Description
Polynomial operators are a natural generalization of linear operators. Equations in such operators are the linear space analog of ordinary polynomials in one or several variables over the fields of real or complex numbers. Such equations encompass a broad spectrum of applied problems including all linear equations. Often the polynomial nature of many nonlinear problems goes unrecognized by researchers. This is more likely due to the fact that polynomial operators - unlike polynomials in a single variable - have received little attention. Consequently, this comprehensive presentation is needed, benefiting those working in the field as well as those seeking information about specific results or techniques. Polynomial Operator Equations in Abstract Spaces and Applications - an outgrowth of fifteen years of the author's research work - presents new and traditional results about polynomial equations as well as analyzes current iterative methods for their numerical solution in various general space settings.Topics include:oSpecial cases of nonlinear operator equationsoSolution of polynomial operator equations of positive integer degree noResults on global existence theorems not related with contractionsoGalois theoryoPolynomial integral and polynomial differential equations appearing in radiative transfer, heat transfer, neutron transport, electromechanical networks, elasticity, and other areasoResults on the various Chandrasekhar equationsoWeierstrass theoremoMatrix representationsoLagrange and Hermite interpolationoBounds of polynomial equations in Banach space, Banach algebra, and Hilbert spaceThe materials discussed can be used for the following studiesoAdvanced numerical analysisoNumerical functional analysisoFunctional analysisoApproximation theoryoIntegral and differential equationsTables includeoNumerical solutions for Chandrasekhar's equation I to VIoError bounds comparisonoAccelerations schemes I and II for Newton's methodoNewton's methodoSecant methodThe self-contained text thoroughly details results, adds exercises for each chapter, and includes several applications for the solution of integral and differential equations throughout every chapter.
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