Handbook of mathematics for engineers and scienteists part 200 pptx

This system can be successively integrated, since the first equation is a Bernoulli equation and the second one is linear in the unknown.. Arbitrary functions depend on the product of po

The first equation in (1) is a separable equation; its solution can be written out in implicit

form The second equation in (1) can be solved using the change of variable ψ = e ζ(it is

reduced to a linear equation for ζ).

Equation (2) admits exact solutions of the form

θ= exp

σ2(t)x2+ σ0(t)

,

where the functions σ n (t) are described by the equations

σ 

2= f (ϕ)σ2+4aσ2

2,

σ 

0= f (ϕ)σ0+2a (n +1)σ2 This system can be successively integrated, since the first equation is a Bernoulli equation and the second one is linear in the unknown

If f = const, equation (2) also has a traveling-wave solution θ = θ(kx – λt).

T10.3.2-3 Arbitrary functions depend on the product of powers of the unknowns

8. ∂u

∂t = a

x n

∂

∂x



x n ∂u

∂x



+ uf (x, u k w m),

∂w

∂t = b

x n

∂

∂x



x n ∂w

∂x



+ wg(x, u k w m).

Multiplicative separable solution:

u = e–mλt y (x), w = e kλt z (x), where λ is an arbitrary constant and the functions y = y(x) and z = z(x) are determined by

the system of ordinary differential equations

ax–n (x n y 

x) x + mλy + yf (x, y k z m) =0,

bx–n (x n z 

x) x – kλz + zg(x, y k z m) =0

9. ∂u

∂t = a

x n

∂

∂x



x n ∂u

∂x



+ u1+kn f u n w m

,

∂w

∂t = b

x n

∂

∂x



x n ∂w

∂x



+ w1–km g u n w m

.

Self-similar solution:

u = (C1t + C2)–kn1 y (ξ), w = (C1t + C2)km1 z (ξ), ξ = √ x

C1t + C2, where C1 and C2 are arbitrary constants, and the functions y = y(ξ) and z = z(ξ) are

determined by the system of ordinary differential equations

aξ–n (ξ n y 

ξ) ξ+

1

2C1ξy  ξ+

C1

kn y + y1+kn f y n z m

=0,

bξ–n (ξ n z 

ξ) ξ+ 1

2C1ξz ξ  –

C1

km z + z1–km g y n z m

=0

10. ∂u

∂t = a

x n

∂

∂x



x n ∂u

∂x



+ cu ln u + uf (x, u k w m),

∂w

∂t = b

x n

∂

∂x



x n ∂w

∂x



+ cw ln w + wg(x, u k w m).

Multiplicative separable solution:

u = exp(Ame ct )y(x), w = exp(–Ake ct )z(x), where A is an arbitrary constant, and the functions y = y(x) and z = z(x) are determined by

the system of ordinary differential equations

ax–n (x n y 

x) x + cy ln y + yf (x, y k z m) =0,

bx–n (x n z 

x) x + cz ln z + zg(x, y k z m) =0

T10.3.2-4 Arbitrary functions depend on u2 w2.

11. ∂u

∂t = a

x n

∂

∂x



x n ∂u

∂x



+ uf (u2+ w2) – wg(u2+ w2 ),

∂w

∂t = a

x n

∂

∂x



x n ∂w

∂x



+ wf (u2+ w2) + ug(u2+ w2 ).

Time-periodic solution:

u = r(x) cos

θ (x) + C1t + C2

, w = r(x) sin

θ (x) + C1t + C2

,

where C1 and C2 are arbitrary constants, and the functions r = r(x) and θ = θ(x) are

determined by the system of ordinary differential equations

ar 

xx – ar(θ  x)2+ an x r x  + rf (r2) =0,

arθ 

xx+2ar 

x θ x  + an x rθ x  + rg(r2) – C1r =0

12. ∂u

∂t = a

x n

∂

∂x



x n ∂u

∂x



+ uf (u2– w2) + wg(u2– w2 ),

∂w

∂t = a

x n

∂

∂x



x n ∂w

∂x



+ wf (u2– w2) + ug(u2– w2 ).

Solution:

u = r(x) cosh

θ (x) + C1t + C2

, w = r(x) sinh

θ (x) + C1t + C2

,

where C1 and C2 are arbitrary constants, and the functions r = r(x) and θ = θ(x) are

determined by the system of ordinary differential equations

ar 

xx + ar(θ  x)2+

an

x r



x + rf (r2) =0,

arθ 

xx+2ar 

x θ x  + an x rθ x  + rg(r2) – C1r =0

T10.3.2-5 Arbitrary functions have different arguments.

13. ∂u

∂t = a

x n

∂

∂x



x n ∂u

∂x



+ uf (u2+ w2) – wg  w

u



,

∂w

∂t = a

x n

∂

∂x



x n ∂w

∂x



+ wf (u2+ w2) + ug  w

u



.

