HANDBOOK OFINTEGRAL EQUATIONS phần 3 potx

The functionsu1x and u2x are expressed in terms of Bessel functions or modified Bessel functions, depending on the sign ofAλ, as follows: whereu1x, u2x is a fundamental system of solutio

30. y(x) + A

 x

0 (x 2 –t2 )eλ(x–t) y(t) dt = f (x).

The substitutionu(x) = e–λx y(x) leads to an equation of the form 2.1.11:

R(x) = 1

n + 1 e λx n

σ k=|An!| n+11 cos 2πk

n + 1

, β k=|An!| n+11 sin 2πk

n + 1

for A > 0.

exp[πb2(x – t)]F (t) dt

,where

–λt µ.Solution:

y(x) = f (x) – A

 x a

y(t) dt = g(x).

This is a special case of equation 2.9.71 withf (z) = ke–λz

For a polynomial right-hand side,g(x) =

f (t) dt.

cosh(λt) exp

A

λ

sinh(λx) – sinh(λt)

This is a special case of equation 2.9.28 withg(t) = A Therefore, solving the original integral

equation is reduced to solving the second-order linear nonhomogeneous ordinary differential

equation with constant coefficients

y xx+Ay x  –λ2y = f xx  –λ2f , f = f (x),

under the initial conditions

y(a) = f (a), y x (a) = f x (a) – Af (a).

Solution:

y(x) = f (x) +

 x a R(x – t)f (t) dt, R(x) = exp

–1

2Ax  A2

2k sinh(kx) – A cosh(kx)

, k =

Solution:

y(x) = f (x) + A

 x a

Solution:

y(x) = f (x) + A

 x a

coshk(λx) coshm(µt)y(t) dt = f (x).

This is a special case of equation 2.9.2 withg(x) = A cosh k(λx) and h(t) = cosh m(µt).

2.3-2 Kernels Containing Hyperbolic Sine

sinh(λx) exp



A λ

cosh(λx) – cosh(λt)

f (t) dt.

sinh(λt) exp



A λ

cosh(λx) – cosh(λt)

This is a special case of equation 2.9.30 withg(x) = A.

1◦ Solution withλ(A – λ) > 0:

y(x) = f (x) – Aλ

k

 x a

sinh[λ1(x – t)]y(t) dt, I2=

 x a

Eliminating I1 andI2 from (1), (3), and (5), we arrive at a fourth-order linear ordinary

differential equation with constant coefficients:

y xxxx  – (λ21+λ22–A1λ1–A2λ2)y xx+ (λ21λ22–A1λ1λ22–A2λ21λ2)y =

f xxxx  – (λ21+λ22)f xx+λ21λ22f (6)

The initial conditions can be obtained by settingx = a in (1)–(4):

y(a) = f (a), y  x(a) = f x (a),

Assume that the discriminant of equation (8) is positive:

Depending on the signs ofz1andz2the following three cases are possible

Case 1 If z1 > 0 andz2 > 0, then the solution of the integral equation has the form

(i = 1, 2):

y(x) = f (x) +

 x a {B1sinh[µ1(x – t)] + B2sinh

µ2(x – t)

f (t) dt, µ i =√

z i,where

µ2(x – t)

f (t) dt, µ i=

|z i|,where the coefficientsB1andB2are found by solving the following system of linear algebraic

µ2(x – t)

f (t) dt, µ i=

|z i|,whereB1andB2are determined from the following system of linear algebraic equations:

19. y(x) +

 x

a

n k=1

By reducing it to a common denominator, we arrive at the problem of determining the roots

of annth-degree characteristic polynomial.

Assume that allz kare real, different, and nonzero Let us divide the roots into two groups

In the case of a nonzero rootz s = 0, we can introduce the new constantD = B s µ s and

proceed to the limitµ s → 0 As a result, the term D(x – t) appears in solution (2) instead of

Solution:

y(x) = f (x) + A

 x a

sinhk(λx) sinhm(µt)y(t) dt = f (x).

This is a special case of equation 2.9.2 withg(x) = A sinh k(λx) and h(t) = sinh m(µt).

whereu1(x), u2(x) is a fundamental system of solutions of the second-order linear ordinary

differential equationu  xx+λ(Ax – λ)u = 0, and W is the Wronskian.

