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Porter D., Stirling D.S.G. Integral equations: a practical treatment, from spectral theory to applications
Porter D., Stirling D.S.G.  Integral equations: a practical treatment, from spectral theory to applications







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: Integral equations: a practical treatment, from spectral theory to applications

: Porter D., Stirling D.S.G.

:

This book gives a rigorous and practical treatment of integral equations. These are significant because they occur in many problems in mathematics, physics and engineering and they offer a powerful (sometimes the only) technique for solving these problems. The book aims to tackle the solution of integral equations using a blend of abstract 'structural' results and more direct, down-to-earth mathematics. The interplay between these two approaches is a central feature of the text and it allows a thorough account to be given of many of the types of integral equation which arise in application areas. Since it is not always possible to find explicit solutions of the problems posed, much attention is devoted to obtaining qualitative information and approximations to the solutions, with the associated error estimates. This treatment is intended for final year mathematics undergraduates, postgraduates and research workers in application areas such as numerical analysis and fluid mechanics.


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: 1990

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: 01.04.2005

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$L_2$-kernel      72 83 104 105 107 120 122 124 132 160
Abels equation      1 23 291
Abels problem      21
Absolute convergence      352
Adjoint equation      61 104 105106
Adjoint operator      102 104105 356357
aerofoil equation      319
Airys equation      9 24
Almost everywhere      66 67 114 358359
alternative      see Fredholm Alternative
approximation by finite-rank operator      146 245 277 287288
approximation by particular classes of function      274 286
approximation by powers of operator      137 279 280
approximation of compact operator      211 212
approximation of continuous kernel      245 277 288
approximation of eigenvalues, eigenvectors etc.      203240
approximation of solution of inhomogeneous equation      241290
Associated Legendre equation      5152
Banach space      351
Banach space, equations in      6364
Bessel function      20 92 93 181 236 337 338 340
Bessels equation      45 48 51 92 93 188 235
Bessels inequality      353
beta function      364
bivariational principle      260
boundary condition, change of      149 162
Boundary condition, mixed      17 18 52 289 335
boundary condition, regular/singular      44
boundary condition, weak      45 52
Boundary-value problem      13 16 34 44 149 150 155 182ff 289 335
Bounded below      164 165
bounded linear map      see Operator
Bounded sequence      351
bounds, complementary      214 215 224 227 232233 261ff 267 285
bounds, pointwise      264 (see also Upper bounds and lower bounds)
Calculus, Fundamental Theorem of      360
Cauchy principal value      304 306 309 365
Cauchy sequence      351
Cauchy singular      7 20
Cauchy singular equation      308ff
change of boundary conditions      149 162
classification of integral equations      2
Closed subset      103 353
Commutativity      133 152 166 352
Compact operator      69 72 75 95138 135 136 211 212
compactness, of operator generated by $L_2$-kernel      72
compactnessof operator generated by weakly singular kernel      76
comparison of operators      176178 227ff
complementary bounds      214 215 224 227 232233 261ff 267 285
completely continuous      see Compact operator
completeness of orthonormal sequence      112 354
completeness, of normed vector space      351
compression of operator      154 182 212
continuity condition on solutions of ODEs      7 8
continuous dependence on data      138 195 243
continuous kernel      126 128131 245
convergence, in the mean      66 120 132 352 355
convergence, in the mean, of sequence of projections      211212
convergence, in the mean, uniform/pointwise      see Uniform or pointwise convergence
conversion of differential equation to integral equation      31ff
Convolution      333
Cosine transform      337338 350
coupled equations      16
Degenerate kernel      57ff 70 94 146 150
Dense set      301 356
derivative of eigenvector      189
diagonalisation      113
differential equations, Airys equation      9 24
