762 research outputs found
Asymptotic soliton like solutions to the singularly perturbed Benjamin-Bona-Mahony equation with variable coefficients
The paper deals with a problem of asymptotic soliton like solutions to the
Benjamin-Bona-Mahony (BBM) equaion with a small parameter at the highest
derivative and variable coefficients depending on the variables , as
well as a small parameter. There is proposed an algorithm of constructing the
solutions and there are proved theorems on accuracy with which the solutions
satisfy the BBM equation.Comment: 19 pages, 44 reference
Spatial Manifestations of Order Reduction in Runge-Kutta Methods for Initial Boundary Value Problems
This paper studies the spatial manifestations of order reduction that occur
when time-stepping initial-boundary-value problems (IBVPs) with high-order
Runge-Kutta methods. For such IBVPs, geometric structures arise that do not
have an analog in ODE IVPs: boundary layers appear, induced by a mismatch
between the approximation error in the interior and at the boundaries. To
understand those boundary layers, an analysis of the modes of the numerical
scheme is conducted, which explains under which circumstances boundary layers
persist over many time steps. Based on this, two remedies to order reduction
are studied: first, a new condition on the Butcher tableau, called weak stage
order, that is compatible with diagonally implicit Runge-Kutta schemes; and
second, the impact of modified boundary conditions on the boundary layer theory
is analyzed.Comment: 41 pages, 9 figure
Painlev\'e III and the Hankel Determinant Generated by a Singularly Perturbed Gaussian Weight
In this paper, we study the Hankel determinant generated by a singularly
perturbed Gaussian weight By using the ladder operator approach associated with the orthogonal
polynomials, we show that the logarithmic derivative of the Hankel determinant
satisfies both a non-linear second order difference equation and a non-linear
second order differential equation. The Hankel determinant also admits an
integral representation involving a Painlev\'e III. Furthermore, we consider
the asymptotics of the Hankel determinant under a double scaling, i.e.
and such that is fixed. The
asymptotic expansions of the scaled Hankel determinant for large and small
are established, from which Dyson's constant appears.Comment: 22 page
A Numerical Slow Manifold Approach to Model Reduction for Optimal Control of Multiple Time Scale ODE
Time scale separation is a natural property of many control systems that can
be ex- ploited, theoretically and numerically. We present a numerical scheme to
solve optimal control problems with considerable time scale separation that is
based on a model reduction approach that does not need the system to be
explicitly stated in singularly perturbed form. We present examples that
highlight the advantages and disadvantages of the method
Boundary layer analysis for nonlinear singularly perturbed differential equations
This paper focuses on the boundary layer phenomenon arising in the study of singularly perturbed differential equations. Our tools include the method of lower and upper solutions combined with analysis of the integral equation associated with the class of nonlinear equations under consideration
A Maximum Entropy Method for Solving the Boundary Value Problem of Second Order Ordinary Differential Equations
We propose a new method to solve the boundary value problem for a class of second order linear ordinary differential equations, which has a non-negative solution. The method applies the maximum entropy principle to approximating the solution numerically. The theoretical analysis and numerical examples show that our method is convergent
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