References for Chapter T10 Barannyk, T., Symmetry and exact solutions for systems of nonlinear reaction-diffusion equations, Proc.. R., Lie symmetries of nonlinear multidimensional react
Trang 11382 SYSTEMS OFPARTIALDIFFERENTIALEQUATIONS
Solution:
u m = ϕ m (t)F n (t)
*
θ (x1, , x n , t) +
g (t, ϕ1, , ϕ
n–1)
F n (t) dt
+ , m=1, , n –1,
u n = F n (t)
*
θ (x1, , x n , t) +
g (t, ϕ1, , ϕ
n–1)
F n (t) dt
+ ,
F n (t) = exp
*
f n (t, ϕ1, , ϕ n–1) dt
+ ,
where the functions ϕ m = ϕ m (t) are described by the nonlinear system of first-order ordinary
differential equations
ϕ
m = ϕ m [f m (t, ϕ1, , ϕ n–1) – f n (t, ϕ1, , ϕ n–1)], m=1, , n –1,
and the function θ = θ(x1, , x n , t) satisfies the linear equation
∂θ
∂t = L[θ].
3. ∂u m
∂t = L[u m] +n
k=1
u k f mk
t, u1
u n , , u n–1
u n
, m = 1, , n.
Here, the system involves n2 arbitrary functions f mk = f mk (t, z1, , z n–1) that depend
on n arguments; L is an arbitrary linear differential operator with respect to the spatial variables x1, , x n(of any order in derivatives), whose coefficients can be dependent on
x1, , x n , t It is assumed that L[const] =0
Solution:
u m (x1, , x n , t) = ϕ m (t)F (t)θ(x1, , x n , t), m=1, , n,
F (t) = exp
n k=1
ϕ k (t)f nk (t, ϕ1, , ϕ n–1) dt
, ϕ n (t) =1,
where the functions ϕ m = ϕ m (t) are described by the nonlinear system of first-order ordinary
differential equations
ϕ
m =
n
k=1
ϕ k f mk (t, ϕ1, , ϕ n–1) – ϕ m
n
k=1
ϕ k f nk (t, ϕ1, , ϕ n–1), m=1, , n –1,
and the function θ = θ(x1, , x n , t) satisfies the linear equation
∂θ
∂t = L[θ].
References for Chapter T10
Barannyk, T., Symmetry and exact solutions for systems of nonlinear reaction-diffusion equations, Proc of
Inst of Mathematics of NAS of Ukraine, Vol 43, Part 1, pp 80–85, 2002.
Barannyk, T A and Nikitin, A G., Proc of Inst of Mathematics of NAS of Ukraine, Vol 50, Part 1, pp 34–39,
2004.
Cherniha, R and King, J R., Lie symmetries of nonlinear multidimensional reaction-diffusion systems: I,
J Phys A: Math Gen., Vol 33, pp 267–282, 7839–7841, 2000.
Trang 2R EFERENCES FOR C HAPTER T10 1383 Cherniha, R and King, J R., Lie symmetries of nonlinear multidimensional reaction-diffusion systems: II,
J Phys A: Math Gen., Vol 36, pp 405–425, 2003.
Nikitin, A G., Group classification of systems of non-linear reaction-diffusion equations with general diffusion
matrix II Diagonal diffusion matrix, From Website arXiv.org (a service of automated e-print archives of
articles), http://arxiv.org/abs/math-ph/0411028.
Nikitin, A G and Wiltshire, R J., Systems of reaction-diffusion equations and their symmetry properties, J.
Math Phys., Vol 42, No 4, pp 1667–1688, 2001.
Polyanin, A D., Exact solutions of nonlinear sets of equations of the theory of heat and mass transfer in
reactive media and mathematical biology, Theor Foundations of Chemical Engineering, Vol 38, No 6,
pp 622–635, 2004.
Polyanin, A D., Exact solutions of nonlinear systems of diffusion equations for reacting media and
mathemat-ical biology, Doklady Mathematics, Vol 71, No 1, pp 148–154, 2005.