differential equations, associated Legendre equation      5152
differential equations, Bessels equation      45 48 51 92 93 188 235
differential equations, Gegenbauers equation      52 230
differential equations, Hills equation      49
differential equations, Laplaces equation      16 18 30 197 335
differential equations, Legendres equation      4647
differential equations, Tricomis equation      17 20
Digamma function      365
Dimension of nullspace      99 106
Dinis theorem      126 136
Dominant eigenvalue      203
Dominated Convergence Theorem      360
dual bounds      see Complementary bounds
dual extremum principles      250 257
dual integral equation      19
Eigenvalue      14 95 99 100 101 108 109 110 120 122 164 206ff 214ff
eigenvalue, approximation of      206ff
eigenvalue, conventions for numbering      141
eigenvalue, inequalities for      146ff 162
eigenvalue, minimax results      142 144 145
Eigenvector      95 101 108 110 120 122 189 206 218ff
eigenvector, approximation of      217ff
eigenvector, completeness of sequence of      112 191 193
eigenvector, conventions for numbering      141
eigenvector, derivative of      189
Elliptic partial differential equation      16 289
equivalence class in $L_2(a, b)$, choice of representative      67 69 114115
equivalence of differential equations and integral equations      31ff 41
Essential supremum      93
Essentially bounded      93 155
estimates for eigenvalues      142ff 163ff
existence of solution of integral equation      116 117 342
expansion of kernel      120 122 126 129
expansion of operator, spectral      109110 118
extension of operator      301
extremum principle      see Maximum or minimum principle
extremum principles, dual      250 257
Finite rank      69 70 94 102 153 353
Finite-dimensional space      353
first kind equation      3 4 116 117 119 259 291ff 295ff 319ff
Fourier cosine transform      337338 350
Fourier series      354 355 362ff
Fourier sine transform      17 335 350
Fourier transform      1 181 301 302 306 307 330ff
Fredholm alternative      61 69 70 78 104 105 114
Fredholm equation      3
free term      2
Fubinis Theorem      74 105 361
Functional      205
Fundamental theorem of calculus      360
Galerkins method      210 269ff
Galerkins method, iterated      277229
Gamma function      364
Gegenbauer polynomial      230
Gegenbauers equation      52 230
Gram Schmidt process      355
Hermitian kernel      107 120 124 128131 158161
Hermitian matrix      208
Hermitian operator      see Self-adjoint
Hilbert space      352
Hilbert transform      307ff
Hills equation      49
Homogeneous equation      5
image (or range) of operator      103 356 357
inclusion principle      216
inequalities for eigenvalues      146ff 162
infimum, attainment of      144
inhomogeneous equation      5
initial values/initial-value problem      7 33 84
Inner product      352
integral      see Lebesgue
integral, principal value      304 306 308 309 365
Integro-differential equation      348
Inverse      69 165
invertible/invertibility      69 87 96 97 133 135 160
iterated Galerkin method      277279
iterated kernel      82
iterated trial function      222 253
Kantorovichs method      276 281
Kelloggs method      217
Kernel      2
kernel in sense of of vectors annihilated      see Nullspace
kernel, $L_{2^-}$      72 83 104 105 107 120 122 124 132 160
Kernel, approximation of      245 277 288
kernel, continuous      126 128131 245
Kernel, degenerate      57ff 70 94 146 150
kernel, differentiable      158 161 277 288
kernel, hermitian      107 120 124 128131 158161
kernel, non-negative-valued      159
kernel, Poisson      134
kernel, resolvent      13 115 130132 194
kernel, restriction of domain      172
kernel, Schmidt      123125 132 195
kernel, Schur      74 105 107 133 135 136 160 173
kernel, skew-symmetric      240
kernel, symmetric      14 108
kernel, weakly singular      6 74 76 173 174 291ff
kind (of equation)      3 (see also First kind second
Lagrange multiplier      205 206
Laplace transform operator      178
Laplaces equation      16 18 30 197 335
latent root      see Eigenvalue
Lebesgue Convergence Theorems      360
Lebesgue integral      358
Lebesgue measure/measurable      358 359
Legendres equation      46
Linear integral equation      5
Linear map      63
Linear transformation      55
logarithmically singular kernel      6 320
Lower bound      214 215 218219 220221 224225 227 232233 234 255 259 261 263 267 285
Maximum principle      206 248 249 250 256257
maximum, attainment of      144
Mean convergence      66 120 132 352 355
Measurable function      358