Solution:

u = r(x, t) cos ϕ(t), w = r(x, t) sin ϕ(t), where the function ϕ = ϕ(t) is determined by the autonomous ordinary differential equation

ϕ 

and the function r = r(x, t) is determined by the differential equation

∂r

∂t = a

x n

∂

∂x



x n ∂r

∂x



The general solution of equation (1) is expressed in implicit form as



dϕ

g (tan ϕ) = t + C.

Equation (2) admits a time-independent exact solution r = r(x).

14. ∂u

∂t = a

x n

∂

∂x



x n ∂u

∂x



+ uf (u2– w2) + wg  w

u



,

∂w

∂t = a

x n

∂

∂x



x n ∂w

∂x



+ wf (u2– w2) + ug  w

u



.

Solution:

u = r(x, t) cosh ϕ(t), w = r(x, t) sinh ϕ(t), where the function ϕ = ϕ(t) is determined by the autonomous ordinary differential equation

ϕ 

and the function r = r(x, t) is determined by the differential equation

∂r

∂t = a

x n

∂

∂x



x n ∂r

∂x



The general solution of equation (1) is expressed in implicit form as



dϕ

g (tanh ϕ) = t + C.

Equation (2) admits a time-independent exact solution r = r(x).

T10.3.3 Systems of the Form Δu = F (u, w), Δw = G(u, w)

T10.3.3-1 Arbitrary functions depend on a linear combination of the unknowns

1. ∂

2u

∂x2 + ∂

2u

∂y2 = uf (au – bw) + g(au – bw),

∂2w

∂x2 + ∂

2w

∂y2 = wf (au – bw) + h(au – bw).

1◦ Solution:

u = ϕ(x) + bθ(x, y), w = ψ(x) + aθ(x, y), where ϕ = ϕ(x) and ψ = ψ(x) are determined by the autonomous system of ordinary

differential equations

ϕ 

xx = ϕf (aϕ – bψ) + g(aϕ – bψ),

ψ 

xx = ψf (aϕ – bψ) + h(aϕ – bψ),

and the function θ = θ(x, y) satisfies the linear Schr¨odinger equation of the special form

∂2θ

∂x2 +

∂2θ

∂y2 = F (x)θ, F (x) = f (au – bw).

Its solutions are determined by separation of variables

2◦ Let us multiply the first equation by a and the second one by –b and add the results

together to obtain

∂2ζ

∂x2 +

∂2ζ

∂y2 = ζf (ζ) + ag(ζ) – bh(ζ), ζ = au – bw. (1) This equation will be considered in conjunction with the first equation of the original system

∂2u

∂x2 +

∂2u

∂y2 = uf (ζ) + g(ζ). (2) Equation (1) can be treated separately An extensive list of exact solutions to equations of

this form for various kinetic functions F (ζ) = ζf (ζ) + ag(ζ) – bh(ζ) can be found in the

book by Polyanin and Zaitsev (2004)

Note two important solutions to equation (1):

(i) In the general case, equation (1) admits an exact, traveling-wave solution ζ = ζ(z), where z = k1x + k2y with arbitrary constants k1and k2

(ii) If the condition ζf (ζ) + ag(ζ) – bh(ζ) = c1ζ + c0 holds, equation (1) is a linear Helmholtz equation

Given a solution ζ = ζ(x, y) to equation (1), the function u = u(x, y) can be determined

by solving the linear equation (2) and the function w = w(x, y) is found as w = (bu – ζ)/c.

2. ∂

2u

∂x2 + ∂

2u

∂y2 = e λu f (λu – σw), ∂

2w

∂x2 + ∂

2w

∂y2 = e σw g(λu – σw).

1◦ Solution:

u = U (ξ) – 2

λln|x + C1|, w = W (ξ) – 2

σ ln|x + C1|, ξ= y + C2

x + C1,

where C1 and C2 are arbitrary constants, and the functions U = U (ξ) and W = W (ξ) are

determined by the system of ordinary differential equations

(1+ ξ2)U ξξ  +2ξU 

ξ+

2

λ = e λU f (λU – σW ),

(1+ ξ2)W ξξ  +2ξW 

ξ+ 2

σ = e σW g (λU – σW ).

2◦ Solution:

u = θ(x, y), w= λ

σ θ (x, y) – k

σ,

where k is a root of the algebraic (transcendental) equation

λf (k) = σe–k g (k), and the function θ = θ(x, y) is described by the solvable equation

∂2θ

∂x2 +

∂2θ

∂y2 = f (k)e λθ.

This equation is encountered in combustion theory; for its exact solutions, see Polyanin and Zaitsev (2004)

T10.3.3-2 Arbitrary functions depend on the ratio of the unknowns

3. ∂

2u

∂x2 + ∂

2u

∂y2 = uf



u w



, ∂2w

∂x2 + ∂

2w

∂y2 = wg



u w



.