The functionsu1(x) and u2(x) are expressed in terms of Bessel functions or modified

Bessel functions, depending on the sign ofAλ, as follows:

whereu1(x), u2(x) is a fundamental system of solutions of the second-order linear ordinary

differential equationu  xx+λ(Ax – λ)u = 0, and W is the Wronskian.

The functionsu1(x), u2(x), and W are specified in 2.3.23.

2.3-3 Kernels Containing Hyperbolic Tangent

tanh(λx)

cosh(λx)

tanh(λt)

cosh(λx)



Y1(x)Y2(t) – Y2(x)Y1(t)

f (t) dt,

whereY1(x), Y2(x) is a fundamental system of solutions of the second-order linear ordinary

differential equation cosh2(λx)Y xx  +AλY = 0, W is the Wronskian, and the primes stand for

the differentiation with respect to the argument specified in the parentheses

As shown in A D Polyanin and V F Zaitsev (1996), the functionsY1(x) and Y2(x) can

be represented in the form

Y1(x) = F α, β, 1; e

λx

1 +e λx

, Y2(x) = Y1(x)

 x

a

dξ

Y2(ξ), W = 1,

whereF (α, β, γ; z) is the hypergeometric function, in which α and β are determined from

the algebraic systemα + β = 1, αβ = –A/λ.

33. y(x) – A

 x

a

tanh(λx) tanh(λt)y(t) dt = f (x).

tanhk(λx) tanhm(µt)y(t) dt = f (x).

This is a special case of equation 2.9.2 withg(x) = A tanh k(λx) and h(t) = tanh m(µt).

This is a special case of equation 2.9.8 withλ = B and g(t) = A tanh(kt).

2.3-4 Kernels Containing Hyperbolic Cotangent

A/λ

f (t) dt.

y(x) = f (x) + A

 x a

cothk(λx) cothm(µt)y(t) dt = f (x).

This is a special case of equation 2.9.2 withg(x) = A coth k(λx) and h(t) = coth m(µt).

2.3-5 Kernels Containing Combinations of Hyperbolic Functions

53. y(x) – A

 x

a

coshk(λx) sinhm(µt)y(t) dt = f (x).

This is a special case of equation 2.9.2 withg(x) = A cosh k(λx) and h(t) = sinh m(µt).

tanhk(λx) cothm(µt)y(t) dt = f (x).

This is a special case of equation 2.9.2 withg(x) = A tanh k(λx) and h(t) = coth m(µt).

2.4 Equations Whose Kernels Contain Logarithmic



u 1(x)u 2(t) – u 2(x)u 1(t)

f (t) dt,

where the primes denote differentiation with respect to the argument specified in the

paren-theses; and u1(x), u2(x) is a fundamental system of solutions of the second-order linear

homogeneous ordinary differential equationu  xx+Ax–1u = 0, with u1(x) and u2(x) expressed

in terms of Bessel functions or modified Bessel functions, depending on the sign ofA:

4. y(x) – A

 x

a

ln(λx) ln(λt)y(t) dt = f (x).

Solution:

y(x) = f (x) + A

 x a

whereC = 0.5772 is the Euler constant.

2.4-2 Kernels Containing Power-Law and Logarithmic Functions

1 + 12 + 13 +· · · + 1

n – lnλ k–C,whereC = 0.5772 is the Euler constant.

2.5 Equations Whose Kernels Contain Trigonometric

cos(λx) exp

A

λ

sin(λx) – sin(λt)

f (t) dt.

3. y(x) + A

 x

a

cos[λ(x – t)]y(t) dt = f (x).

This is a special case of equation 2.9.34 with g(t) = A Therefore, solving this integral

equation is reduced to solving the following second-order linear nonhomogeneous ordinary

differential equation with constant coefficients:

y xx+Ay  x+λ2y = f xx  +λ2f , f = f (x),

with the initial conditions

y(a) = f (a), y x (a) = f x (a) – Af (a).

1◦ Solution with|A| > 2|λ|:

y(x) = f (x) +

 x

a R(x – t)f (t) dt, R(x) = exp

–1

2Ax  A2

2k sinh(kx) – A cosh(kx)

, k =

A kcos[λk(x – t)]



y(t) dt = f (x).

