298 Polynomial Approximation of Differential Equations gottlieb d.(1981), The stabil<strong>it</strong>y of pseudospectral Chebyshev methods, Math. Comp., 36, pp.107-118. gottlieb d., gunzburger m. & turkel e.(1982), On numerical boundary treatment for hyperbolic systems, SIAM J. Numer. Anal., 19, pp.671-697. gottlieb d., hussaini m.y. & orszag s.a.(1984), Theory and application of spectral methods, in Spectral Methods for Partial Differential Equations (R.G.Voigt, D.Gottlieb & M.Y.Hussaini Eds.), SIAM-CBMS, Philadelphia, pp.1-54. gottlieb d.& lustman.l.(1983), The spectrum of the Chebyshev collocation operator for the heat equation, SIAM J. Numer. Anal., 20, n.5, pp.909-921. gottlieb d., lustman.l.& tadmor e.(1987a), Stabil<strong>it</strong>y analysis of spectral methods for hyperbolic in<strong>it</strong>ialboundary value problems, SIAM J. Numer. Anal., 24, pp.241-256. gottlieb d., lustman.l.& tadmor e.(1987b), Convergence of spectral methods for hyperbolic in<strong>it</strong>ialboundary value problems, SIAM J. Numer. Anal., 24, pp.532-537. gottlieb d. & orszag s.a.(1977), Numerical Analysis of Spectral Methods, Theory and Applications, CBMS-NSF Regional Conference Series in Applied Mathematics, SIAM, Philadelphia. gottlieb d. & tadmor e.(1990), The CFL cond<strong>it</strong>ion for spectral approximations to hyperbolic in<strong>it</strong>ialboundary value problems, ICASE (Hampton VA) report n. 90-42. gottlieb d. & turkel e.(1985), Topics in spectral methods for time dependent problems, in Numerical Methods in Fluid Dynamics (F. Brezzi Ed.), Springer-Verlag, pp.115-155. greenspan d.(1961), Introduction to Partial Differential Equations, McGraw-Hill, New York. grosch c.e. & orszag s.a.(1977), Numerical solution of problems in unbounded regions: coordinate transformations, J. Comput. Phys., 25, pp.273-296. guelfand j.m., graev m.i. & vilenkin n.j.(1970), Les Distributions, 5 Volumes, Dunod, Paris. hackbusch w.(1985), Multi-Grid Methods and Applications, Spriger Series in Computational Mathematics, Springer-Verlag, Heidelberg. hageman l.a. & young d.m.(1981), Applied Iterative Methods, Academic Press, Orlando. haldenwang p., labrosse g., abboudi s. & deville m.(1984), Chebyshev 3-D spectral and 2-D pseudospectral solvers for the Helmholtz equation, J. Comput. Phys., 55, pp.115-128. halmos p.r.(1950), Measure Theory, D.Van Nostrand Company Inc., New York. hartman s. & mikusiński j.(1961), The Theory of Lebesgue Measure and Integration, Pergamon Press, Oxford. heinrichs w.(1988), Line relaxation for spectral multigrid methods, J. Comput. Phys., 77, n.1, pp.166-182. heinrichs w.(1989), Improved cond<strong>it</strong>ion number for spectral methods, Math. Comp., 53, n.187, pp.103-119. hildebrandt t.h.(1963), Theory of Integration, Academic Press, London. hill j.m. & dewynne j.n.(1987), Heat Conduction, Blackwell Scientific Publications, Oxford. hochstadt h.(1971), The functions of Mathematical Physics, Wiley-Interscience, New York. hochstadt h.(1973), Integral Equations, John Wiley & Sons, New York. hoffman k. & kunze r.(1971), Linear Algebra, Second ed<strong>it</strong>ion, Prentice-Hall, Englewood Cliffs NJ. ioakimidis n.i.(1981), On the weighted Galerkin method of numerical solution of Cauchy type singular integral equations, SIAM J. Numer. Anal., 18, n.6, pp.1120-1127. isaacson e. & keller h.b.(1966), Analysis of Numerical Methods, John Wiley & Sons, New York. jackson d.(1911), Über Genauigke<strong>it</strong> der Annäherung stetiger Funktionen durch ganze rationale Funktionen gegebenen Grades und trigonometrische Summen gegebener Ordnung, Dissertation, Göttingen.
