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On the initial-boundary value problems for soliton equations
Journal of Experimental and Theoretical Physics Letters, 2001We present a novel approach to solving initial-boundary value problems on the segment and the half line for soliton equations. Our method is illustrated by solving a prototypal and widely applied dispersive soliton equation—the celebrated nonlinear Schroedinger equation. It is well known that the basic difficulty associated with boundaries is that some
Paolo Maria Santini +4 more
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Semigroups for Initial-Boundary Value Problems
2003In the foreword of what later became the classic in semigroup theory, E. Hille [Hi48] wrote in 1948: “The analytical theory of semi-groups is a recent addition to the ever-growing list of mathematical disciplines. It was my good fortune to take an early interest in this discipline and to see it reach maturity. It has been a pleasant association: I hail
Delio Mugnolo +2 more
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Initial-Boundary Value and LC Problems
2013Evaluation of European-style derivatives can be reduced to solving initial value or initial-boundary value problems of parabolic partial differential equations. This chapter discusses numerical methods for such problems. If an American option problem is formulated as a linear complementarity problem, then the only difference between solving a European ...
Xiaonan Wu +3 more
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On an adjoint initial-boundary value problem
Differential Equations, 2008We study an adjoint initial-boundary value problem for linear parabolic equations; which arises when modeling the unidirectional motion of two viscous fluids with a common interface under the action of a pressure gradient. Under some conditions on the pressure gradient, we obtain a priori estimates and show that the solution enters a stationary mode ...
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Some initial-boundary value problems
2016In this chapter we will study initial-boundary value problems and their treatment by methods of quaternionic analysis in combination with classical analytic numerical techniques. We start with a brief discussion of strategies for the treatment of time-dependent parabolic problems. Three methods should be considered here: the horizontal method of lines,
Wolfgang Sprößig +2 more
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Travelling waves in an initial-boundary value problem
Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 1981SynopsisIn this paper we consider the initial-boundary value problem for the semihnear diffusion equation ul=uxx+f(u) on the half-line x>0, when for 0<a<1 f(0)=f(a)=f(1)=0 and f(u)<0 on (0, a), f(u)>0 on (a, 1). For a wide class of initial and boundary values a uniformly valid asymptotic expression is given to which the solution ...
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Difference approximations for initial boundary-value problems
Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences, 1971The problem of the stability of difference approximations for initial boundary-value problems is examined and recent results obtained in this field by Gustafsson, Kreiss & Sundström (1971) are presented. It is shown that even for scalar equations the Ryabenkii-Godunow condition is not sufficient for stability and a procedure is outlined which leads
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On the initial-boundary value problem for the Boltzmann equation
Archive for Rational Mechanics and Analysis, 1992zbMATH Open Web Interface contents unavailable due to conflicting licenses.
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Initial boundary value problems for the method of lines
Journal of Computational Physics, 1980This paper treats the stability of the initial boundary value problem for the method of lines applied to hyperbolic and parabolic partial differential equations in one space dimension. The theory treats the case of variable coefficients and allows for very general boundary conditions. Several examples are given which illustrate the theory.
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The solvability of the initial boundary-value problem for magnetohydrodynamics
Journal of Soviet Mathematics, 1978The three-dimensional, nonlinear, time-dependent problem of magnetohydrodynamics is considered in the case where displacement currents cannot be neglected. The “local” unique solvability of the initial boundary-value problem for this system of equations is proved.
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