Handbook of mathematics for engineers and scienteists part 202 ppsx

The coefficients of the linear differential operator L can be dependent on x1,.. 1 This equation will be considered in conjunction with the first equation of the original system ∂u Equat

1◦ Solution:

u = ϕ(t) + c exp



f (t, bϕ – cψ) dt



θ (x, t), w = ψ(t) + b exp



f (t, bϕ – cψ) dt



θ (x, t), where ϕ = ϕ(t) and ψ = ψ(t) are determined by the system of ordinary differential equations

ϕ 

t = ϕf (t, bϕ – cψ) + g(t, bϕ – cψ),

ψ 

t = ψf (t, bϕ – cψ) + h(t, bϕ – cψ),

and the function θ = θ(x1, , x n , t) satisfies linear equation

∂θ

∂t = L[θ].

Remark 1. The coefficients of the linear differential operator L can be dependent on x1, , x n , t.

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

together to obtain

∂ζ

∂t = L[ζ] + ζf (t, ζ) + bg(t, ζ) – ch(t, ζ), ζ = bu – cw. (1) This equation will be considered in conjunction with the first equation of the original system

∂u

Equation (1) can be treated separately Given a solution of this equation, ζ = ζ(x1, , x n , t), the function u = u(x1, , x n , t) can be determined by solving the linear equation (2) and the function w = w(x1, , x n , t) is found as w = (bu – ζ)/c.

Remark 2. Let L be a constant-coefficient differential operator with respect to the independent variable

x = x1and let the condition

∂

∂t

*

ζf (t, ζ) + bg(t, ζ) – ch(t, ζ)

+

= 0

hold true (for example, it is valid if the functions f , g, h are not implicitly dependent on t) Then equation (1) admits an exact, traveling-wave solution ζ = ζ(z), where z = kx – λt with arbitrary constants k and λ.

2. ∂u

∂t = L1[u] + uf



u w



∂t = L2[w] + wg



u w



.

Here, L1and L2are arbitrary constant-coefficient linear differential operators (of any order)

with respect to x.

1◦ Solution:

u = e kx–λt y (ξ), w = e kx–λt z (ξ), ξ = βx – γt, where k, λ, β, and γ are arbitrary constants and the functions y = y(ξ) and z = z(ξ) are

determined by the system of ordinary differential equations

M1[y] + λy + yf (y/z) =0, M2[z] + λz + zg(y/z) =0,

M1[y] = e–kx L1[e kx y (ξ)], M2[z] = e–kx L2[e kx z (ξ)].

To the special case k = λ =0there corresponds a traveling-wave solution

2◦ If the operators L1and L2contain only even derivatives, there are solutions of the form

u = [C1sin(kx) + C2cos(kx)]ϕ(t), w = [C1sin(kx) + C2cos(kx)]ψ(t);

u = [C1exp(kx) + C2exp(–kx)]ϕ(t), w = [C1exp(kx) + C2exp(–kx)]ψ(t);

u = (C1x + C2)ϕ(t), w = (C1x + C2)ψ(t),

where C1, C2, and k are arbitrary constants Note that the third solution is degenerate.

3. ∂u

∂t = L[u] + uf



w



∂t = L[w] + wg



w



.

Here, L is an arbitrary linear differential operator with respect to the coordinates x1, , x n (of any order in derivatives), whose coefficients can be dependent on x1, , x n , t:

L [u] =

A k1 k n (x1, , x n , t) ∂ k1

+···+k n u

∂x k1

1 ∂x k n n

1◦ Solution:

u = ϕ(t) exp



g (t, ϕ(t)) dt



θ (x1, , x n , t),

w= exp



g (t, ϕ(t)) dt



θ (x1, , x n , t),

(2)

where the function ϕ = ϕ(t) is described by the first-order nonlinear ordinary differential

equation

ϕ 

and the function θ = θ(x1, , x n , t) satisfies the linear equation

∂θ

∂t = L[θ].

