By Mats Gyllenberg, Lars-Erik Persson
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Additional info for Analysis, Algebra and Computers in Mathematical Research
The selection of the proper method is, in fact, one of the most diﬃcult tasks of applied mathematics and mainly depends on the structure of the problem to be solved and, to a minor extent, on the aims of the simulation. Mathematical Models and Problems 29 A well-known solution technique of nonlinear initial-boundary value problems for nonlinear partial diﬀerential equations is the generalized collocation method originally called the diﬀerential quadrature method. The original literature can be found in Bellman et al.
Such an analysis requires a detailed knowledge of the theory of partial diﬀerential equations, but this is not one of the aims of this book. , Lions and Magenes (1969); Evans (1998). Here, it is only pointed out that the study of the well-posedness ﬁrst requires a precise deﬁnition of the class of functions to which the solution belongs. Roughly speaking, the following guidelines apply: − The auxiliary conditions imposed must not be too many or a solution will not exist. − The auxiliary conditions imposed must not be too few or the solution will not be unique.
5) (continuous line). Unlike the previous example, now the behavior of the sinc interpolation is relatively more accurate than the one obtained by Lagrange interpolation. The reason is the behavior of the sinc function which allows us to capture the oscillations of the function u(x), in particular near the boundaries, as testiﬁed by the behavior of its derivatives. 5) (continuous line). 5) (continuous line). in this case the error decreases very fast with an increasing number of nodes, even if the Lagrange interpolation shows a residual error which is signiﬁcantly diﬀerent from zero.
Analysis, Algebra and Computers in Mathematical Research by Mats Gyllenberg, Lars-Erik Persson