This integral equation is reduced to a linear nonhomogeneous ordinary differential equation

of order 2n with constant coefficients Set

sin[λ k(x – t)]y(t) dt,

I k =y x (x) – λ2

k

 x a

With the aid of (1), the integral equation can be rewritten in the form

y(x) + n

Differentiating (6) with respect tox twice followed by eliminating I n–1from the resulting

expression with the aid of (6) yields a similar equation whose left-hand side is a

fourth-order differential operator (acting on y) with constant coefficients plus the sum

n–2

k=1

B k I k.Successively eliminating the termsI n–2,I n–3, using double differentiation and formula (3),

we finally arrive at a linear nonhomogeneous ordinary differential equation of order 2n with

constant coefficients

The initial conditions fory(x) can be obtained by setting x = a in the integral equation

and all its derivative equations

5. y(x) – A

 x

a

cos(λx) cos(λt)y(t) dt = f (x).

Solution:

y(x) = f (x) + A

 x a

Solution:

y(x) = f (x) + A

 x a

cosk(λx) cosm(µt)y(t) dt = f (x).

This is a special case of equation 2.9.2 withg(x) = A cos k(λx) and h(t) = cos m(µt).

2.5-2 Kernels Containing Sine

f (t) dt.

16. y(x) + A

 x

a

sin[λ(x – t)]y(t) dt = f (x).

This is a special case of equation 2.9.36 withg(t) = A.

1◦ Solution withλ(A + λ) > 0:

y(x) = f (x) – Aλ

k

 x a

This equation can be solved by the same method as equation 2.3.18, by reducing it to a

fourth-order linear ordinary differential equation with constant coefficients

Consider the characteristic equation

z2+ (λ21+λ22+A1λ1+A2λ2)z + λ21λ22+A1λ1λ22+A2λ21λ2= 0, (1)whose roots,z1andz2, determine the solution structure of the integral equation

Assume that the discriminant of equation (1) is positive:

Depending on the signs ofz1andz2the following three cases are possible

Case 1 If z1 > 0 andz2 > 0, then the solution of the integral equation has the form

(i = 1, 2):

y(x) = f (x) +

 x

a {B1sinh[µ1(x – t)] + B2sinh

Case 2 If z1< 0 andz2< 0, then the solution of the integral equation has the form

y(x) = f (x) +

 x a {B1sin[µ1(x – t)] + B2sin

µ2(x – t)

f (t) dt, µ i=

|z i|,whereB1andB2are determined from the system

B1µ1

λ2

1–µ2 1

+ B2µ2

λ2

1–µ2 2

– 1 = 0, B1µ1

λ2

2–µ2 1

+ B2µ2

λ2

2–µ2 2

– 1 = 0

Case 3 If z1> 0 andz2< 0, then the solution of the integral equation has the form

y(x) = f (x) +

 x a {B1sinh[µ1(x – t)] + B2sin

µ2(x – t)

f (t) dt, µ i=

|z i|,whereB1andB2are determined from the system

B1µ1

λ2

1+µ2 1

+ B2µ2

λ2

1–µ2 2

– 1 = 0, B1µ1

λ2

2+µ2 1

+ B2µ2

λ2

2–µ2 2

– 1 = 0

Remark The solution of the original integral equation can be obtained from the solution

of equation 2.3.18 by performing the following change of parameters:

λ k → iλ k, µ k → iµ k, A k → –iA k, B k → –iB k, i2= –1 (k = 1, 2).

19. y(x) +

 x

a

n k=1

A ksin[λk(x – t)]



y(t) dt = f (x).

1◦ This integral equation can be reduced to a linear nonhomogeneous ordinary differential

equation of order 2n with constant coefficients Set

I k(x) =

 x a

y(x) + n

Eliminating the integralI nfrom (4) and (5), we obtain

Differentiating (6) with respect tox twice followed by eliminating I n–1from the resulting

expression with the aid of (6) yields a similar equation whose left-hand side is a

fourth-order differential operator (acting on y) with constant coefficients plus the sum

n–2

k=1

B k I k.Successively eliminating the termsI n–2,I n–3, using double differentiation and formula (3),

we finally arrive at a linear nonhomogeneous ordinary differential equation of order 2n with

constant coefficients

The initial conditions fory(x) can be obtained by setting x = a in the integral equation

and all its derivative equations

2◦ Let us find the rootsz kof the algebraic equation

By reducing it to a common denominator, we arrive at the problem of determining the roots

of annth-degree characteristic polynomial.