Measure      358
Mercers theorem      126 130 171
minimax results      142 144 145
Minimum principle      206 250 256257 259
minimum, attainment of      144
mixed boundary condition      17 52 289 335
mixed boundary-value problem      17 18 52 289 335
Monotone Convergence Theorem      360
Neumann series      78ff
Non-negative operator      126 129 150 160 163ff
non-self-adjoint equation, bounds for      266267
Non-singular      see Invertible
Norm      64 69 351 352
norm-preserving operator      168 200
Normal operator      199
NullSpace      103 106 356 357
Operator      69 72ff 352
operator, simplifying      294 309 323 339 345 348
operator, simplifying, Volterra      83 291ff
order of integration, reversal of      105 345 361
ordinary differential equation, boundary-value problem      13 34 182ff
ordinary differential equation, conventions about solution      8
ordinary differential equation, conversion to integral equation      7ff 31ff
ordinary differential equation, equivalence to integral equation      41
ordinary differential equation, initial-value problem      7 33 84
Orthogonal complement      356
Orthogonal projection      356 (see also Projection)
Orthogonal vectors      108 112 353
orthogonality with respect to weight function      192
Orthonormal sequence      99 110 111 112 353 354 355
orthonormal sequence, complete      112 354 355
Parsevals theorem      73 354
Partial differential equation      16 30 112 289 335
piecewise constant function      275
Piecewise linear function      275
Poincare Bertrand formula      345
pointwise bounds      264
Pointwise convergence      124
Poisson kernel      134
Positive operator      163202 248 254 259
positive operator, comparison results      176ff
Positive semi-definite      see Non-negative
Power method      220
Principal value      304 306 309 365
principal value integral, reversal of integration order      345
Principles      see Maximum principle minimum
Projection      153 154 160 211 212 356
Projection methods      270
Psi function      365
radius (of spectrum)      86
range (or image) of operator      103 356 357
Rank      69
Rayleigh quotient      139 203 206
Rayleigh Ritz method      207 210 213 214 260 267 272
reciprocal principle      260 268
recovery of properties of solution      67 69 114115
regular boundary value problem      44
residual error      251 271
resolvent kernel/operator      13 115 130132 194
restriction of kernel to smaller domain      172
reversal of order of integration      74 105 345 361
Riesz Fischer theorem      116 120 355
Scalar product      see Inner product
Schmidt kernel      123125 132 195
Schmidts Theorem      124
Schur kernel      74 105 107 133 135 136 160 173
Schwarz quotient      221
Schwarzs inequality      359
Schwarzs method      220
second kind equation, definition of      3 5
second-order accuracy      207 246 261
Self-adjoint operator      107ff
Separation of variables      112
simple eigenvalues      225
simplifying operator      294 309 323 339 345 348
Sine transform      17 335 350
singular boundary point      45
singular boundary value problem      44ff
Singular integral      see Principal value integral
Singular integral equation      3 6 7 20 308ff 330ff
Singular value      118 122
Skew-adjoint      135 240
Spectral radius      86
Spectral theorem      109
Spectrum      96 97 108 109 133 135 280 332
Spectrum of compact operator      98 101 280
Square root      166 196
square-integrable      359
Stationary point      205207 260
stationary principle      206 260 266
Stationary value      204207 261 266
strongly singular (Cauchy singular)      7 20 308ff
Sturm Liouville problem      34 150 155 161 182ff
successive substitution      11
supremum, attainment of      144
Symmetric kernel      14 108
Test function      207
third kind equation      3 4 5
Tonellis Theorem      361
Trace      120 129 223 225
Trial function      207
Tricomis equation      17 20
Uniform convergence      124 125 126 127 195
Uniqueness of solutions      see Fredholm Alternative
Unitary operator      177 200 303
Upper bound      214 215 219 222223 224225 227 232233 256 261 263 267 285
variation-iteration      222
variational methods, accuracy of      207
Variational Principle      203ff 246ff
Volterra equation      3 9 83
Volterra operator      83 291ff
weak boundary condition      45 52
weakly convergent sequence      99100
weakly singular equation/kernel      6 74 76 173 174 291ff
Weight function      192
Well-posedness      138 195 243
wronskian      32
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