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Preconditioners Based on Fundamental Solutions
BIT Numerical Mathematics, 2005A new type of preconditioner is used in solving systems of linear algebraic equations obtained from the finite difference discretization of partial differential equations. This preconditioner is a discretization of an approximate inverse given by a convolution-like operator with the fundamental solution as a kernel.
Brandén, H., Sundqvist, P.
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Modified Fundamental Solutions
2011As we have seen in Chapters 6 and 7, the integral equations of the second kind that arise in the classical direct and indirect boundary integral formulations may have multiple solutions for certain values of the oscillation frequency ω. In Chapter 7 it was shown that, for the exterior problems, a unique solution does exist if we operate with a special ...
Gavin R. Thomson, Christian Constanda
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2004
Trajectories of a dynamical system, starting from a particular initial state, might evolve towards a steady state of the system. A steady state can be an equilibrium of the system but can also be a (quasi-)periodic motion. The stability of equilibria is (for the hyperbolic case) determined by the eigenvalues of the local linearization of the system ...
Remco I. Leine, Henk Nijmeijer
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Trajectories of a dynamical system, starting from a particular initial state, might evolve towards a steady state of the system. A steady state can be an equilibrium of the system but can also be a (quasi-)periodic motion. The stability of equilibria is (for the hyperbolic case) determined by the eigenvalues of the local linearization of the system ...
Remco I. Leine, Henk Nijmeijer
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Some new fundamental solutions
Mathematical Methods in the Applied Sciences, 1990AbstractThe fundamental solutions of non‐decomposable evolution operators are represented by multi‐dimensional parameter integration formulae. The method is applied to operators occurring in the theories of elasticity, magnetohydrodynamics and heat conduction.
Norbert Ortner, Peter Wagner
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1992
Up to now, all applications of the Dual Reciprocity Method considered is this book were related to the Laplace operator. This means that the governing partial differential equations were initially recast as some kind of Poisson’s equation and the fundamental solution of Laplace’s equation employed to obtain an equivalent integral formulation.
P. W. Partridge +2 more
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Up to now, all applications of the Dual Reciprocity Method considered is this book were related to the Laplace operator. This means that the governing partial differential equations were initially recast as some kind of Poisson’s equation and the fundamental solution of Laplace’s equation employed to obtain an equivalent integral formulation.
P. W. Partridge +2 more
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1993
Abstract We can apply this result in the theory of Diophantine inequalities to the basic plan for classifying Haken manifolds. Remember that the 3-manifold M consists of tetrahedrons—that is, a set of sets of four vertices. Furthermore, any normal surface consists of squares and triangles in the tetrahedrons.
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Abstract We can apply this result in the theory of Diophantine inequalities to the basic plan for classifying Haken manifolds. Remember that the 3-manifold M consists of tetrahedrons—that is, a set of sets of four vertices. Furthermore, any normal surface consists of squares and triangles in the tetrahedrons.
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2003
In Section 3.2, we saw how, using various tricks, solutions in rational x and y of x 2 — dy 2 = 1 could be obtained from two solutions of an equation x 2 - dy 2 = k. Sometimes, the rational numbers turned out to be integers. The chances of this happening would apparently improve with the number of solutions of x 2 - dy 2 = k for a particular k.
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In Section 3.2, we saw how, using various tricks, solutions in rational x and y of x 2 — dy 2 = 1 could be obtained from two solutions of an equation x 2 - dy 2 = k. Sometimes, the rational numbers turned out to be integers. The chances of this happening would apparently improve with the number of solutions of x 2 - dy 2 = k for a particular k.
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Distributions and Fundamental Solutions
2015This chapter is an introduction to distribution theory illustrated by the verification of fundamental solutions of the classical operators \(\Delta _{n}^{k},(\lambda -\Delta _{n})^{k},(\Delta _{n}+\lambda )^{k},\partial _{\bar{z}}, (\partial _{t}^{2} - \Delta _{n})^{k},\partial _{1}\cdots \partial _{k},(\partial _{t} -\lambda \Delta _{n})^{k},(\partial
Norbert Ortner, Peter Wagner
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