Assume that allz kare real, different, and nonzero Let us divide the roots into two groups

s k=1

In the case of a nonzero rootz s = 0, we can introduce the new constantD = B s µ s and

proceed to the limitµ s → 0 As a result, the term D(x – t) appears in solution (8) instead of

Solution:

y(x) = f (x) + A

 x a

22. y(x) – A

 x

a

sink(λx) sinm(µt)y(t) dt = f (x).

This is a special case of equation 2.9.2 withg(x) = A sin k(λx) and h(t) = sin m(µt).

whereu1(x), u2(x) is a fundamental system of solutions of the second-order linear ordinary

differential equationu  xx+λ(Ax + λ)u = 0, and W is the Wronskian.

Depending on the sign ofAλ, the functions u1(x) and u2(x) are expressed in terms of

Bessel functions or modified Bessel functions as follows:

x

u1(x)u2(t) – u2(x)u1(t)

f (t) dt,

whereu1(x), u2(x) is a fundamental system of solutions of the second-order linear ordinary

differential equationu  xx+λ(Ax + λ)u = 0, and W is the Wronskian.

The functionsu1(x), u2(x), and W are specified in 2.5.23.

e B(x–s) G(s) ds, G(x) = exp

–A

k cos(kx)



e B(t–s) G(s) ds, G(x) = exp

–A

k cos(kx)



2.5-3 Kernels Containing Tangent



Y1(x)Y2(t) – Y2(x)Y1(t)

f (t) dt,

whereY1(x), Y2(x) is a fundamental system of solutions of the second-order linear ordinary

differential equation cos2(λx)Y xx  +AλY = 0, W is the Wronskian, and the primes stand for

the differentiation with respect to the argument specified in the parentheses

As shown in A D Polyanin and V F Zaitsev (1995, 1996), the functionsY1(x) and Y2(x)

can be expressed via the hypergeometric function

33. y(x) – A

 x

a

tan(λx) tan(λt)y(t) dt = f (x).

34. y(x) – A

 x

a

tan(λt) tan(λx)y(t) dt = f (x).

tank(λx) tanm(µt)y(t) dt = f (x).

This is a special case of equation 2.9.2 withg(x) = A tan k(λx) and h(t) = tan m(µt).

This is a special case of equation 2.9.8 withλ = B and g(t) = A tan(kt).

2.5-4 Kernels Containing Cotangent

43. y(x) – A

 x

a

cot(λt) cot(λx)y(t) dt = f (x).

This is a special case of equation 2.9.8 withλ = B and g(t) = A cot(kt).

2.5-5 Kernels Containing Combinations of Trigonometric Functions

48. y(x) – A

 x

a

cosk(λx) sinm(µt)y(t) dt = f (x).

This is a special case of equation 2.9.2 withg(x) = A cos k(λx) and h(t) = sin m(µt).

tank(λx) cotm(µt)y(t) dt = f (x).

This is a special case of equation 2.9.2 withg(x) = A tan k(λx) and h(t) = cot m(µt).

2.6 Equations Whose Kernels Contain Inverse

y(x) = f (x) + A

 x a

Solution:

y(x) = f (x) + A

 x a

This is a special case of equation 2.9.8 withλ = B and g(t) = A arccos(kt).

2.6-2 Kernels Containing Arcsine

This is a special case of equation 2.9.8 withλ = B and g(t) = A arcsin(kt).

2.6-3 Kernels Containing Arctangent

Solution:

y(x) = f (x) + A

 x a

This is a special case of equation 2.9.8 withλ = B and g(t) = A arctan(kt).

2.6-4 Kernels Containing Arccotangent

This is a special case of equation 2.9.8 withλ = B and g(t) = A arccot(kt).

2.7 Equations Whose Kernels Contain Combinations of

(µ –12A)x  A2

2k sinh(kx) – A cosh(kx)

, k =

λ2+ 14A2

e µ(x–t)

u 1(x)u 2(t) – u 2(x)u 1(t)

f (t) dt,

where the primes stand for the differentiation with respect to the argument specified in the

parentheses, andu1(x), u2(x) is a fundamental system of solutions of the second-order linear

homogeneous ordinary differential equationu  xx+Ax–1u = 0, with u1(x) and u2(x) expressed

in terms of Bessel functions or modified Bessel functions, depending on the sign ofA:

W = π1, u1(x) = √

x J1

2√

Ax, u2(x) = √

This is a special case of equation 2.9.62 withK(x) = ae λxln(–x).

