Freiling G., Yurko V. Lectures on Differential Equations of Mathematical Physics: A First Course

Подождите немного. Документ загружается.

292 G.

Freiling and V. Yurko

5) G(x,y) =

2π

∞

∑

n=−∞

|x −y

−ln

|x − ˜y

πn + y

), ˜y

= (y

,2πn −y

);

6) G(x,y) =

2π

∞

∑

n=−∞

∑

k=0

|x −y

n,k

−ln

|x − ˜y

n,k

((−1

)

,2πn + y

), ˜y

n,k

= ((−1)

,2πn −y

6.4.24.

1) G(z,ζ) =

2π

|z −

ζ|

|z −ζ|

;

2) G(z,ζ)

2π

−

−ζ

the

function ω

(z,ζ) is a superposition of the function z

and a linear-fractional

function;

3) G(z,ζ) =

2π

−

−ζ

;

4) G(z,ζ)

2π

|z −

ζ|

−z

ζ|

|z −ζ|

−z

ζ|

the function ω(z,ζ) is a combination of the functions: z

= −

, z

(z +

) ,

= e

iα

−ζ

;

5) G(z,ζ)

2π

−

||R

−(zζ)

−ζ

||R

−(zζ)

;

7) G(z,ζ)

2π

|chz −

chζ|

|chz −ch ζ|

6.4.28.

1. u(x

)

= e

−

√

cosx

;

2. u(x

)

−

√

−e

√

)cos x

cosx

;

6.4.29.

1. u(x

)

+ 1

)

;

2. u(x

) =

cosx

sh(π −x

)

shπ

Exercises 293

6.4.30. No.

6.4.31. f (x

) = −20

+ x

6.4.32. Show that in polar coordinates the function V (x

) is a solution of the problem

∆V (ρ) =

1, ρ ≤1,

0, ρ > 1,

V (1 + 0) = V (1 −0), V

(1 + 0) = V

(1 −0).

6.4.33. Show that in spherical coordinates the function V (x

) is a solution of the

problem

∆V (ρ) =

−1, ρ ≤1,

0, ρ > 1,

V (1 −0) = V (1 + 0), V

(1 −0) = V

(1 + 0).

6.4.34. −

6.4.35. f (x

)

= 2

πx

, V (x

) =

πx

4|x|

. F

solving the exterior Dirichlet problem

introduce polar coordinates and then apply the method of separation of variables.

6.4.36.

V (x) =











3|x|

, |x|

≥ R

−

|x|

, |x| < R;

1. In

spherical

coordinates the problem is reduced to calculating the integral

sinθ dθdζ

+ ρ

−2ζρcosθ

which

can

be computed by the change of the variable

ψ =

+ ρ

−2ζρcosθ; (instead

of θ ).

Using the spherical symmetry of the potential, write the Laplace operator in the form

∆V (ρ) =

∂

∂ρ

∂V

∂ρ

Inte

grating

the corresponding equations inside and outside the domain one gets

V (ρ) =











, ρ ≥R,

−

, ρ < R.

The

constants C

(

i =

1,3) are

found

by using the continuity of the function V (ρ)

and its normal derivatives and lim

ρ→∞

V (ρ) = 0.

294 G.

Freiling and V. Yurko

6.4.37.

1. V (x) =











−

−a

), |x| < a,

−

|x|+

|x|

, a < |x| < b;

−a

)

|x|

, |x| > b.

2. V (x)











|x|

+ f

−b

)

, |x| < a,

−b

)

|x|

), a < |x| < b;

−b

)

+ a

3|x|

, |x| > c.

6.4.39.

1. W (ρ,θ)

2π

(ρ

−r

)g(ζ)dζ

+ ρ

−2rρcos(ζ −θ)

, where ρ,θ are

polar

coordinates;

2. W (x

) =

∞

−∞

g(ζ)dζ

−3)

+ x

;

3. W (x

)

2π

∞

−∞

∞

−∞

g(ζ,η)dζdη

[(x

−ζ)

−η

)

+ x

]

3/2

6.4.40.

