Untitled - Cdm.unimo.it

More documents

Recommendations

Info

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].
Page 7 and 8:
PREFACE This book is devoted to the
Page 9 and 10:
CONTENTS 1. Special families of pol
Page 11 and 12:
Contents ix 7. Derivative Matrices
Page 13 and 14:
1 SPECIAL FAMILIES OF POLYNOMIALS A
Page 15 and 16:
Special Families of Polynomials 3 n
Page 17 and 18:
Special Families of Polynomials 5 B
Page 19 and 20:
Special Families of Polynomials 7 1
Page 21 and 22:
Special Families of Polynomials 9 (
Page 23 and 24:
Special Families of Polynomials 11
Page 25 and 26:
Page 27 and 28:
Page 29 and 30:
Page 31 and 32:
Page 33 and 34:
2 ORTHOGONALITY Inner products and
Page 35 and 36:
Orthogonality 23 The function w is
Page 37 and 38:
Orthogonality 25 As we promised, we
Page 39 and 40:
Orthogonality 27 Let us now define
Page 41 and 42:
Orthogonality 29 Therefore, one obt
Page 43 and 44:
Orthogonality 31 From (2.3.8), we g
Page 45 and 46:
Orthogonality 33 only mention the r
Page 47 and 48:
3 NUMERICAL INTEGRATION As we shall
Page 49 and 50:
Numerical Integration 37 In the cas
Page 51 and 52:
Numerical Integration 39 (3.1.9) z
Page 53 and 54:
Numerical Integration 41 We give th
Page 55 and 56:
Numerical Integration 43 where 1
Page 57 and 58:
Numerical Integration 45 (3.2.10)
Page 59 and 60:
Numerical Integration 47 This means
Page 61 and 62:
Numerical Integration 49 (3.4.2) w
Page 63 and 64:
Numerical Integration 51 3.5 Gauss-
Page 65 and 66:
Numerical Integration 53 The proof
Page 67 and 68:
Numerical Integration 55 3.7 Clensh
Page 69 and 70:
Numerical Integration 57 the first
Page 71 and 72:
Numerical Integration 59 (3.8.14) p
Page 73 and 74:
Numerical Integration 61 (3.9.5) p
Page 75:
Numerical Integration 63 (3.10.5)
Page 78 and 79:
66 Polynomial Approximation of Diff
Page 80 and 81:
Page 82 and 83:
Page 84 and 85:
Page 86 and 87:
Page 88 and 89:
Page 90 and 91:
Page 92 and 93:
Page 94 and 95:
Page 96 and 97:
Page 98 and 99:
Page 100 and 101:
Page 102 and 103:
Page 104 and 105:
Page 106 and 107:
Page 108 and 109:
Page 110 and 111:
Page 112 and 113:
100 Polynomial Approximation of Dif
Page 114 and 115:
Page 116 and 117:
Page 118 and 119:
Page 120 and 121:
Page 122 and 123:
Page 124 and 125:
Page 126 and 127:
Page 128 and 129:
Page 130 and 131:
Page 132 and 133:
Page 134 and 135:
Page 136 and 137:
Page 138 and 139:
Page 140 and 141:
Page 142 and 143:
Page 144 and 145:
Page 146 and 147:
Page 148 and 149:
Page 150 and 151:
Page 152 and 153:
Page 154 and 155:
Page 156 and 157:
Page 158 and 159:
Page 160 and 161:
Page 162 and 163:
Page 164 and 165:
Page 166 and 167:
Page 168 and 169:
Page 170 and 171:
Page 172 and 173:
Page 174 and 175:
Page 176 and 177:
Page 178 and 179:
Page 180 and 181:
Page 182 and 183: 170 Polynomial Approximation of Dif
Page 282 and 283:
Page 284 and 285:
Page 286 and 287:
Page 288 and 289:
Page 290 and 291:
Page 292 and 293:
Page 294 and 295:
Page 296 and 297:
Page 298 and 299:
Page 300 and 301:
Page 302 and 303:
Page 305 and 306:
REFERENCES adams r.(1975), Sobolev
Page 307 and 308:
References 295 canuto c., hussaini
Page 309 and 310:
References 297 friedman a.(1959), F
Page 311 and 312:
References 299 jackson d.(1930), Th
Page 313 and 314:
References 301 pavoni d.(1988), Sin
Page 315:
References 303 vainikko g.m.(1964),
show all

Untitled - Cdm.unimo.it

Create successful ePaper yourself

Delete template?

Save as template?