2 Equation 1 under the boundary conditions 2 determines the solution of the original integral equation.. P., Integral Transforms and Operational Calculus, Pergamon Press, New York, 1965.
Trang 12◦ The solution bounded at the endpoint x =1and unbounded at the endpoint x = –1:
y (x) = Af (x) – B
π
1
–1
g (x)
g (t)
f (t) dt
t – x , g (x) = (1 + x) α(1– x)–α, (3)
where α is the solution of the trigonometric equation (2) on the interval –1 < α <0
3◦ The solution unbounded at the endpoints:
y (x) = Af (x) – B
π
1
–1
g (x)
g (t)
f (t) dt
t – x + Cg(x), g (x) = (1 + x) α(1– x)–1–α,
where C is an arbitrary constant and α is the solution of the trigonometric equation (2) on
the interval –1< α <0
4. y(x) – λ
1
0
1
t – x –
1
x + t – 2xt
y(t) dt = f (x), 0 < x < 1.
Tricomi’s equation.
Solution:
y (x) = 1
1+ λ2π2
f (x) +
1
0
t α(1– x) α
x α(1– t) α
t – x –
1
x + t –2xt
f (t) dt
+ C(1– x) β
x1+β ,
α= 2
π arctan(λπ) (–1 < α <1), tan βπ
2 = λπ (–2 < β <0),
where C is an arbitrary constant.
5. y(x) + λ
∞
0
e–|x–t|y(t) dt = f (x).
Solution for λ > –12:
y (x) = f (x) – √ λ
1+2λ
0 exp –
√
1+2λ|x – t| f (t) dt
+
1– λ+1
√
1+2λ
0 exp
–√
1+2λ (x + t)
f (t) dt.
6. y(x) – λ
∞
–∞ e
–|x–t|y(t) dt= 0, λ> 0.
Lalesco–Picard equation.
Solution:
y (x) =
⎧
⎪
⎪
C1exp x √
1–2λ
+ C2exp –x √
1–2λ
for 0< λ < 12,
C1cos x √
2λ–1 + C2sin x √
2λ–1 for λ > 12,
where C1and C2are arbitrary constants
Trang 21404 INTEGRALEQUATIONS
7. y(x) + λ
∞
–∞ e
–|x–t|y(t) dt = f (x).
1◦ Solution for λ > –1
2:
y (x) = f (x) – √ λ
1+2λ
–∞exp –
√
1+2λ|x – t| f (t) dt.
2◦ If λ≤–12, for the equation to be solvable the conditions
–∞ f (x) cos(ax) dx =0, ∞
–∞ f (x) sin(ax) dx =0,
where a = √
–1–2λ, must be satisfied In this case, the solution has the form
y (x) = f (x) – a
2+1
2a
0 sin(at)f (x + t) dt (–∞ < x < ∞).
In the class of solutions not belonging to L2(–∞, ∞), the homogeneous equation (with
f (x)≡ 0) has a nontrivial solution In this case, the general solution of the corresponding
nonhomogeneous equation with λ≤–12 has the form
y (x) = C1sin(ax) + C2cos(ax) + f (x) – a
2+1
4a
–∞ sin(a| x – t|)f (t) dt.
8. y(x) + A
b
a e
λ|x–t|y(t) dt = f (x).
1◦ The function y = y(x) obeys the following second-order linear nonhomogeneous
ordi-nary differential equation with constant coefficients:
y
xx + λ(2 A – λ)y = f xx (x) – λ2f (x). (1) The boundary conditions for (1) have the form
y
x (a) + λy(a) = f x (a) + λf (a),
y
x (b) – λy(b) = f x (b) – λf (b). (2) Equation (1) under the boundary conditions (2) determines the solution of the original integral equation
2◦ For λ(2 A – λ) <0, the general solution of equation (1) is given by
y (x) = C1cosh(kx) + C2sinh(kx) + f (x) – 2Aλ
k
x
a sinh[k(x – t)] f (t) dt,
k=
λ (λ –2A),
(3)
where C1and C2are arbitrary constants
For λ(2 A – λ) >0, the general solution of equation (1) is given by
y (x) = C1cos(kx) + C2sin(kx) + f (x) – 2Aλ
k
x
a sin[k(x – t)] f (t) dt,
k=
λ(2A – λ).
