [y0] =0 (s =1, , k).

Then the substitution

w = ϕ(x)

y0(x)y x  – y0 (x)y

(12.5.2.5)

with an arbitrary function ϕ(x) reduces by one the order of equation (12.5.2.4).

Example Consider the third-order equation

xy xxx  = f (xy x  – 2y).

It can be represented in the form (12.5.2.4) with

k= 2 , F(x, u, w) = xu – f (w), L1[y] = y  xxx, L2[y] = xy x  – 2y.

The linear equations Lk [y] =0 are

y  xxx= 0 , xy x  – 2y= 0

These equations have a common particular solution y0 = x2 Therefore, the substitution w = xy x  – 2y (see

formula (12.5.2.5) with ϕ(x) =1/x) leads to a second-order autonomous equation: w xx  = f (w) For the

solution of this equation, see Example 1 in Subsection 12.3.2.

12.4.2-7 Solution of equations... (x)≠ and D(x)≠ are satisfied, the leading terms

of the asymptotic expansions of the fundamental system of solutions of equation (12.4.3.1) are described by the formulas

y... fulfillment of

the conditions P λ (x, λ m)≠ Then the leading terms of the asymptotic expansions of the

fundamental system of solutions of equation