References 299 jackson d.(1930), The Theory of Approximation, American Mathematical Society, New York. jain m.k.(1984), Numerical Solution of Differential Equations, Second ed<strong>it</strong>ion, John Wiley & Sons, New York. kantorovich l.v. & krylov v.i.(1964), Approximate methods of Higher Analysis, Noordhoff, Groningen. karageorghis a. & phillips t.n.(1989), On the coefficients of differential expansions of ultraspherical polynomials, ICASE (Hampton VA) report n. 89-65. kavian o.(1990), Manuscript. kolmogorov a.n. & fomin s.v.(1961), Elements of the Theory of Functions and Functional Analysis, Graylock Press, Albany NY. kopriva d.a.(1986), A spectral multidomain method for the solution of hyperbolic systems, Applied Numer. Math., 2, pp.221-241. kosloff d. & tal-ezer h.(1989), Modified Chebyshev pseudospectral method w<strong>it</strong>h O(N −1 ) time step restriction, ICASE (Hampton VA) report n. 89-71. kreiss h.o. & lorenz j.(1989), In<strong>it</strong>ial-Boundary Value Problems and the Navier-Stokes Equation, Academic Press, London. krenk s.(1975), On quadrature formulas for singular integral equations of the first and the second kind, Quart. Appl. Math., 33, pp.225-232. krenk s.(1978), Quadrature formulae of closed type for solution of singular integral equations, J. Inst. Math. Appl., 22, pp.99-107. kufner a.(1980), Weighted Sobolev Spaces, Teubner Verlagsgesellschaft, Leipzig. ladyzhenskaya o.a.(1969), The Mathematical Theory of Viscous Incompressible Flow (translated from russian), Second ed<strong>it</strong>ion, Gordon and Breach Science Publishers, New York. lagerstrom p.a.(1988), Matched Asymptotic Expansions, Spinger-Verlag, New York. lanczos c.(1956), Applied Analysis, Prentice-Hall, Englewood Cliffs NJ. lapidus l. & seinfeld j.h.(1971), Numerical Solutions of Ordinary Differential Equations, Academic Press, New York. lax p.d. & milgram n.(1954), Parabolic equations, Contribution to the theory of partial differential equations, Ann. of Math. Stud., 33, pp.167-190. lions j.l. & magenes e.(1972), Non-Homogeneous Boundary Value Problems and Applications, Springer-Verlag, New York. lowenthal f.(1975), Linear Algebra w<strong>it</strong>h Linear Differential Equations, John Wiley & Sons, New York. luke y.l.(1969), The Special Functions and their Approximations, Academic Press, New York. lustman l.(1986), The time evolution of spectral discretization of hyperbolic equations, SIAM J. Numer. Anal., 23, pp.1193-1198. lynch r.e., rice j.r. & thomas d.h.(1964), Direct solution of partial differential equations by tensor product methods, Numer. Math., 6, pp.185-199. macaraeg m. & streett c.l.(1986), Improvements in spectral collocation through a multiple domain technique, Applied Numer. Math., 2, pp.95-108. maday y.(1990), Analysis of spectral projectors in one-dimensional domains, Math. Comp., 55, n.192, pp.537-562. maday y.(1991), Résultats d’approximation optimaux pur les opérateurs d’interpolation polynomiale, C.R. Acad. Sci. Paris, t.312, Série I, pp.705-710.