Polyanin, A D., Systems of Partial Differential Equations, From Website EqWorld—The World of Mathematical
Equations, http://eqworld.ipmnet.ru/en/solutions/syspde.htm.
Polyanin, A D and Vyaz’mina, E A., New classes of exact solutions to nonlinear systems of reaction-diffusion
equations, Doklady Mathematics, Vol 74, No 1, pp 597–602, 2006.
Polyanin, A D and Zaitsev, V F., Handbook of Exact Solutions for Ordinary Differential Equations, 2nd
Edition, Chapman & Hall/CRC Press, Boca Raton, 2004.
Trang 4Chapter T11
Integral Equations
T11.1 Linear Equations of the First Kind
with Variable Limit of Integration
1.
x
a (x – t)y(t) dt = f(x), f (a) = f x (a) = 0.
Solution: y(x) = f xx (x).
2.
x
a (Ax + Bt + C)y(t) dt = f(x), f (a)= 0.
1◦ Solution for B ≠–A:
y (x) = d
dx
(A + B)x + C– A
A+B
x
a
(A + B)t + C– B
A+B f t (t) dt
2◦ Solution for B = –A:
y (x) = 1
C
d dx
exp –A
C x
x
a exp
A
C t
f
t (t) dt
3.
x
a (x – t) n y(t) dt = f (x), n = 1, 2,
It is assumed that the right-hand side of the equation satisfies the conditions f (a) = f x (a) =
· · · = f x(n)(a) =0
Solution: y(x) = 1
n!f
(n+ 1 )
x (x).
4.
x
a
√
x – t y(t) dt = f (x), f (a)= 0.
Solution: y(x) = 2
π
d2
dx2
x
a
f (t) dt
√
x – t.
5.
x
a
y(t) dt
√
x – t = f (x).
Abel equation.
Solution:
y (x) = 1
π
d dx
x
a
f (t) dt
√
x – t =
f (a)
π √
x – a +
1
π
x
a
f
t (t) dt
√
x – t.
1385
Trang 51386 INTEGRALEQUATIONS
6.
x
a (x – t) λ y(t) dt = f (x), f (a)= 0, 0 < λ < 1.
Solution: y(x) = sin(πλ)
πλ
d2
dx2
x
a
f (t) dt (x – t) λ.
7.
x
a
y(t) dt
(x – t) λ = f (x), 0 < λ < 1.
Generalized Abel equation.
Solution:
y (x) = sin(πλ)
π
d dx
x
a
f (t) dt (x – t)1–λ =
sin(πλ)
π
f (a) (x – a)1–λ +
x
a
f
t (t) dt
(x – t)1–λ
8.
x
a e
λ(x–t) y(t) dt = f (x), f (a)= 0.
Solution: y(x) = f x (x) – λf (x).
9.
x
a e
λx+βt y(t) dt = f (x), f (a) = 0.
Solution: y(x) = e–(λ+β)x
f
x (x) – λf (x)
10.
x
a
e λ(x–t)– 1
y(t) dt = f (x), f (a) = f x (a) = 0.
Solution: y(x) = λ1f
xx (x) – f x (x).
11.
x
a
e λ(x–t) + b
y(t) dt = f (x), f (a) = 0.
For b = –1, see equation T11.1.10
Solution for b≠–1:
y (x) = f
x (x)
b+1 –
λ
(b +1)2
x
a exp
λb
b+1(x – t)
f
t (t) dt.
12.
x
a
e λ(x–t) – e μ(x–t)
y(t) dt = f (x), f (a) = f x (a) = 0.
Solution:
y (x) = 1
λ – μ
f
xx – (λ + μ)f x + λμf
, f = f (x).
13.
x
a
y(t) dt
√
e λx – e λt = f (x), λ > 0.
Solution: y(x) = λ
π
d dx
x
a
e λt f (t) dt
√
e λx – e λt.