2◦ The transformation

u = a1(t)U + b1(t)W , w = a2(t)U + b2(t)W , where a n (t) and b n (t) are arbitrary functions (n =1,2), leads to an equation of the similar

form for U and W

Remark. The coefficients of the operator (1) can also depend on the ratio of the unknowns u/w, A k1 k n=

A k1 k n (x1, , x n , t, u/w) (in this case, L will be a quasilinear operator) Then there also exists a solution

of the form (2), where ϕ = ϕ(t) is described by the ordinary differential equation (3) and θ = θ(x1, , x n , t)

satisfies the linear equation

∂θ

∂t = L ◦ [θ], L ◦ = L u/w=ϕ.

4. ∂u

∂t = L[u] + uf



u w



+ g



u w



∂t = L[w] + wf



u w



+ h



u w



.

Here, L is an arbitrary linear differential operator with respect to x1, , x n(of any order

in derivatives), whose coefficients can be dependent on x1, , x n , t:

L [u] =

A k1 k n (x1, , x n , t) ∂ k1

+···+k n u

∂x k1

1 ∂x k n n

,

where k1+· · · + k n≥ 1

Let k be a root of the algebraic (transcendental) equation

1◦ Solution if f (k)≠ 0:

u (x, t) = k



exp[f (k)t]θ(x, t) – h (k)

f (k)



, w (x, t) = exp[f (k)t]θ(x, t) – h (k)

f (k),

where the function θ = θ(x1, , x n , t) satisfies the linear equation

∂θ

2◦ Solution if f (k) =0:

u (x, t) = k[θ(x, t) + h(k)t], w (x, t) = θ(x, t) + h(k)t, where the function θ = θ(x1, , x n , t) satisfies the linear equation (2).

5. ∂u

∂t = L[u]+ uf



w



+ u



w



, ∂w

∂t = L[w]+ wg



w



+ h



w



.

Solution:

u = ϕ(t)G(t)



θ (x1, , x n , t) +



h (t, ϕ)

G (t) dt



, G (t) = exp



g (t, ϕ) dt



,

w = G(t)



θ (x1, , x n , t) +



h (t, ϕ)

G (t) dt



,

where the function ϕ = ϕ(t) is described by the first-order nonlinear ordinary differential

equation

ϕ 

t = [f (t, ϕ) – g(t, ϕ)]ϕ,

and the function θ = θ(x1, , x n , t) satisfies the linear equation

∂θ

∂t = L[θ].

6. ∂u

∂t = L[u] + uf



w



ln u + ug



w



,

∂w

∂t = L[w] + wf



w



ln w + wh



w



.

Solution:

u (x, t) = ϕ(t)ψ(t)θ(x1, , x n , t), w (x, t) = ψ(t)θ(x1, , x n , t),

where the functions ϕ = ϕ(t) and ψ = ψ(t) are determined by solving the ordinary differential

equations

ϕ 

t = ϕ[g(t, ϕ) – h(t, ϕ) + f (t, ϕ) ln ϕ],

ψ 

t = ψ[h(t, ϕ) + f (t, ϕ) ln ψ],

(1)

and the function θ = θ(x1, , x n , t) is determined by the differential equation

∂θ

Given a solution to the first equation in (1), the second equation can be solved with the change

of variable ψ = e ζ by reducing it to a linear equation for ζ If L is a constant-coefficient one-dimensional operator (n = 1) and f = const, then equation (2) has a traveling-wave solution θ = θ(kx – λt).

7. F1



∂x , , ∂ m w

u k

∂w

u

∂u

∂x , , 1

u



= 0,

F2



∂x , , ∂ m w

u k

∂w

u

∂u

∂x , , 1

u



= 0.

Solution:

w = W (z), u = [ϕ  (t)]1/k U (z), z = x + ϕ(t), where ϕ(t) is an arbitrary function, and the functions W (z) and U (z) are determined by the

autonomous system of ordinary differential equations

F1 W , W z  , , W z(m) , W z  /U k , U 

z /U , , U z(n) /U

=0,

F2 W , W z  , , W z(m) , W z  /U k , U 

z /U , , U z(n) /U

=0

T10.4.3 Nonlinear Systems of Two Equations Involving the Second

Derivatives in t

1. ∂

2u

∂t2 = L[u] + uf (t, au – bw) + g(t, au – bw),

∂t2 = L[w] + wf (t, au – bw) + h(t, au – bw).