- 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:
Special Families of Polynomials 13
- Page 27 and 28:
Special Families of Polynomials 15
- Page 29 and 30:
Special Families of Polynomials 17
- Page 31 and 32:
Special Families of Polynomials 19
- 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:
68 Polynomial Approximation of Diff
- Page 82 and 83:
70 Polynomial Approximation of Diff
- Page 84 and 85:
72 Polynomial Approximation of Diff
- Page 86 and 87:
74 Polynomial Approximation of Diff
- Page 88 and 89:
76 Polynomial Approximation of Diff
- Page 90 and 91:
78 Polynomial Approximation of Diff
- Page 92 and 93:
80 Polynomial Approximation of Diff
- Page 94 and 95:
82 Polynomial Approximation of Diff
- Page 96 and 97:
84 Polynomial Approximation of Diff
- Page 98 and 99:
86 Polynomial Approximation of Diff
- Page 100 and 101:
88 Polynomial Approximation of Diff
- Page 102 and 103:
90 Polynomial Approximation of Diff
- Page 104 and 105:
92 Polynomial Approximation of Diff
- Page 106 and 107:
94 Polynomial Approximation of Diff
- Page 108 and 109:
96 Polynomial Approximation of Diff
- Page 110 and 111:
98 Polynomial Approximation of Diff
- Page 112 and 113:
100 Polynomial Approximation of Dif
- Page 114 and 115:
102 Polynomial Approximation of Dif
- Page 116 and 117:
104 Polynomial Approximation of Dif
- Page 118 and 119:
106 Polynomial Approximation of Dif
- Page 120 and 121:
108 Polynomial Approximation of Dif
- Page 122 and 123:
110 Polynomial Approximation of Dif
- Page 124 and 125:
112 Polynomial Approximation of Dif
- Page 126 and 127:
114 Polynomial Approximation of Dif
- Page 128 and 129:
116 Polynomial Approximation of Dif
- Page 130 and 131:
118 Polynomial Approximation of Dif
- Page 132 and 133:
120 Polynomial Approximation of Dif
- Page 134 and 135:
122 Polynomial Approximation of Dif
- Page 136 and 137:
124 Polynomial Approximation of Dif
- Page 138 and 139:
126 Polynomial Approximation of Dif
- Page 140 and 141:
128 Polynomial Approximation of Dif
- Page 142 and 143:
130 Polynomial Approximation of Dif
- Page 144 and 145:
132 Polynomial Approximation of Dif
- Page 146 and 147:
134 Polynomial Approximation of Dif
- Page 148 and 149:
136 Polynomial Approximation of Dif
- Page 150 and 151:
138 Polynomial Approximation of Dif
- Page 152 and 153:
140 Polynomial Approximation of Dif
- Page 154 and 155:
142 Polynomial Approximation of Dif
- Page 156 and 157:
144 Polynomial Approximation of Dif
- Page 158 and 159:
146 Polynomial Approximation of Dif
- Page 160 and 161:
148 Polynomial Approximation of Dif
- Page 162 and 163:
150 Polynomial Approximation of Dif
- Page 164 and 165:
152 Polynomial Approximation of Dif
- Page 166 and 167:
154 Polynomial Approximation of Dif
- Page 168 and 169:
156 Polynomial Approximation of Dif
- Page 170 and 171:
158 Polynomial Approximation of Dif
- Page 172 and 173:
160 Polynomial Approximation of Dif
- Page 174 and 175:
162 Polynomial Approximation of Dif
- Page 176 and 177:
164 Polynomial Approximation of Dif
- Page 178 and 179:
166 Polynomial Approximation of Dif
- Page 180 and 181:
168 Polynomial Approximation of Dif
- Page 182 and 183:
170 Polynomial Approximation of Dif
- Page 184 and 185:
172 Polynomial Approximation of Dif
- Page 186 and 187:
174 Polynomial Approximation of Dif
- Page 188 and 189:
176 Polynomial Approximation of Dif
- Page 190 and 191:
178 Polynomial Approximation of Dif