1◦ A space-periodic solution in multiplicative form (another solution is obtained by

inter-changing x and y):

u = [C1sin(kx) + C2cos(kx)]ϕ(y),

w = [C1sin(kx) + C2cos(kx)]ψ(y), where C1, C2, and k are arbitrary constants and the functions ϕ = ϕ(y) and ψ = ψ(y) are

determined by the autonomous system of ordinary differential equations

ϕ 

yy = k2ϕ + ϕf (ϕ/ψ),

ψ 

yy = k2ψ + ψg(ϕ/ψ).

2◦ Solution in multiplicative form:

u = [C1exp(kx) + C2exp(–kx)]U (y),

w = [C1exp(kx) + C2exp(–kx)]W (y), where C1, C2, and k are arbitrary constants and the functions U = U (y) and W = W (y) are

determined by the autonomous system of ordinary differential equations

U 

yy = –k2U + U f (U/W ),

W 

yy = –k2W + W g(U/W ).

3◦ Degenerate solution in multiplicative form:

u = (C1x + C2)U (y),

w = (C1x + C2)W (y), where C1and C2are arbitrary constants and the functions U = U (y) and W = W (y) are

determined by the autonomous system of ordinary differential equations

U 

yy = U f (U/W ),

W 

yy = W g(U/W ).

Remark. The functions f and g in Items1◦– 3◦ can be dependent on y.

4◦ Solution in multiplicative form:

u = e a1x+b1y ξ (z), w = e a1 x+b1y η (z), z = a2x + b2y,

where a1, a2, b1, and b2are arbitrary constants, and the functions ξ = ξ(z) and η = η(z) are

determined by the autonomous system of ordinary differential equations

(a22+ b22)ξ zz  +2(a1a2+ b1b2)ξ z  + (a21+ b21)ξ = ξf (ξ/η), (a22+ b22)η  zz+2(a1a2+ b1b2)η  z + (a21+ b21)η = ηg(ξ/η).

5◦ Solution:

u = kθ(x, y), w = θ(x, y), where k is a root of the algebraic (transcendental) equation f (k) = g(k), and the function

θ = θ(x, y) is described by the linear Helmholtz equation

∂2θ

∂x2 +

∂2θ

∂y2 = f (k)θ.

For its exact solutions, see Subsection T8.3.3

4. ∂

2u

∂x2 + ∂

2u

∂y2 = uf



u w



+ u

w h



u w



, ∂2w

∂x2 + ∂

2w

∂y2 = wg



u w



+ h



u w



.

Solution:

u = kw, w = θ(x, y) – h (k)

f (k), where k is a root of the algebraic (transcendental) equation

f (k) = g(k), and the function θ = θ(x, y) satisfies the linear Helmholtz equation

∂2θ

∂x2 +

∂2θ

∂y2 = f (k)w.

5. ∂

2u

∂x2 + ∂

2u

∂y2 = u n f  u

w



, ∂2w

∂x2 + ∂

2w

∂y2 = w n g  u

w



.

For f (z) = kz–m and g(z) = –kz n–m , the system describes an nth-order chemical reaction

(of order n – m in the component u and of order m in the component w); to n =2and m =1 there corresponds a second-order reaction, which often occurs in applications

1◦ Solution:

u = r1 –2n U (θ), w = r1 –2n W (θ), r=



(x + C1)2+ (y + C2)2, θ= y + C2

x + C1, where C1 and C2 are arbitrary constants, and the functions y = y(ξ) and z = z(ξ) are

determined by the autonomous system of ordinary differential equations

U 

θθ+ 4 (1– n)2U = U

n f

U W



,

W 

θθ+

4 (1– n)2W = W

n g

U W



2◦ Solution:

u = kζ(x, y), w = ζ(x, y), where k is a root of the algebraic (transcendental) equation

k n–1f (k) = g(k),

and the function ζ = ζ(x, y) satisfies the equation with a power-law nonlinearity

∂2ζ

∂x2 +

∂2ζ

∂y2 = g(k)ζ n.

T10.3.3-3 Other systems

6. ∂

2u

∂x2 + ∂

2u

∂y2 = uf (u n w m), ∂2w

∂x2 + ∂

2w

∂y2 = wg(u n w m).

Solution in multiplicative form:

u = e m(a1x+b1 y) ξ (z), w = e–n(a1x+b1 y) η (z), z = a2x + b2y,

where a1, a2, b1, and b2are arbitrary constants, and the functions ξ = ξ(z) and η = η(z) are

determined by the autonomous system of ordinary differential equations

(a22+ b22)ξ zz  +2m (a1a2+ b1b2)ξ  z + m2(a21+ b21)ξ = ξf (ξ n η m),

(a22+ b22)η  zz–2n (a1a2+ b1b2)η z  + n2(a21+ b21)η = ηg(ξ n η m).