## Methods for Solving Mathematical Physics ProblemsThe aim of the book is to present to a wide range of readers (students, postgraduates, scientists, engineers, etc.) basic information on one of the directions of mathematics, methods for solving mathematical physics problems. The authors have tried to select for the book methods that have become classical and generally accepted. However, some of the current versions of these methods may be missing from the book because they require special knowledge. The book is of the handbook-teaching type. On the one hand, the book describes the main definitions, the concepts of the examined methods and approaches used in them, and also the results and claims obtained in every specific case. On the other hand, proofs of the majority of these results are not presented and they are given only in the simplest (methodological) cases. Another special feature of the book is the inclusion of many examples of application of the methods for solving specific mathematical physics problems of applied nature used in various areas of science and social activity, such as power engineering, environmental protection, hydrodynamics, elasticity theory, etc. This should provide additional information on possible applications of these methods. To provide complete information, the book includes a chapter dealing with the main problems of mathematical physics, together with the results obtained in functional analysis and boundary-value theory for equations with partial derivatives. |

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### Contents

33 Transverse vibrations of an infinite circular membrane | 146 |

4 USING INTEGRAL TRANSFORMS IN HEAT CONDUCTIVITY PROBLEMS | 147 |

42 Solution of a heat conductivity problem using Fourier transforms | 148 |

43 Temperature regime of a spherical ball | 149 |

51 The solution of the equation of deceleration of neutrons for a moderator of infinite dimensions | 150 |

6 APPLICATION OF INTEGRAL TRANSFORMATIONS TO HYDRODYNAMIC PROBLEMS | 151 |

62 The flow of the ideal liquid through a slit | 152 |

63 Discharge of the ideal liquid through a circular orifice | 153 |

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312 Equations of oscillations | 24 |