(4)
For λ =2A, the general solution of equation (1) is given by
y (x) = C1+ C2x + f (x) –4A2 x
a (x – t)f (t) dt. (5)
The constants C1and C2in solutions (3)–(5) are determined by conditions (2)
Trang 33◦ In the special case a =0and λ(2 A – λ) >0, the solution of the integral equation is given
by formula (4) with
C1= (λ – A) sin μ – k cos μ A (kIc– λIs) , C2 = –λ k (λ – A) sin μ – k cos μ A (kIc– λIs) ,
k=
λ(2A – λ), μ = bk, Is=
b
0 sin[k(b – t)]f (t) dt, Ic=
b
0 cos[k(b – t)]f (t) dt.
9. y(x) + λ
∞
–∞
y(t) dt
cosh[b(x – t)] = f (x).
Solution for b > π| λ|:
y (x) = f (x) – 2λb
√
b2– π2λ2
–∞
sinh[2k (x – t)]
sinh[2b (x – t)] f (t) dt, k=
b
π arccosπλ
b
10. y(x) – λ
∞
0
cos(xt)y(t) dt = f(x).
Solution:
y (x) = f (x)
1– π2λ2 +
λ
1– π2λ2
0 cos(xt)f (t) dt, λ≠ 2/π
11. y(x) – λ
∞
0
sin(xt)y(t) dt = f(x).
Solution:
y (x) = f (x)
1– π2λ2 +
λ
1– π2λ2
0 sin(xt)f (t) dt, λ≠ 2/π
12. y(x) – λ
∞
–∞
sin(x – t)
x – t y(t) dt = f (x).
Solution:
y (x) = f (x) + √ λ
2π – πλ
–∞
sin(x – t)
x – t f (t) dt, λ≠2/π
13. Ay(x)– B
2π
2π
0
cot t – x
2
y(t) dt = f (x), 0 ≤x≤2π.
Here the integral is understood in the sense of the Cauchy principal value Without loss of
generality we may assume that A2+ B2=1
Solution:
y (x) = Af (x) + B
2π
2π
0 cot
t – x
2
f (t) dt + B
2
2πA
2π
0 f (t) dt.
Trang 41406 INTEGRALEQUATIONS
14. y(x) – λ
∞
0
e μ(x–t) cos(xt)y(t) dt = f(x).
Solution:
y (x) = f (x)
1– π2λ2 +
λ
1– π2λ2
0 e
μ(x–t) cos(xt)f (t) dt, λ≠ 2/π
15. y(x) – λ
∞
0
e μ(x–t) sin(xt)y(t) dt = f(x).
Solution:
y (x) = f (x)
1– π2λ2 +
λ
1– π2λ2
0 e
μ(x–t) sin(xt)f (t) dt, λ≠ 2/π
16. y(x)–
∞
–∞K(x – t)y(t) dt = f (x).
The Fourier transform is used to solve this equation
Solution:
y (x) = f (x) +
–∞ R (x – t)f (t) dt,
where
R (x) = 1
√
2π
–∞
2
R (u)e iux du, 2R (u) = K2(u)
1–√
2π 2 K (u), 2K (u) =
1
√
2π
–∞ K (x)e
–iuxdx.
17. y(x)–
∞
0
K(x – t)y(t) dt = f (x).
Wiener–Hopf equation of the second kind This equation is discussed in the books by Noble
(1958), Gakhov and Cherskii (1978), and Polyanin and Manzhirov (1998) in detail
References for Chapter T11
Bitsadze, A V., Integral Equation of the First Kind, World Scientific Publishing Co., Singapore, 1995 Ditkin, V A and Prudnikov, A P., Integral Transforms and Operational Calculus, Pergamon Press, New
York, 1965.
Gakhov, F D., Boundary Value Problems, Dover Publications, New York, 1990.