1. W (x) =











, |x| < 1,

, |x| = 1,

0, |x| > 1.

follo

ws from (6.4.8) that W

(x) = ν

, W

−

(x) = 0. The solution of the interior

Dirichlet problem

∂

∂ρ

∂W

∂ρ

∂

∂θ

= 0,

ρ=1

= ν

( ρ, θ are

polar

coordinates) is found with the help of the method of separation of

variables:

2. W (x

) =











, |x|< 1,

0, |x| = 1,

−

2|x|

, |x| > 1.

aking

(6.4.8) into account solve the interior and exterior Dirichlet problems:

∂

∂ρ

∂W

∂ρ

∂

∂θ

= 0, W

ρ=1

cosθ;

Exercises 295

∂

∂ρ

∂W

∂ρ

∂

∂θ

= 0, W

ρ=1

= −

cosθ,

the

method of separation of variables.

6.4.42.

u(x

) =







, |x| ≤ 1,

|x|

, |x| > 1;

1. Introduce

spherical

coordinates and instead of θ ( θ has the range 0 ≤θ ≤ π ) use a

new variable ψ =

+ 1 −2ρcos θ.

2. Using

the

spherical symmetry of the potential write the Laplace equation in the form

∂

∂ρ

∂u

∂ρ

= 0.

ﬁnding the constants use (6.4.20), (6.4.21), the continuity properties for u(ρ)

and

lim

ρ→∞

u(ρ) = 0.

6.4.43.

u(x) =

(

−Rln R, |x| ≤ R,

−Rln |x|, |x| > R.

The problem is reduced to ﬁnding the constants appearing in the integration of the equation

∂

∂ρ

∂u

∂ρ

= 0.

6.4.44. u(x

)

= −

. Introduce

polar

coordinates and use the method of separation of

variables.

6.4.45. µ(x

) = 2x

+ 8x

. Use the relations (4.17)-(4.18).

6.4.46.

u(ρ,ϕ) =

2π

g(ψ)ln

+ ρ

−2rρcos(θ −ψ)

dψ +C.

Determine

the

solution of the problem

∂

∂ρ

∂

∂θ

= 0,

∂u

∂ρ

ρ=r

= g(θ)

the

form

u(ρ,ϕ) =

2π

µ(ψ)ln

+ ρ

−2rρcos(θ −ψ)

dψ +C,

296 G.

Freiling and V. Yurko

hence

∂u

∂ρ

ρ=r

= −

4π

2π

µ(τ)dτ.

Using

(6.4.17)

and the condition

2π

f (τ) dτ = 0 for the solvability of the interior Neu-

mann problem one has µ(τ) = 2g(τ).

Refer

ences

[1]

Tikhonov, A. N. and Samarski

ı, A. A. Equations of mathematical physics. Translated

from the Russian by A. R. M. Robson and P. Basu. Reprint of the 1963 translation.

Dover Publications, Inc., New York, 1990, xvi+765 pp.

[2] Koshlyakov, N. S., Smirnov, M. M., and Gliner, E. B. Differential equations of math-

ematical physics. Translated by Scripta Technica, Inc.; translation editor: Herbert J.

Eagle North-Holland Publishing Co., Amsterdam; Interscience Publishers John Wiley

and Sons New York 1964, xvi+701 pp.

[3] Sobolev, S. L. and Dawson, E. R. Partial differential equations of mathematical

physics. Translated from the third Russian edition by E. R. Dawson; English trans-

lation edited by T. A. A. Broadbent Pergamon Press, Oxford-Edinburgh-New York-

Paris-Frankfurt; Addison-Wesley Publishing Co., Inc., Reading, Mass.-London 1964,

x+427pp.

[4] Petrovsky, I. G. Lectures on partial differential equations. Translated from the Russian

by A. Shenitzer. Reprint of the 1964 English translation. Dover Publications, Inc.,

New York, 1991, x+245 pp.