2.7-3 Kernels Containing Exponential and Trigonometric Functions

(µ – 12A)x  A2

2k sinh(kx) – A cosh(kx)

, k =

(µ –1

2A)x  A2

2k sin(kx) – A cos(kx)

, k =

n k=1

te µ(x–t)

u1(x)u2(t) – u2(x)u1(t)

f (t) dt,

whereu1(x), u2(x) is a fundamental system of solutions of the second-order linear ordinary

differential equationu  xx+λ(Ax + λ)u = 0, and W is the Wronskian.

Depending on the sign ofAλ, the functions u1(x) and u2(x) are expressed in terms of

Bessel functions or modified Bessel functions as follows:

xe µ(x–t)

u1(x)u2(t) – u2(x)u1(t)

f (t) dt,

whereu1(x), u2(x) is a fundamental system of solutions of the second-order linear ordinary

differential equationu  xx+λ(Ax + λ)u = 0, and W is the Wronskian.

The functionsu1(x), u2(x), and W are specified in 2.7.22.

e B(x–s) G(s) ds, G(x) = exp

–A

k cos(kx)



k cos(kx)



This is a special case of equation 2.9.2 withg(x) = Ae µxandh(t) = cot(λt).

2.7-4 Kernels Containing Hyperbolic and Logarithmic Functions

2.7-5 Kernels Containing Hyperbolic and Trigonometric Functions

42. y(x) – A

 x

a

coshk(λx) cosm(µt)y(t) dt = f (x).

This is a special case of equation 2.9.2 withg(x) = A cosh k(λx) and h(t) = cos m(µt).

43. y(x) – A

 x

a

coshk(λt) cosm(µx)y(t) dt = f (x).

This is a special case of equation 2.9.2 withg(x) = A cos m(µx) and h(t) = cosh k(λt).

44. y(x) – A

 x

a

coshk(λx) sinm(µt)y(t) dt = f (x).

This is a special case of equation 2.9.2 withg(x) = A cosh k(λx) and h(t) = sin m(µt).

45. y(x) – A

 x

a

coshk(λt) sinm(µx)y(t) dt = f (x).

This is a special case of equation 2.9.2 withg(x) = A sin m(µx) and h(t) = cosh k(λt).

46. y(x) – A

 x

a

sinhk(λx) cosm(µt)y(t) dt = f (x).

This is a special case of equation 2.9.2 withg(x) = A sinh k(λx) and h(t) = cos m(µt).

47. y(x) – A

 x

a

sinhk(λt) cosm(µx)y(t) dt = f (x).

This is a special case of equation 2.9.2 withg(x) = A cos m(µx) and h(t) = sinh k(λt).

48. y(x) – A

 x

a

sinhk(λx) sinm(µt)y(t) dt = f (x).

This is a special case of equation 2.9.2 withg(x) = A sinh k(λx) and h(t) = sin m(µt).

49. y(x) – A

 x

a

sinhk(λt) sinm(µx)y(t) dt = f (x).

This is a special case of equation 2.9.2 withg(x) = A sin m(µx) and h(t) = sinh k(λt).

50. y(x) – A

 x

a

tanhk(λx) cosm(µt)y(t) dt = f (x).

This is a special case of equation 2.9.2 withg(x) = A tanh k(λx) and h(t) = cos m(µt).

51. y(x) – A

 x

a

tanhk(λt) cosm(µx)y(t) dt = f (x).

This is a special case of equation 2.9.2 withg(x) = A cos m(µx) and h(t) = tanh k(λt).

52. y(x) – A

 x

a

tanhk(λx) sinm(µt)y(t) dt = f (x).

This is a special case of equation 2.9.2 withg(x) = A tanh k(λx) and h(t) = sin m(µt).

53. y(x) – A

 x

a

tanhk(λt) sinm(µx)y(t) dt = f (x).

This is a special case of equation 2.9.2 withg(x) = A sin m(µx) and h(t) = tanh k(λt).