762 research outputs found

    Asymptotic soliton like solutions to the singularly perturbed Benjamin-Bona-Mahony equation with variable coefficients

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    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 xx, tt 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

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    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 IIIβ€²' and the Hankel Determinant Generated by a Singularly Perturbed Gaussian Weight

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    In this paper, we study the Hankel determinant generated by a singularly perturbed Gaussian weight w(x,t)=eβˆ’x2βˆ’tx2,β€…β€Šβ€…β€Šx∈(βˆ’βˆž,∞),β€…β€Šβ€…β€Št>0. w(x,t)=\mathrm{e}^{-x^{2}-\frac{t}{x^{2}}},\;\;x\in(-\infty, \infty),\;\;t>0. 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 IIIβ€²'. Furthermore, we consider the asymptotics of the Hankel determinant under a double scaling, i.e. nβ†’βˆžn\rightarrow\infty and tβ†’0t\rightarrow 0 such that s=(2n+1)ts=(2n+1)t is fixed. The asymptotic expansions of the scaled Hankel determinant for large ss and small ss 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

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    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

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    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

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    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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