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Bitsadze A.V. Equations of mathematical physics
Bitsadze A.V.  Equations of mathematical physics







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: Equations of mathematical physics

: Bitsadze A.V.

:

The present book consists of an introduction and six chapters. The introduction discusses basic notions and definitions of the traditional course of mathematical physics and also mathematical models of some phenomena in physics and engineering.
Chapters 1 and 2 are devoted to elliptic partial differential equations. Here much emphasis is placed on the Cauchy-Riemann system of partial differential equations, that is on fundamentals of the theory of analytic functions, which facilitates the understanding of the role played in mathematical physics by the theory of functions of a complex variable.
In Chapters 3 and 4 the structural properties of the solutions of hyperbolic and parabolic partial differential equations are studied and much attention is paidj to basic problems of the theory of wave equation and heat conduction equation.
In Chapter 5 some elements of the theory of linear integral equations are given. A separate section of this chapter is devoted to singular integral equations which are frequently used in applications.
Chapter 6 is devoted to basic practical methods for the solution of partial differential equations. This chapter contains a number of typical examples demonstrating the essence of the Fourier method of separation of variables, the method of integral transformations, tho fi è He-difference method, the method of asymptotic expansions and also the variational methods.
To study the book it is sufficient for the reader to be familiar with an ordinary classical course on mathematical analysis studied in colleges. Since such a course usually does not involve functional analysis, the embedding theorems for function spaces are not included in the present book.
A.V. Bitsadze


