"Symbolic-Algebraic Methods for Linear Partial Differential Operators"
Symbolic-Algebraic Methods for Linear Partial Differential Operators
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This thesis is devoted to the study of symbolic-algebraic factorization, classification, and integration methods for Linear Partial Differential Operators (LPDOs). A new theoretical notion, an obstacle to factorizations of LPDOs of general form, that simplifies the considerations of factorization algorithms is introduced. A full system of invariants for third-order bivariate hyperbolic LPDOs is found. The factorizations of LPDOs of orders two, three, and four with completely factorable symbols and without any additional requirement are studied. We prove that ``irreducible'' parametric factorizations can exist only for a few certain types of factorizations. For these cases explicit examples are given. For operators of orders two and three, it is shown that a family may be parameterized by at most one function in one variable. New transformations (Generalized Laplace Transformations) of bivariate hyperbolic second order LPDOs are introduced. The important application is the possibility to extend the class of analytically solvable partial differential equations. Examples are given. The results have been obtained with the help of a specially created Maple-package. Also the procedures for computing the obstacles to factorizations and invariants are implemented in the package.