Trang 6T11.1 L INEAR E QUATIONS OF THE F IRST K IND WITH V ARIABLE L IMIT OF I NTEGRATION 1387 14.
x
a cosh[λ(x – t)]y(t) dt = f(x), f (a)= 0.
Solution: y(x) = f x (x) – λ2
x
a f (x) dx.
15.
x
a
55
cosh[λ(x – t)] – 166
y(t) dt = f (x), f (a) = f x (a) = f
xx (x) = 0.
Solution: y(x) = 1
λ2f xxx (x) – f x (x).
16.
x
a
55
cosh[λ(x – t)] + b66
y(t) dt = f (x), f (a)= 0.
For b =0, see equation T11.1.14 For b = –1, see equation T11.1.15
1◦ Solution for b(b +1) <0:
y (x) = f x (x)
b+1 –
λ2
k (b +1)2
x
a sin[k(x – t)]f
t (t) dt, where k = λ b –b+1.
2◦ Solution for b(b +1) >0:
y (x) = f x (x)
b+1 –
λ2
k (b +1)2
x
a sinh[k(x – t)]f
t (t) dt, where k = λ b+b1.
17.
x
a cosh 2[λ(x – t)]y(t) dt = f(x), f (a) = 0.
Solution:
y (x) = f x (x) – 2λ2
k
x
a sinh[k(x – t)]f
t (t) dt, where k = λ
√
2
18.
x
a sinh[λ(x – t)]y(t) dt = f(x), f (a) = f x (a) = 0.
Solution: y(x) = 1
λ f
xx (x) – λf (x).
19.
x
a
55
sinh[λ(x – t)] + b66
y(t) dt = f (x), f (a)= 0.
For b =0, see equation T11.1.18
Solution for b≠ 0:
y (x) = 1
b f
x (x) +
x
a R (x – t)f
t (t) dt,
R (x) = λ
b2 exp
–λx
2b
λ
2bk sinh(kx) – cosh(kx)
√
1+4b2
2b .
Trang 71388 INTEGRALEQUATIONS
20.
x
a sinh λ √
x – t
y(t) dt = f (x), f (a) = 0.
Solution: y(x) = 2
πλ
d2
dx2
x
a
cos λ √
x – t
√
x – t f (t) dt.
21.
x
0
ln(x – t)y(t) dt = f(x).
Solution:
y (x) = –
x
0 f
tt (t) dt
0
(x – t) z e–Cz
Γ(z +1) dz – f
x(0)
0
x z e–Cz
Γ(z +1) dz, where C = lim
k→∞
1+ 1
2 +· · · +
1
k+1 – ln k
=0.5772 . is the Euler constant andΓ(z) is
the gamma function
22.
x
a [ln(x – t) + A]y(t) dt = f(x).
Solution:
y (x) = – d
dx
x
a ν A (x – t)f (t) dt, ν A (x) =
d dx
0
x z e(A–C)z
Γ(z +1) dz, whereC =0.5772 . is the Euler constant andΓ(z) is the gamma function.
For a =0, the solution can be written in the form
y (x) = –
x
0 f
tt (t) dt
0
(x – t) z e(A–C)z
Γ(z +1) dz – f
x(0)
0
x z e(A–C)z
Γ(z +1) dz.
23.
x
a (x – t)
ln(x – t) + A
y(t) dt = f (x), f (a) = 0.
Solution:
y (x) = – d2
dx2
x
a ν A (x – t)f (t) dt, ν A (x) =
d dx
0
x z e(A–C)z
Γ(z +1) dz, whereC =0.5772 . is the Euler constant andΓ(z) is the gamma function.
24.
x
a cos[λ(x – t)]y(t) dt = f(x), f (a) = 0.
Solution: y(x) = f x (x) + λ2
x
a f (x) dx.
25.
x
a
55
cos[λ(x – t)] – 166
y(t) dt = f (x), f (a) = f x (a) = f xx (a) = 0.
Solution: y(x) = – 1
λ2f xxx (x) – f x (x).