[5] Zauderer, E. Partial differential equations of applied mathematics. John Wiley &

Sons, New York, 1989.

[6] Zachmanoglou, E. C. and Thoe, Dale W. Introduction to partial differential equations

with applications. Second edition. Dover Publications, Inc., New York, 1986.

[7] Blanchard, Philippe and Br

uning, Erwin. Variational methods in mathematical

physics. A uniﬁed approach. Translated from the German by Gillian M. Hayes. Texts

and Monographs in Physics. Springer-Verlag, Berlin, 1992.

[8] Godunov S. K. Equations of Mathematical Physics. Moscow, Nauka, 1979.

[9] Mikhlin, S. G. Variational methods in mathematical physics. Translated by T. Bod-

dington; editorial introduction by L. I. G. Chambers. A Pergamon Press Book The

Macmillan Co., New York 1964, xxxii+582 pp.

[10] Gantmakher, F. Theory of matrix. Moscow, Nauka, 1988.

[11] Zygmund A. Trigonometric Series. Cambridge University Press, 1959.

298 References

[12] Conway

J. B. Functions of One Complex Variable, 2nd ed., vol.I, Springer-Verlag,

New York, 1995.

[13] Naimark M. A. Linear differential operators. 2nd ed., Nauka, Moscow, 1969; English

transl. of 1st ed., Parts I,II, Ungar, New York, 1967, 1968.

[14] Jerri A. J. Introduction to Integral Equations with Applications. Dekker, New York,

1985.

[15] Marchenko V. A. Sturm-Liouville Operators and their Applications. Naukova Dumka,

Kiev, 1977; English transl., Birkhauser, 1986.

[16] Levitan B. M. Inverse Sturm-Liouville Problems. Nauka, Moscow, 1984; English

transl., VNU Sci.Press, Utrecht, 1987.

[17] Freiling G. and Yurko V. A. Inverse Sturm-Liouville Problems and their Applications.

NOVA Science Publishers, New York, 2001, 305pp.

[18] Yurko V. A. Method of Spectral Mappings in the Inverse Problem Theory. Inverse and

Ill-posed Problems Series. VSP, Utrecht, 2002, 303pp.

[19] Yurko V. A. Inverse Spectral Problems for Linear Differential Operators and their

Applications. Gordon and Breach, New York, 2000.

[20] Romanov V. G. Inverse Problems in Mathematical Physics. Nauka, Moscow, 1984;

English transl.: VNU Scince Press, Utrecht, 1987.

[21] Chadan K. and Sabatier P. C. Inverse Problems in Quantum Scattering Theory, 2nd

ed., Texts and Monographs in Physics. Springer-Verlag, New York-Berlin, 1989.

[22] Prilepko A. I., Orlovsky D. G. and Vasin I. A. Methods for Solving Inverse Problems

in Mathematical Physics. Marcel Dekker, New York, 2000.

[23] Ambarzumian V. A.

Uber eine Frage der Eigenwerttheorie, Zs. f. Phys. 53 (1929),

690-695.

[24] Levinson N. The inverse Sturm-Liouville problem, Math. Tidsskr. 13 (1949), 25-30.

[25] Levitan B. M. and Sargsjan I. S. Introduction to Spectral Theory. Nauka, Moscow,

1970; English transl.: AMS Transl. of Math. Monographs. 39, Providence, RI, 1975.

[26] Yurko V. A. An inverse spectral problem for singular non-selfadjoint differential sys-

tems. Mat. Sbornik 195, no. 12 (2004), 123-156 (Russian); English transl. in Sbornik:

Mathematics 195, no. 12 (2004), 1823-1854.

[27] Young R. M. An introduction to Nonharmonic Fourier Series. Academic Press, New

York, 1980.

[28] Gardner G.; Green J.; Kruskal M. and Miura R., A method for solving the Korteweg-

de Vries equation, Phys. Rev. Letters, 19 (1967), 1095-1098.

