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220 Polynomial Approximation of Differential Equations The differential problems examined in this chapter are mainly of academic interest. Closer to physics applications are the examples of chapter twelve. It is our opinion, however, that a correct implementation of more realistic problems starts from a full understanding of the basic patterns. For simplicity, boundary value problems in two or more variables are not considered. Actually, although this extension is sometimes straightforward (see chapter thirteen), most of the times it yields a rather involved theoretical analysis, even for simple model problems, which contrasts with the elementary exposition to which we have tried to adhere.
10 TIME-DEPENDENT PROBLEMS The purpose of this chapter is to analyze the approximation, using spectral methods, of differential problems where the solution is time-dependent. We consider parabolic and hyperbolic partial differential equations in one space variable. Finite-differences are usually employed for the treatment of the time variable, hence part of the analysis will be devoted to this topic. 10.1 The Gronwall inequality A basic result, known as Gronwall lemma, will be used in the following. Several versions exist, but we recall only one of them here. Theorem 10.1.1 - Let T > 0 and C ≥ 0. Let g and h be two continuous functions in the interval [0,T], such that (10.1.1) g(t) ≤ g(0) + Then, we have (10.1.2) g(t) ≤ e Ct t 0 g(0) + [Cg(s) + h(s)]ds, ∀t ∈ [0,T]. t 0 h(s)e −Cs ds , ∀t ∈ [0,T].
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PREFACE This book is devoted to the
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CONTENTS 1. Special families of pol
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Contents ix 7. Derivative Matrices
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1 SPECIAL FAMILIES OF POLYNOMIALS A
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Special Families of Polynomials 3 n
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Special Families of Polynomials 5 B
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Special Families of Polynomials 7 1
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Special Families of Polynomials 9 (
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Special Families of Polynomials 11
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2 ORTHOGONALITY Inner products and
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Orthogonality 23 The function w is
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Orthogonality 25 As we promised, we
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Orthogonality 27 Let us now define
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Orthogonality 29 Therefore, one obt
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Orthogonality 31 From (2.3.8), we g
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Orthogonality 33 only mention the r
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3 NUMERICAL INTEGRATION As we shall
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Numerical Integration 37 In the cas
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Numerical Integration 39 (3.1.9) z
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Numerical Integration 41 We give th
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Numerical Integration 43 where 1
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Numerical Integration 45 (3.2.10)
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Numerical Integration 47 This means
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Numerical Integration 49 (3.4.2) w
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Numerical Integration 51 3.5 Gauss-
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Numerical Integration 53 The proof
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Numerical Integration 55 3.7 Clensh
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Numerical Integration 57 the first
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Numerical Integration 59 (3.8.14) p
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Numerical Integration 61 (3.9.5) p
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Numerical Integration 63 (3.10.5)
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66 Polynomial Approximation of Diff
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100 Polynomial Approximation of Dif
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REFERENCES adams r.(1975), Sobolev
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References 295 canuto c., hussaini
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References 297 friedman a.(1959), F
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References 299 jackson d.(1930), Th
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References 301 pavoni d.(1988), Sin
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References 303 vainikko g.m.(1964),
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