An Approximation Theory for Conjugate Surfaces and Solutions of Elliptic Multiple Integral Problems: Application to Numerical Solutions of Generalized Laplace's Equation
An Approximation Theory is Given for a Class of Elliptic Quadratic Forms Which Include the Study of Conjugate Surfaces for Elliptic Multiple Integral Problems. These Ideas Follow from the Quadratic Form Theory of Hestenes, Applied to Multiple Integral Problems by Dennemeyer, and Extended with Applications for Approximation Problems by Gregory. the Application of This Theory to a Variety of Approximation Problem Areas in This Setting is Given. These Include Conjugate Surfaces and Conjugate Solutions in the Calculus of Variations, Oscillation Problems for Elliptic Partial Differential Equations, Eigenvalue Problems for Compact Operators, Numerical Approximation Problems, And, Finally, the Intersection of These Problem Areas. in the Final Part of This Paper the Ideas Are Specifically Applied to the Construction and Counting of Negative Vectors in Order to Obtain New Numerical Methods for Solving Laplace-Type Equations and to Obtain the "Euler-Lagrange Equations" for Symmetric-Banded Tridiagonal Matrices. in This New Result (Which Will Allow the Reexamination of Both the Theory and Applications of Symmetric banded Matrices) One Can Construct, in a Meaningful Way, Negative Vectors, Oscillation Vectors, Eigenvectors, and Extremal Solutions of Classical Problems as Well as Efficient Algorithms for the Numerical Solution of Partial Differential Equations. Numerical Examples (Test Runs) Are Given. © 1982.
J. Gregory and R. W. Wilkerson, "An Approximation Theory for Conjugate Surfaces and Solutions of Elliptic Multiple Integral Problems: Application to Numerical Solutions of Generalized Laplace's Equation," Journal of Mathematical Analysis and Applications, vol. 88, no. 1, pp. 231 - 244, Elsevier, Jan 1982.
The definitive version is available at https://doi.org/10.1016/0022-247X(82)90189-5
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01 Jan 1982