313 Helmholtz equation | 26 |

315 Maxwell and telegraph equations | 27 |

316 Transfer equation | 28 |

317 Gas and hydrodynamic equations | 29 |

32 Formulation of the main problems of mathematical physics | 32 |

322 The Cauchy problem | 33 |

323 The boundaryvalue problem for the elliptical equation | 34 |

324 Mixed problems | 35 |

33 Generalized formulations and solutions of mathematical physics problems | 37 |

331 Generalized formulations and solutions of elliptical problems | 38 |

332 Generalized formulations and solution of hyperbolic problems | 41 |

333 The generalized formulation and solutions of parabolic problems | 43 |

34 Variational formulations of problems | 45 |

342 Variational formulation of the problem in the case of positive operators | 46 |

343 Variational formulation of the basic elliptical problems | 47 |

35 Integral equations | 49 |

352 Volterra integral equations | 50 |

353 Integral equations with a polar kernel | 51 |

355 Integral equation with the Hermitian kernel | 52 |

BIBLIOGRAPHIC COMMENTARY | 54 |

METHODS OF POTENTIAL THEORY | 56 |

1 INTRODUCTION | 57 |

2 FUNDAMENTALS OF POTENTIAL THEORY | 58 |

213 Formulae from the field theory | 59 |

214 Main properties of harmonic functions | 60 |

22 Potential of volume masses or charges | 61 |

223 Potential of a homogeneous sphere | 62 |

23 Logarithmic potential | 63 |

233 The logarithmic potential of a circle with constant density | 64 |

242 The properties of the simple layer potential | 65 |

243 The potential of the homogeneous sphere | 66 |

25 Double layer potential | 67 |

253 The logarithmic double layer potential and its properties | 69 |

3 USING THE POTENTIAL THEORY IN CLASSIC PROBLEMS OF MATHEMATICAL PHYSICS | 70 |

312 Solution of the Dirichlet problem in space | 71 |

313 Solution of the Dirichlet problem on a plane | 72 |

314 Solution of the Neumann problem | 73 |

315 Solution of the third boundaryvalue problem for the Laplace equation | 74 |

316 Solution of the boundaryvalue problem for the Poisson equation | 75 |

32 The Green function of the Laplace operator | 76 |

323 Solution of the Dirichlet problem for simple domains | 77 |

33 Solution of the Laplace equation for complex domains | 78 |

332 The sweep method | 80 |

4 OTHER APPLICATIONS OF THE POTENTIAL METHOD | 81 |

412 Boundaryvalue problems for the Helmholtz equations | 82 |

413 Green function | 84 |

414 Equation vλv 0 | 85 |

42 Nonstationary potentials | 86 |

422 Heat sources in multidimensional case | 88 |

423 The boundaryvalue problem for the wave equation | 90 |

BIBLIOGRAPHIC COMMENTARY | 92 |

EIGENFUNCTION METHODS | 94 |

2 EIGENVALUE PROBLEMS | 95 |

22 Eigenvalue problems for differential operators | 98 |

23 Properties of eigenvalues and eigenfunctions | 99 |

24 Fourier series | 100 |

25 Eigenfunctions of some onedimensional problems | 102 |

3 SPECIAL FUNCTIONS | 103 |

32 Legendre polynomials | 105 |

33 Cylindrical functions | 106 |

34 Chebyshef Laguerre and Hermite polynomials | 107 |

35 Mathieu functions and hypergeometrical functions | 109 |

4 EIGENFUNCTION METHOD | 110 |

42 The eigenfunction method for differential equations of mathematical physics | 111 |

43 Solution of problems with nonhomogeneous boundary conditions | 114 |

5 EIGENFUNCTION METHOD FOR PROBLEMS OF THE THEORY OF ELECTROMAGNETIC PHENOMENA | 115 |

52 Electrostatic field inside an infinite prism | 117 |

54 The field inside a ball at a given potential on its surface | 118 |

55 The field of a charge induced on a ball | 120 |

6 EIGENFUNCTION METHOD FOR HEAT CONDUCTIVITY PROBLEMS | 121 |

62 Stationary distribution of temperature in an infinite prism | 122 |

63 Temperature distribution of a homogeneous cylinder | 123 |

7 EIGENFUNCTION METHOD FOR PROBLEMS IN THE THEORY OF OSCILLATIONS | 124 |

72 Oscillations of the string with a moving end | 125 |

73 Problem of acoustics of free oscillations of gas | 126 |

74 Oscillations of a membrane with a fixed end | 127 |

75 Problem of oscillation of a circular membrane | 128 |

BIBLIOGRAPHIC COMMENTARY | 129 |

4 METHODS OF INTEGRAL TRANSFORMS | 130 |

1 INTRODUCTION | 131 |

2 MAIN INTEGRAL TRANSFORMATIONS | 132 |

211 The main properties of Fourier transforms | 133 |

212 Multiple Fourier transform | 134 |

222 The inversion formula for the Laplace transform | 135 |

24 Hankel transform | 136 |

25 Meyer transform | 138 |

27 MellerFock transform | 139 |

28 Hilbert transform | 140 |

210 Bochner and convolution transforms wavelets and chain transforms | 141 |

3 USING INTEGRAL TRANSFORMS IN PROBLEMS OF OSCILLATION THEORY | 143 |

7 USING INTEGRAL TRANSFORMS IN ELASTICITY THEORY | 155 |

72 Bussinesq problem for the half space | 157 |

73 Determination of stresses in a wedge | 158 |

8 USING INTEGRAL TRANSFORMS IN COAGULATION KINETICS | 159 |

82 Violation of the mass conservation law | 161 |

BIBLIOGRAPHIC COMMENTARY | 162 |

5 METHODS OF DISCRETISATION OF MATHEMATICAL PHYSICS PROBLEMS | 163 |

1 INTRODUCTION | 164 |

2 FINITEDIFFERENCE METHODS | 166 |

212 General definitions of the net method The convergence theorem | 170 |

213 The net method for partial differential equations | 173 |

22 The method of arbitrary lines | 182 |

222 The method of arbitrary lines for hyperbolic equations | 184 |

223 The method of arbitrary lines for elliptical equations | 185 |

23 The net method for integral equations the quadrature method | 187 |