Here, L is an arbitrary linear differential operator (of any order) with respect to the spatial variables x1, , x n , whose coefficients can be dependent on x1, , x n , t It is assumed that L[const] =0

1◦ Solution:

u = ϕ(t) + aθ(x1, , x n , t), w = ψ(t) + bθ(x1, , x n , t),

where ϕ = ϕ(t) and ψ = ψ(t) are determined by the system of ordinary differential equations

ϕ 

tt = ϕf (t, aϕ – bψ) + g(t, aϕ – bψ),

ψ 

tt = ψf (t, aϕ – bψ) + h(t, aϕ – bψ),

and the function θ = θ(x1, , x n , t) satisfies linear equation

∂2θ

∂t2 = L[θ] + f (t, aϕ – bψ)θ.

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

together to obtain

∂2ζ

∂t2 = L[ζ] + ζf (t, ζ) + ag(t, ζ) – bh(t, ζ), ζ = au – bw. (1) This equation will be considered in conjunction with the first equation of the original system

∂2u

Equation (1) can be treated separately Given a solution ζ = ζ(x, t) to equation (1), the function u = u(x1, , x n , t) can be determined by solving the linear equation (2) and the function w = w(x1, , x n , t) is found as w = (au – ζ)/b.

Note three important cases where equation (1) admits exact solutions:

(i) Equation (1) admits a spatially homogeneous solution ζ = ζ(t).

(ii) Suppose the coefficients of L and the functions f , g, h are not implicitly dependent

on t Then equation (1) admits a steady-state solution ζ = ζ(x1, , x n)

(iii) If the condition ζf (t, ζ) + bg(t, ζ) – ch(t, ζ) = k1ζ + k0holds, equation (1) is linear

If the operator L is constant-coefficient, the method of separation of variables can be used

to obtain solutions

2. ∂

2u

∂t2 = L1[u] + uf



u w



, ∂2w

∂t2 = L2[w] + wg



u w



.

Here, L1and L2are arbitrary constant-coefficient linear differential operators (of any order)

with respect to x It is assumed that L1[const] =0and L2[const] =0

1◦ Solution in the form of the product of two waves traveling at different speeds:

u = e kx–λt y (ξ), w = e kx–λt z (ξ), ξ = βx – γt, where k, λ, β, and γ are arbitrary constants, and the functions y = y(ξ) and z = z(ξ) are

determined by the system of ordinary differential equations

γ2y 

ξξ+2λγy

ξ + λ2y = M1[y] + yf (y/z), γ2z ξξ  +2λγz

ξ + λ2z = M2[z] + zg(y/z),

M1[y] = e–kx L1[e kx y (ξ)], M2[z] = e–kx L2[e kx z (ξ)].

To the special case k = λ =0there corresponds a traveling-wave solution

2◦ Periodic multiplicative separable solution:

u = [C1sin(kt) + C2cos(kt)]ϕ(x), w = [C1sin(kt) + C2cos(kt)]ψ(x),

where C1, C2, and k are arbitrary constants and the functions ϕ = ϕ(x) and ψ = ψ(x) are

determined by the system of ordinary differential equations

L1[ϕ] + k2ϕ + ϕf (ϕ/ψ) =0,

L2[ψ] + k2ψ + ψg(ϕ/ψ) =0

3◦ Multiplicative separable solution:

u = [C1sinh(kt) + C2cosh(kt)]ϕ(x), w = [C1sinh(kt) + C2cosh(kt)]ψ(x), where C1, C2, and k are arbitrary constants and the functions ϕ = ϕ(x) and ψ = ψ(x) are

determined by the system of ordinary differential equations

L1[ϕ] – k2ϕ + ϕf (ϕ/ψ) =0,

L2[ψ] – k2ψ + ψg(ϕ/ψ) =0

4◦ Degenerate multiplicative separable solution:

u = (C1t + C2)ϕ(x), w = (C1t + C2)ψ(x),

where C1 and C2 are arbitrary constants and the functions ϕ = ϕ(x) and ψ = ψ(x) are

determined by the system of ordinary differential equations

L1[ϕ] + ϕf (ϕ/ψ) =0, L2[ψ] + ψg(ϕ/ψ) =0

Remark 1. The coefficients of L1, L2and the functions f and g in Items2◦– 4◦ can be dependent on x.