Gakhov, F D and Cherskii, Yu I., Equations of Convolution Type [in Russian], Nauka Publishers, Moscow,
1978.
Gorenflo, R and Vessella, S., Abel Integral Equations: Analysis and Applications, Springer-Verlag, Berlin,
1991.
Krasnov, M L., Kiselev, A I., and Makarenko, G I., Problems and Exercises in Integral Equations, Mir
Publishers, Moscow, 1971.
Lifanov, I K., Poltavskii, L N., and Vainikko, G., Hypersingular Integral Equations and Their Applications,
Chapman & Hall/CRC Press, Boca Raton, 2004.
Mikhlin, S G and Pr¨ossdorf, S., Singular Integral Operators, Springer-Verlag, Berlin, 1986.
Muskhelishvili, N I., Singular Integral Equations: Boundary Problems of Function Theory and Their
Appli-cations to Mathematical Physics, Dover PubliAppli-cations, New York, 1992.
Noble, B., Methods Based on Wiener–Hopf Technique for the Solution of Partial Differential Equations,
Pergamon Press, London, 1958.
Polyanin, A D and Manzhirov, A V., Handbook of Integral Equations, CRC Press, Boca Raton, 1998.
Trang 5Prudnikov, A P., Brychkov, Yu A., and Marichev, O I., Integrals and Series, Vol 5, Inverse Laplace
Transforms, Gordon & Breach, New York, 1992.
Sakhnovich, L A., Integral Equations with Difference Kernels on Finite Intervals, Birkh¨auser Verlag, Basel,
1996.
Samko, S G., Kilbas, A A., and Marichev, O I., Fractional Integrals and Derivatives Theory and
Applica-tions, Gordon & Breach, New York, 1993.
Zabreyko, P P., Koshelev, A I., et al., Integral Equations: A Reference Text, Noordhoff International
Publish-ing, Leyden, 1975.
Trang 7Functional Equations
T12.1 Linear Functional Equations in One Independent
Variable
T12.1.1 Linear Difference and Functional Equations Involving
Unknown Function with Two Different Arguments
T12.1.1-1 First-order linear difference equations involving y(x) and y(x + a).
1. y(x + 1) – y(x) = 0.
This functional equation may be treated as a definition of periodic functions with unit
period
1◦ Solution:
y (x) = Θ(x),
whereΘ(x) = Θ(x +1) is an arbitrary periodic function with unit period
2◦ A periodic function Θ(x) with period 1 that satisfies the Dirichlet conditions can be
expanded into a Fourier series:
Θ(x) = a20 +
∞
n=1
a ncos(2πnx ) + b nsin(2πnx)
,
where
a n=2 1
0 Θ(x) cos(2πnx ) dx, b n=2 1
0 Θ(x) sin(2πnx ) dx.
2. y(x + 1) – y(x) = f (x).
Solution:
y (x) = Θ(x) + ¯y(x),
whereΘ(x) = Θ(x +1) is an arbitrary periodic function with period 1, and¯y(x) is any
par-ticular solution of the nonhomogeneous equation Table T12.1 presents parpar-ticular solutions
of the nonhomogeneous equation for some specific f (x).
3. y(x + 1) – ay(x) = 0.
A homogeneous first-order constant-coefficient linear difference equation.
1◦ Solution for a >0:
y (x) = Θ(x)a x,
whereΘ(x) = Θ(x +1) is an arbitrary periodic function with period 1
ForΘ(x) ≡const, we have a particular solution y(x) = Ca x , where C is an arbitrary
constant
1409
... Technique for the Solution of Partial Differential Equations,Pergamon Press, London, 1958.
Polyanin, A D and Manzhirov, A V., Handbook of Integral... equation of the second kind This equation is discussed in the books by Noble
(1958), Gakhov and Cherskii (1978), and Polyanin and Manzhirov (1998) in detail
References for. ..
Bitsadze, A V., Integral Equation of the First Kind, World Scientific Publishing Co., Singapore, 1995 Ditkin, V A and Prudnikov, A P., Integral Transforms and Operational Calculus, Pergamon