The Hypercircle in Mathematical Physics: A Method for the Approximate Solution of Boundary Value Problems

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Cambridge University Press, Mar 22, 2012 - Mathematics - 440 pages
Originally published in 1957, this book was written to provide physicists and engineers with a means of solving partial differential equations subject to boundary conditions. The text gives a systematic and unified approach to a wide class of problems, based on the fact that the solution may be viewed as a point in function-space, this point being the intersection of two linear subspaces orthogonal to one another. Using this method the solution is located on a hypercircle in function-space, and the approximation is improved by reducing the radius of the hypercircle. The complexities of calculation are illuminated throughout by simple, intuitive geometrical pictures. This book will be of value to anyone with an interest in solutions to boundary value problems in mathematical physics.
 

Contents

Representation of numbers p 7 Representation of functions p
8
Addition subtraction and multiplication by a scalar p 19 Logic
23
Straight lines p 24 Linear dependence of two Fvectors p 24 Linear
34
Conditions to be satisfied by the scalar product p 37 Examples
43
Length and distance in Fspace p 44 Example p 44 Unit Fvectors
50
Orthonormality and linear independence p 50 Orthogonal transforma
67
No closed boxes in functionspacep 76 nspheres
84
Linear subspaces and orthogonality p 98 The vertices V V of two non
102
section p 224 Approximation to stress and warping p 225 Torsion
233
General plan p 241 Use of approximate solutions for the weights of
269
General plan p 270 Use of approximate solutions for the weights of
280
The approximation n 8 p 283 The approximation n 16 p
286
VARIOUS BOUNDARY VALUE PROBLEMS
292
unknown p 292 Splitting the differential equation p 293 The scalar
299
Pyramid Fvectors for threedimensional Pspace p 301 First mixed
311
viscous flow in a channel p 312 Flow problem
332

Summary of formulae for application of the method of the hypercircle
118
Use of nonlinear subspaces p 121 Bounds on S2 obtained with
124
Statement of the Dirichlet problem the Neumann problem and the mixed
131
Case Where the boundary value function f is only piecewise continuous
137
The Greens Fvector p 155 Bounds for the solution p 157 Checks
163
Pyramid functions p 168 Pyramid Fvectors of the first class p
170
Pyramid Fvectors of the second class p 171 Triangulation and poly
176
Summary of plan for the use of pyramid Fvectors p 178 The linear
185
Definition and normalization p 188 Scalar products p 190 Determina
199
Definition and normalization p 200 Scalar products p 202 Determina
209
The torsion problem stated p 214 Torsional rigidity p 216 Transforma
219
Equations of equilibrium and compatibility p 336 Splitting the problem
345
The Greens tensor of elasticity p 347 Bounds at a point in elastic
354
Biharmonic functions and their conjugates p 355 The biharmonic
366
Résuméof available facts p 371 Null Fvectors p 372 Null cones p
372
Orthogonality and orthogonal projection p 373 Orthonormalization
380
The vertices V V of two nonintersecting linear subspaces of finite
386
The separation of two straight lines p 389 The separation of two linear
392
The scalar wave equation p 394 Splitting the problem p 395 Eigenvalue
402
Stationary principles for elastic vibrations p 405 Maxwells equations
409
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