: en

: / //

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ed2k: ed2k stats

: 1980

: 318

: 23.04.2005

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Abel, N.H.      45 227
Abels integral equation      45 227
Adjoint equation      477 193
Adjoint operator      86 193
Admitted functions      303
Analytic continuation      146
Analytic function      100
Analytic function of one complex variable      98ff
Analytic function of real variables      470
Analytic function of several complex variables      165
Angle-preserving property      106 473
Antiderivative      124
Area mass density      41
Asymptotic expansion      291
Bessel, W.F.      256 261 270 271
Bessels equation      259
Bessels equation, (cylindrical) functions      261
Bessels equation, (cylindrical) functions of the first kind      259
Bessels equation, (cylindrical) functions of the second kind (or Neumanns functions)      271
Bessels equation, (cylindrical) functions of the third kind (or Hankers functions)      270
Bessels equation, inequality      256
Boundary, condition      44
Boundary-value problem with oblique derivative boundary condition      94
Boundary-value problem, first (or Dirichlets)      34
Boundary-value problem, Poincares      91
Boundary-value problem, second (or Neumanns)      83 91
Boundary-value problem, third (mixed)      91
Branch of a many-valued function      113
Branch point      114
Bubnov Galerkin method      310
Bubnov, I.G.      310
Cauchy Dirichlet problem      204ff
Cauchy Hadamard formula      103 Cauchy Riemann partial differential equations for two independent variables
Cauchy Riemann partial differential equations for several independent variables      165
Cauchy, A.L.      40 97 103 118 150 165 168
Cauchy-type integral      124
Cauchys integral formula      121 168
Cauchys principal value of a singular integral      150ff
Cauchys problem (or initial-value problem)      34
Cauchys problem (or initial-value problem) for Laplaces equation      192
Cauchys problem (or initial-value problem), general      188ff
Cauchys Theorem      118ff
Characteristic curve      21
Characteristic determinant      19
Characteristic direction      21
Characteristic form      15
Characteristic value (number)      217
Chebyshev, P.L.      261
Chebyshevs equation      261
Chebyshevs functions      261
Chebyshevs polynomials      262
Circle of convergence      103
Circulation      38
Closed contour, negatively oriented      116
Closed contour, positively oriented      116
Coefficient of thermal conductivity      43
Complex integration      116ff
Complex integration of a power function      119ff
Complex vector space      163
Composite function      108
Composition formula for singular integrals      240
Condition for solvability of Neumann problem      83
conformal      87
Conformal mapping      108
Conformal mapping in Euclidean space      172ff
Conformal mapping of a circle onto a circle      111
Conformal mapping of a half-plane onto a circle      111
Conformal mapping of a half-plane onto a half-plane      110
Conformal mapping, angle-preserving property of      106 173
Conformal mapping, Gauss      173
Conjugate harmonic functions      126
Continuity principle for analytic functions      147
Convolution (or German faltung)      281
Derivative(s) of a volume potential      67ff
Derivative(s) of analytic function of one variable      101
Derivative(s) of analytic function of several variables      166
Derivative(s), normal, of a double layer potential      80
Derivative(s), normal, of a single-layer potential      82
Derivative(s), tangential, of a single-layer potential      151
Differential of an analytic function of one variable      101
Differential, total, of an analytic function of several variables      165
Dipole      75
Dipole, axis of      75
Dipole, polarization (or dipole moment) of      75
Dirac, P.A.M.      284
Diracs delta function      284ff
Dirichlet, P.G.L.      34 42 91 303
Dirichlets integral      142 303
Dirichlets principle      305
Dirichlets problem (or first boundary-value problem)      34 91
Domain of dependence      181
Domain of influence      182
Domain of propagation of solution of wave equation      182
DAlembert, J.      180
DAlemberts formula      180
Eigenfunction      218 248
Eigenvalue      248
Eigenvalue problem      305ff
Elliptic partial differential equations      16 18 47ff
Entire function      141
Entire function, transcendental      141
Equation of oscillation of a membrane      251
Equation of oscillation of a string      180 246ff 264
Equations of mathematical physics      37
Essential singular point of an analytic function      138
Euler, L.      41 50 303
Eulers equation      41 303
Eulers gamma function      50 272
Exponential function      114
Extended complex plane      98
Extremum principle for elliptic partial differential equations      92f
Extremum principle for harmonic functions      53
Extremum principle for heat conduction equation      200
Field intensity      38
Finite-difference approximation of a differential equation      286ff
Flux through a contour      38
Force function of a field      39
Fourier, J.B.J.      43 273
Fouriers coefficients      255
Fouriers inverse transformation      273
Fouriers law of heat conduction      43
Fouriers method (or method of separation of variables)      246ff
Fouriers transform of a function      273
Fouriers transform of a function, inverse      273f
Fouriers transformation      273
Fredholm, E.I.      36 210 217 218 219 221
Fredholms alternative      221
Fredholms first theorem      217
Fredholms integral equation      36
Fredholms integral equation of the first kind      36
Fredholms integral equation of the second kind      36f 210
Fredholms integral equation of the third kind      36
Fredholms second theorem      218
Fredholms third theorem      219
Function(s) of a complex variable      98
Function(s) of a complex variable, continuous      99
Function(s) of a complex variable, monogenic      99
Function(s), analytic      100
Function(s), exponential      114
Function(s), Hankels (or Bessels, of the third kind)      270
Function(s), harmonic      26 47f
Function(s), harmonic, conjugate      126
Function(s), infinite-sheeted      115
