Least Squares Fourier Series Solutions to Boundary Value Problems

Fourier series solutions are obtained for mixed boundary value problems for the Laplace equation in the plane and in the sphere. The nature of the boundary conditions presents difficulties in obtaining the Fourier coefficients, thus suggesting the use of variational procedures, e.g., least squares, to obtain the coefficients in the separated variable solutions. It is shown that for certain types of mixed boundary conditions numerically useful solutions can be obtained employing a hand held, programmable calculator, whereas other mixed boundary conditions require more computing power, say that needed to invert a $16 \times 16$ matrix. Modern applications of mixed boundary value problems are given, e,g., cryogenic thermal insulation, nuclear power generation, nondestructive testing, transportation of liquified natural gas, etc. This material is suitable for presentation in the first (applied) course in partial differential equations.

[1]  Robert B. Kelman,et al.  Algorithms for classic dual trigonometric equations , 1977 .

[2]  R. B. Kelman.  Perturbational analysis of dual trigonometric series associated with boundary conditions of first and third kind , 1978, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[3]  James Barber.  THE DISTURBANCE OF A UNIFORM STEADY-STATE HEAT FLUX BY A PARTIALLY CONDUCTING PLANE CRACK , 1976 .

[4]  R. Collin.  Field theory of guided waves , 1960 .

[5]  P. R. Barrett.  An iterative method for assessing the importance of hairline cracks in fuel rods upon the thermoelastic hoop stress in surrounding cans , 1977 .

[6]  W. Lord,et al.  LEAKAGE FIELD METHODS OF DEFECT DETECTION. , 1972 .

[7]  S. M. Marcus,et al.  Minimum Size and Maximum Packing Density of Nonredundant Semiconductor Devices , 1962, Proceedings of the IRE.

[8]  E. Hobson.  The Theory of Spherical and Ellipsoidal Harmonics , 1955 .

[9]  S. G. Mikhlin,et al.  The problem of the minimum of a quadratic functional , 1965 .

[10]  Temperature Rise in a Heat-Producing Solid behind a Surface Defect , 1952 .

[11]  R. A. Burton,et al.  A limiting stationary configuration with partially contacting surfaces , 1973 .

[12]  M. Cooper,et al.  Thermal contact conductance , 1969 .

[13]  B. Finlayson.  The method of weighted residuals and variational principles : with application in fluid mechanics, heat and mass transfer , 1972 .

[14]  A. C. Rapier,et al.  The thermal conductance of uranium dioxide/stainless steel interfaces , 1963 .

[15]  R.W. Keyes,et al.  Physical limits in digital electronics , 1975, Proceedings of the IEEE.