3 VARIATIONAL METHODS | 188 |

312 Concepts of the direct methods in calculus of variations | 189 |

32 The Ritz method | 190 |

322 The Ritz method in energy spaces | 192 |

323 Natural and main boundaryvalue conditions | 194 |

33 The method of least squares | 195 |

34 Kantorovich Courant and Trefftz methods | 196 |

343 Trefftz method | 197 |

35 Variational methods in the eigenvalue problem | 199 |

4 PROJECTION METHODS | 201 |

412 The BubnovGalerkin method A A0 +B | 202 |

42 The moments method | 204 |

43 Projection methods in the Hilbert and Banach spaces | 205 |

432 The GalerkinPetrov method | 206 |

434 The collocation method | 208 |

5 METHODS OF INTEGRAL IDENTITIES | 210 |

52 The method of Marchuks integral identity | 211 |

53 Generalized formulation of the method of integral identities | 213 |

532 The difference method of approximating the integral identities | 214 |

533 The projection method of approximating the integral identities | 215 |

54 Applications of the methods of integral identities in mathematical physics problems | 218 |

542 The solution of degenerating equations | 219 |

BIBLIOGRAPHIC COMMENTARY | 223 |

6 SPLITTING METHODS | 224 |

2 INFORMATION FROM THE THEORY OF EVOLUTION EQUATIONS AND DIFFERENCE SCHEMES | 225 |

212 The nonhomogeneous evolution equation | 228 |

213 Evolution equations with bounded operators | 229 |

22 Operator equations in finitedimensional spaces | 231 |

222 Stationarisation method | 232 |

23 Concepts and information from the theory of difference schemes | 233 |

232 Stability | 239 |

233 Convergence | 240 |

234 The sweep method | 241 |

3 SPLITTING METHODS | 242 |

31 The method of component splitting the fractional step methods | 243 |

32 Methods of twocyclic multicomponent splitting | 245 |

322 Method of twocyclic component splitting for quasilinear problems | 246 |

331 The implicit splitting scheme with approximate factorisation of the operator | 247 |

332 The stabilisation method the explicitimplicit schemes with approximate factorisation of the operator | 248 |

34 The predictorcorrector method | 250 |

342 The predictorcorrector method | 251 |

35 The alternatingdirection method and the method of the stabilising correction | 252 |

352 The method of stabilising correction | 253 |

36 Weak approximation method | 254 |

37 The splitting methods iteration methods of solving stationary problems | 255 |

372 Iteration algorithms | 256 |

4 SPLITTING METHODS FOR APPLIED PROBLEMS OF MATHEMATICAL PHYSICS | 257 |

41 Splitting methods of heat conduction equations | 258 |

421 Locally onedimensional schemes | 259 |

42 Splitting methods for hydrodynamics problems | 262 |

422 The fractional steps method for the shallow water equations | 263 |

43 Splitting methods for the model of dynamics of sea and ocean flows | 268 |

432 The splitting method | 270 |

BIBLIOGRAPHIC COMMENTARY | 272 |

7 METHODS FOR SOLVING NONLINEAR EQUATIONS | 273 |

1 INTRODUCTION | 274 |

2 ELEMENTS OF NONLINEAR ANALYSIS | 276 |

212 Derivative and gradient of the functional | 277 |

213 Differentiability according to Fréchet | 278 |

22 Adjoint nonlinear operators | 279 |

222 Symmetry and skew symmetry | 280 |

24 Variational method of examining nonlinear equations | 282 |

243 Main concept of the variational method | 283 |

25 Minimising sequences | 284 |

252 Correct formulation of the minimisation problem | 285 |

32 Main concept of the steepest descent methods | 286 |

33 Convergence of the method | 287 |

4 THE RITZ METHOD | 288 |

41 Approximations and Ritz systems | 289 |

42 Solvability of the Ritz systems | 290 |

43 Convergence of the Ritz method | 291 |

52 The convergence of the Newton iteration process | 292 |

6 THE GALERKINPETROV METHOD FOR NONLINEAR EQUATIONS | 293 |

62 Relation to projection methods | 294 |

63 Solvability of the Galerkin systems | 295 |

7 PERTURBATION METHOD | 296 |

72 Justification of the perturbation algorithms | 299 |

73 Relation to the method of successive approximations | 301 |

8 APPLICATIONS TO SOME PROBLEM OF MATHEMATICAL PHYSICS | 302 |

82 The Galerkin method for problems of dynamics of atmospheric processes | 306 |

83 The Newton method in problems of variational data assimilation | 308 |

BIBLIOGRAPHIC COMMENTARY | 311 |

References | 313 |

317 | |

### Common terms and phrases

adjoint algorithm approximate solution arbitrary assumed Banach space boundary conditions boundary-value conditions boundary-value problem bounded domain Cauchy problem coefficients Consequently const constant continuous convergence denotes density determined difference scheme Dirichlet problem domain of definition double layer potential eigenfunctions eigenvalues elements equal examine finite formulation Fourier series function f given grad grid heat conductivity Hilbert space homogeneous inequality initial conditions integral equations integral identities integral transform kernel Laplace equation linear operator mathematical physics matrix mixed problem Moscow Nauka Neumann problem non-linear norm normalized space obtain orthogonal oscillations perturbation positive definite problems of mathematical referred respect Ritz method scalar product second order sequence simple layer potential solution of equation splitting methods symmetric system of equations Theorem theory unique solution values variable variational vector zero φ φ φφ Ω Ω