- Page 192 and 193:
180 Polynomial Approximation of Dif
- Page 194 and 195:
182 Polynomial Approximation of Dif
- Page 196 and 197:
184 Polynomial Approximation of Dif
- Page 198 and 199:
186 Polynomial Approximation of Dif
- Page 200 and 201:
188 Polynomial Approximation of Dif
- Page 202 and 203:
190 Polynomial Approximation of Dif
- Page 204 and 205:
192 Polynomial Approximation of Dif
- Page 206 and 207:
194 Polynomial Approximation of Dif
- Page 208 and 209:
196 Polynomial Approximation of Dif
- Page 210 and 211:
198 Polynomial Approximation of Dif
- Page 212 and 213:
200 Polynomial Approximation of Dif
- Page 214 and 215:
202 Polynomial Approximation of Dif
- Page 216 and 217:
204 Polynomial Approximation of Dif
- Page 218 and 219:
206 Polynomial Approximation of Dif
- Page 220 and 221:
208 Polynomial Approximation of Dif
- Page 222 and 223:
210 Polynomial Approximation of Dif
- Page 224 and 225:
212 Polynomial Approximation of Dif
- Page 226 and 227:
214 Polynomial Approximation of Dif
- Page 228 and 229:
216 Polynomial Approximation of Dif
- Page 230 and 231:
218 Polynomial Approximation of Dif
- Page 232 and 233:
220 Polynomial Approximation of Dif
- Page 234 and 235:
222 Polynomial Approximation of Dif
- Page 236 and 237:
224 Polynomial Approximation of Dif
- Page 238 and 239:
226 Polynomial Approximation of Dif
- Page 240 and 241:
228 Polynomial Approximation of Dif
- Page 242 and 243:
230 Polynomial Approximation of Dif
- Page 244 and 245:
232 Polynomial Approximation of Dif
- Page 246 and 247:
234 Polynomial Approximation of Dif
- Page 248 and 249:
236 Polynomial Approximation of Dif
- Page 250 and 251:
238 Polynomial Approximation of Dif
- Page 252 and 253:
240 Polynomial Approximation of Dif
- Page 254 and 255:
242 Polynomial Approximation of Dif
- Page 256 and 257:
244 Polynomial Approximation of Dif
- Page 258 and 259:
246 Polynomial Approximation of Dif
- Page 260 and 261: 248 Polynomial Approximation of Dif
- Page 262 and 263: 250 Polynomial Approximation of Dif
- Page 264 and 265: 252 Polynomial Approximation of Dif
- Page 266 and 267: 254 Polynomial Approximation of Dif
- Page 268 and 269: 256 Polynomial Approximation of Dif
- Page 270 and 271: 258 Polynomial Approximation of Dif
- Page 272 and 273: 260 Polynomial Approximation of Dif
- Page 274 and 275: 262 Polynomial Approximation of Dif
- Page 276 and 277: 264 Polynomial Approximation of Dif
- Page 278 and 279: 266 Polynomial Approximation of Dif
- Page 280 and 281: 268 Polynomial Approximation of Dif
- Page 282 and 283: 270 Polynomial Approximation of Dif
- Page 284 and 285: 272 Polynomial Approximation of Dif
- Page 286 and 287: 274 Polynomial Approximation of Dif
- Page 288 and 289: 276 Polynomial Approximation of Dif
- Page 290 and 291: 278 Polynomial Approximation of Dif
- Page 292 and 293: 280 Polynomial Approximation of Dif
- Page 294 and 295: 282 Polynomial Approximation of Dif
- Page 296 and 297: 284 Polynomial Approximation of Dif
- Page 298 and 299: 286 Polynomial Approximation of Dif
- Page 300 and 301: 288 Polynomial Approximation of Dif
- Page 302 and 303: 290 Polynomial Approximation of Dif
- Page 305 and 306: REFERENCES adams r.(1975), Sobolev
- Page 307 and 308: References 295 canuto c., hussaini
- Page 309: References 297 friedman a.(1959), F
- Page 313 and 314: References 301 pavoni d.(1988), Sin
- Page 315: References 303 vainikko g.m.(1964),