24,654 research outputs found
Grassmannian flows and applications to nonlinear partial differential equations
We show how solutions to a large class of partial differential equations with
nonlocal Riccati-type nonlinearities can be generated from the corresponding
linearized equations, from arbitrary initial data. It is well known that
evolutionary matrix Riccati equations can be generated by projecting linear
evolutionary flows on a Stiefel manifold onto a coordinate chart of the
underlying Grassmann manifold. Our method relies on extending this idea to the
infinite dimensional case. The key is an integral equation analogous to the
Marchenko equation in integrable systems, that represents the coodinate chart
map. We show explicitly how to generate such solutions to scalar partial
differential equations of arbitrary order with nonlocal quadratic
nonlinearities using our approach. We provide numerical simulations that
demonstrate the generation of solutions to
Fisher--Kolmogorov--Petrovskii--Piskunov equations with nonlocal
nonlinearities. We also indicate how the method might extend to more general
classes of nonlinear partial differential systems.Comment: 26 pages, 2 figure
Differential-Algebraic Equations and Beyond: From Smooth to Nonsmooth Constrained Dynamical Systems
The present article presents a summarizing view at differential-algebraic
equations (DAEs) and analyzes how new application fields and corresponding
mathematical models lead to innovations both in theory and in numerical
analysis for this problem class. Recent numerical methods for nonsmooth
dynamical systems subject to unilateral contact and friction illustrate the
topicality of this development.Comment: Preprint of Book Chapte
Partial differential systems with nonlocal nonlinearities: Generation and solutions
We develop a method for generating solutions to large classes of evolutionary
partial differential systems with nonlocal nonlinearities. For arbitrary
initial data, the solutions are generated from the corresponding linearized
equations. The key is a Fredholm integral equation relating the linearized flow
to an auxiliary linear flow. It is analogous to the Marchenko integral equation
in integrable systems. We show explicitly how this can be achieved through
several examples including reaction-diffusion systems with nonlocal quadratic
nonlinearities and the nonlinear Schrodinger equation with a nonlocal cubic
nonlinearity. In each case we demonstrate our approach with numerical
simulations. We discuss the effectiveness of our approach and how it might be
extended.Comment: 4 figure
Local Fractional Operator for the Solution of Quadratic Riccati Differential Equation with Constant Coefficients
In this paper, we consider approximate solutions
of fractional Riccati differential equations via the application of
local fractional operator in the sense of Caputo derivative. The
proposed semi-analytical technique is built on the basis of the
standard Differential Transform Method (DTM). Some
illustrative examples are given to demonstrate the effectiveness
and robustness of the proposed technique; the approximate
solutions are provided in the form of convergent series. This
shows that the solution technique is very efficient, and reliable;
as it does not require much computational work, even without
given up accuracy
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