Function(s), infinite-valued      115
Function(s), inverse      105 113
Function(s), limit of      99
Function(s), logarithmic      113
Function(s), many-sheeted      113
Function(s), many-valued      113
Function(s), meromorphic      141
Function(s), metaharmonic      259
Function(s), monogenic      99
Function(s), Neumanns (or Bessols, of the second kind)      271
Function(s), one-valued      98
Function(s), power      111
Function(s), special      261
Function(s), univalent (or one-sheeted)      104
Fundamental (or elementary) solution      14
Fundamental (or elementary) solution of heat conduction equation      33
Fundamental (or elementary) solution of Laplaces equation      27
Fundamental (or elementary) solution, principal      94
Galerkin, B.G.      310
Gauss Ostrogradsky formula      31
Gauss, K.F.      31 73 173
Gauss formula      73
Generalized potential function for a double layer      96
Generalized potential function for surface distribution of mass      95f
Generalized potential function for volume distribution of mass      89
Goursat problem      192
Goursat, E.      192
Green, G.      55 87 234
Greens function      55ff 234
Greens function, symmetry of      55
Greens Theorem      87
Grid-point      286
Grid-size      286
Hadamard, J.S.      103 192
Hamilton, R.W.      26 41
Hamiltonian operator (or nabla or del)      26
Hamiltons principle      41
Hankel, H.      270
Hankels function (or Bessels function of the third kind)      270
Harmonic function(s)      26 471
Harmonic function(s), conjugate      126
Harmonic function(s), regular at infinity      48
Harmonic polynomial      28
Harnack, A.      64
Harnacks theorem      64
Hartogs, F.      166
Hartogs theorem      166
Heat conduction equation      32 200ff
Heat conduction equation, non-homogeneous      208
Helmholtz equation      94 252
Helmholtz, H.L.F.      94 252
Hilbert, D.      240
Hilberts integral equation      240
Hilberts inverse transformation      241
Hilberts transformation      241
Hilberts transformation on a finite interval      243
Holder, O.      150
Holders condition for a function of one variable      150
Holders condition for a function of several variables      236
Huygens, C.      178
Huygens principle      178
Hyperbolic partial differential equation      16 18 176ff
Image of a point (under a mapping)      105
Image, (or transform) of a function under an integral transformation      272
Improperly posed (or not well-posed) problem      33 192f
Indefinite integral      123f
Infinite-sheeted function      115
Infinite-valued function      115
Initial conditions      44
Integral equation      35ff 2l0ff
Integral equation with kernel having logarithmic singularity      244
Integral equation, Abels      45 227
Integral equation, Fredholms      36
Integral equation, Fredholms of the first kind      36
Integral equation, Fredholms of the second kind      36 210ff
Integral equation, Fredholms of the third kind      36
Integral equation, Fredholms with degenerate kernel      215ff
Integral equation, Hilberts      240
Integral equation, linear      35
Integral equation, linear, homogeneous      35
Integral equation, linear, non-homogeneous      35
Integral equation, singular      236f
Integral equation, singular of the theory of the wing of an airplane      242
Integral equation, singular with logarithmic singularity      244f
Integral equation, Volterras      37
Integral equation, Volterras of the first kind      226
Integral equation, Volterras of the second kind      37 211
Integral equation, Volterras with multiple integral      225
Integral operator      35
Integral representation of a harmonic function      50f
Inverse function      105 113
inversion      84 175
Isolated singular point      138
Isolated singular point at infinity      140
Iterative method (or method of successive approximations)      210ff
Jordan curve      76
Jordan, C.      76
Kernel of an integral equation      35
Kernel of an integral equation, Cauchys      240
Kernel of an integral equation, degenerate      215
Kernel of an integral equation, Hilberts      240
Kernel of an integral equation, iterated      219
Kernel of an integral equation, resolvent      219
Kernel of an integral equation, symmetric      224
Kirchhoff, G.      177
Kirchhoffs formula      177
Lagrange adjoint operator      267
Lagrange, J.L.      267
Laguerre, E.N.      262
Laguerres equation      262
Laplace, P.S.      26 263 272
Laplaces equation      26
Laplaces inverse transform (or inverse image or the original)      272
Laplaces operator      26
Laplaces surface (spherical) harmonics      263
Laplaces transform (or image) of a function      272
Laplaces transformation      272
Laplaces transformation, inverse      272
Laurent, P.M.II.      135
Laurents expansion (series)      136
Laurents expansion (series), principal (or singular) part of      137
Laurents expansion (series), regular part of      137
Laurents expansion (series), theorem      135
Lebesgue, H.      49
Lebesgues integral      49
Legendre, A.M.      264
Legendres equation      264
Legendres functions      264
Legendres functions, associated      264
Leibniz, G.W.      124
Limiting values of Cauchy-type, integral      154ff
Linear partial differential equation      14
Linear-fractional function      102
Liouville, J.      63 134 247
Liouvilles theorem for analytic functions      134
Liouvilles theorem for harmonic functions      63
Lyapunov curve      158
Lyapunov, A.M.      158
Maximum Modulus Principle      127
Mean-square deviation      255
Mean-value formula for a ball      53
Mean-value formula for a sphere      53
Mellin, R.H.      274
Mellins transform of a function      275
Mellins transform of a function, inverse      275
Mellins transformation      274
Meromorphic function      141
Metaharmonic equation      259
Metaharmonic function      259
Method(s) of integral transformations      266ff
Method(s) of separation of variables (or Fouriers method)      246ff
Method(s), finite-difference      286ff
Method(s), saddle-point      301
Method(s), variational      303ff
Minimizing sequence      307
Mixed partial differential equation      17
Mixed problem (or boundary-initial-value problem)      248
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