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This thesis considers algebraic properties of differential equations, and can be divided into two parts. The major distinction among them is that the first part deals with the theory of linear ordinary differential equations, while the second part deals with the nonlinear partial differential equations. In the first part, we present a method to transform the Green's operator into the Green's function. This transformation is already known in the classical case of well-posed two-point boundary value problems, here we extend it to the whole class of Stieltjes boundary problems. In comparison, Stieltjes boundary problems have more freedom from which stems more difficulties. In view of the specification of the boundary conditions: (1) they allow more than two evaluation points. (2) they allow derivatives of arbitrary order; (3) global terms in the form of definite integrals are also allowed. Our results show that the resulting Green's function is not only a piecewise function but also a distribution. Using suitable differential and Rota-Baxter structures, we aim to provide the algebraic underpinning for symbolic computation systems handling such objects. In particular, we show that the Green's function of regular boundary problems (for linear ordinary differential equations) can be expressed naturally in the new setting and that it is characterized by the corresponding distributional differential equation known from analysis. In the second part we concern ourselves with integrable systems. A system of partial differential equations is called \(\textit{integrable}\) if it exhibits infinitely many symmetries. Master symmetries provide a tool which guarantees the existence of infinitely many symmetries and thus help in determining proof of integrability. Using the \(\textit{O}\)-scheme developed by Wang (2015), we compute master symmetries for three new two-component third order Burgers' type systems with non-diagonal constant matrix of leading order terms. These systems can be found in the work of Talati and Turhan (2016). Two more systems with the same dimension are also presented from the ongoing work of Wang et al. In the end, we compute a master symmetry for a Davey-Stewartson type system which is a (2+1)-dimensional system. |