References 299

[29] Ablowitz

M. J. and Segur H. Solitons and the Inverse Scattering Transform. SIAM,

Philadelphia, 1981.

[30] Lax P. Integrals of nonlinear equations of evolution and solitary waves, Comm. Pure

Appl. Math. 21 (1968), 467-490.

[31] Takhtadjan L.A. and Faddeev L.D. Hamiltonian Methods in the Theory of Soli-

tons. Nauka, Moscow, 1986; English transl.: Springer Series in Soviet Mathematics.

Springer-Verlag, Berlin-New York, 1987.

[32] Zakharov V. E., Manakov S. V., Novikov S. P. and Pitaevskii L.P. Theory of solitons.

The inverse scattering method. Nauka, Moscow, 1980; English transl.: Contemporary

Soviet Mathematics. Consultants Bureau [Plenum], New York-London, 1984.

[33] Coddington E. and Levinson N. Theory of Ordinary Differential Equations. McGraw-

Hill Book Company, New York, 1955.

Index

2D, 184, 273

Aβ, 85, 96, 97, 98, 99, 100, 130, 131, 163, 217, 266,

287

AC, 39, 87, 218

acoustic, 3

acoustic waves, 3

Ag, 136, 153

algorithm, 65, 74, 84, 88, 113, 123, 125, 142, 147,

154

alternative, 210

AMS, 302

Amsterdam, 301

appendix, 190

applied mathematics, vii, 301

assumptions, 85

asymptotic, 33, 34, 81, 90

asymptotics, 35, 46, 114

barrier, 194, 195

behavior, 12, 90, 126, 279

bending, 2

Bessel, 55, 288

boundary conditions, 12, 13, 23, 26, 29, 31, 42, 77,

78, 79, 81, 237, 238, 239, 240, 241, 242, 243, 244,

245, 246, 252, 254, 256, 288, 290

boundary value problem, viii, 23, 30, 31, 34, 38, 42,

44, 65, 66, 68, 70, 72, 73, 74, 77, 78, 80, 81, 83,

88, 89, 167, 236, 237, 241, 242, 243, 244, 254,

255, 260, 276, 279

calculus, vii, 216

capacity, 3

Cauchy problem, vii, viii, 12, 13, 15, 16, 17, 18, 19,

21, 28, 51, 54, 55, 61, 62, 64, 65, 69, 73, 117, 151,

152, 154, 155, 157, 162, 163, 164, 166, 222, 223,

225, 233, 234, 236, 245, 246, 247, 249, 251, 256,

258, 259, 289

chemical, 5

classes, 78, 84, 103, 109, 114

classical, vii, viii, 16, 51, 219, 220

classification, vii, 6

closure, 217

Co, 301

complex numbers, 118, 121

composite, 263

computer, vii

computer science, vii

concentration, 5

concrete, 13

conduction, 4, 11, 157, 225, 251

conductivity, 3

configuration, 193

construction, 24, 45, 160, 192, 270, 273, 294

continuity, 59, 145, 168, 171, 206, 207, 208, 209,

296, 298

convergence, 121, 188, 192, 221, 224

convex, 195

cosine, 120, 183, 185, 278

covering, 189

definition, 103, 130, 218

degree, 248, 261

delta, 178

demand, vii

density, 2, 4, 5, 168, 196, 199, 209, 245, 246, 247,

261, 275, 276, 277, 278, 279

derivatives, 1, 6, 16, 26, 39, 47, 96, 106, 113, 127,

151, 160, 164, 167, 180, 183, 205, 208, 212, 221,

230, 296

differential equations, vii, viii, 1, 3, 15, 65, 231, 262,

292, 301

differentiation, 23, 26, 33, 38, 93, 103, 123, 160,

164, 170, 177, 183, 205, 268, 274, 285

diffusion, viii, 4, 5, 157, 251

diffusion process, viii, 4, 5, 157, 251

dipole, 199, 277

displacement, 2, 15, 22, 234, 239, 240, 247

distribution, 5, 13, 199, 253, 256, 277