Remark 2. If L1 and L2contain only even derivatives, there are solutions of the form

u = [C1sin(kx) + C2cos(kx)]U (t), w = [C1sin(kx) + C2cos(kx)]W (t);

u = [C1 exp(kx) + C2 exp(–kx)]U (t), w = [C1 exp(kx) + C2 exp(–kx)]W (t);

u = (C1x + C2)U (t), w = (C1x + C2)W (t),

where C1, C2, and k are arbitrary constants Note that the third solution is degenerate.

3. ∂

2u

∂t2 = L[u] + uf



w



, ∂2w

∂t2 = L[w] + wg



w



.

Here, L is an arbitrary linear differential operator with respect to the coordinates x1, , x n

(of any order in derivatives), whose coefficients can be dependent on the coordinates Solution:

u = ϕ(t)θ(x1, , x n),

w = ψ(t)θ(x1, , x n),

where the functions ϕ = ϕ(t) and ψ = ψ(t) are described by the nonlinear system of

second-order ordinary differential equations

ϕ 

tt = aϕ + ϕf (t, ϕ/ψ),

ψ 

tt = aψ + ψg(t, ϕ/ψ),

a is an arbitrary constant, and the function θ = θ(x1, , x n) satisfies the linear steady-state equation

L [θ] = aθ.

4. ∂

2u

∂t2 = L[u] + uf



u w



+ g



u w



, ∂2w

∂t2 = L[w] + wf



u w



+ h



u w



.

Here, L is an arbitrary linear differential operator with respect to the coordinates x1, , x n (of any order in derivatives), whose coefficients can be dependent on x1, , x n , t.

Solution:

u = kθ(x1, , x n , t), w = θ(x1, , x n , t), where k is a root of the algebraic (transcendental) equation g(k) = kh(k) and the function

θ = θ(x, t) satisfies the linear equation

∂2θ

∂t2 = L[θ] + f (k)w + h(k).

5. ∂

2u

∂t2 = L[u] + au ln u + uf



w



, ∂2w

∂t2 = L[w] + aw ln w + wg



w



.

Here, L is an arbitrary linear differential operator with respect to the coordinates x1, , x n

(of any order in derivatives), whose coefficients can be dependent on the coordinates

u = ϕ(t)θ(x1, , x n),

w = ψ(t)θ(x1, , x n),

where the functions ϕ = ϕ(t) and ψ = ψ(t) are described by the nonlinear system of

second-order ordinary differential equations

ϕ 

tt = aϕ ln ϕ + bϕ + ϕf (t, ϕ/ψ),

ψ 

tt = aψ ln ψ + bψ + ψg(t, ϕ/ψ),

b is an arbitrary constant, and the function θ = θ(x1, , x n) satisfies the steady-state equation

L [θ] + aθ ln θ – bθ =0

T10.4.4 Nonlinear Systems of Many Equations Involving the First

Derivatives in t

1. ∂u m

∂t = L[u m ] + u m f (t, u1– b1u n , , u n–1 – b n–1 u n)

+ g m (t, u1– b1u n , , u n–1 – b n–1 u n), m = 1, , n.

The system involves n +1 arbitrary functions f , g1, , g n 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 = ϕ m (t) + exp

*

f (t, ϕ1– b1ϕ n , , ϕ n–1– b n–1ϕ n ) dt

+

θ (x1, , x n , t).

Here, the functions ϕ m = ϕ m (t) are determined by the system of ordinary differential

equations

ϕ 

m = ϕ m f (t, ϕ1– b1ϕ n , , ϕ n–1– b n–1ϕ n ) + g m (t, ϕ1– b1ϕ n , , ϕ n–1– b n–1ϕ n),

where m =1, , n, the prime denotes the derivative with respect to t, and the function

θ = θ(x1, , x n , t) satisfies the linear equation

∂θ

∂t = L[θ].

2. ∂u m

∂t = L[u m ] + u m f m



t, u1

u n , , u n–1



+ u m



t, u1

u n , , u n–1



,

∂t = L[u n ] + u n f n



t, u1

u n , , u n–1



+ g

t, u1

u n , , u n–1



.

Here, m = 1, , n –1 and the system involves n +1 arbitrary functions f1, , f n , g 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