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296 Polynomial Approximation of Differential Equations dettman j.w.(1969), Mathematical Methods in Physics and Engineering, McGraw-Hill, New York. de veronico m.c.(1991), Approssimazione dell’equazione di Schrödinger mediante polinomi algebrici, Thesis, University of Pavia, Italy. de veronico m.c., funaro d. & reali g.c.(1991), A novel numerical technique to investigate nonlinear guided waves: approximation of nonlinear Schrödinger equation by nonperidic pseudospectral methods, pre-print n.800, Istituto di Analisi Numerica del C.N.R., Pavia, Italy. deville m. & mund e.(1985), Chebyshev pseudospectral solution of second-order elliptic equations with finite element pre-conditioning, J. Comput. Phys., 60, pp.517-533. ditzian z. & totik v.(1987), Moduli of Smoothness, Springer-Verlag, New York. douglas j. Jr. & rachford h.h. Jr.(1956), On the numerical solution of heat conduction problems in two or three space variables, Trans. Amer. Math. Soc., 82, pp.421-439. dryja m. & widlund o.b.(1990), Towards a unified theory of domain decomposition algorithms for elliptic problems, in Domain Decomposition Methods for P.D.E. (T.F. Chan, R. Glowinski, J. Periaux & O.B. Widlund Eds.), SIAM, Philadelphia. dubiner m.(1987), Asymptotic analysis of spectral methods, J. Sci. Comput., 2, pp.3-32. duff i.s., erisman a.m. & reid j.k.(1986), Direct Methods for Sparse Matrices, Clarendon Press, Oxford. duffin r.j. & schaeffer a.c.(1941), A refinement of an inequality of the brothers Markoff, Trans. Amer. Math. Soc., 50, pp.517-528. durand l.(1985), Lagrangian differentiation, Gauss-Jacobi integration and Sturm-Liouville eigenvalue problems, Int. Symp. on Orthogonal Polynomials and their Applications, Bar-le-Duc (France), october 1984, Lecture Notes in Mathematics, Springer-Verlag. eckhaus w.(1979), Asymptotic Analysis of Singular Perturbations, North-Holland, Amsterdam. elliott d.(1981), A convergence theorem for singular integral equations, J. Austral. Math. Soc., B22, pp.539-552. elliott d.(1982), The classical collocation method for solving singular integral equations, SIAM J. Numer. Anal., 19, pp.816-832. elliott c.m. & ockendon j.r.(1982), Weak and Variational Methods for Moving Boundary Problems, Pitman, London. engels h.(1980), Numerical Quadrature and Cubature, Academic Press, London. epstein b.(1962), Partial Differential Equations, An Introduction, McGraw-Hill, New York. erdogan f., gupta g.d. & cook t.s.(1973), Numerical solution of singular integral equations, in Mechanics of Fracture, Vol.1 (G.S. Sih Ed.), Noordhoff, Leyden, pp.368-425. erdös p.(1961), Problems and results on the theory of interpolation. II, Acta Math. Acad. Sci. Hungar, 12, pp.235-244. escobedo m. & kavian o.(1987), Variational problems related to self-similar solutions of the heat equation, Nonlinear Anal., 11, n.10, pp.1103-1133. fasano a. & primicerio m.(1972), Il problema di Stefan con condizioni al contorno non lineari, Ann. Scuola Norm. Sup. Pisa, 26, pp.711-737. feinerman r.p. & newman d.j.(1974), Polynomial Approximation, Waverly Press, Baltimore MD. fletcher c.a.j.(1984), Computational Galerkin Methods, Springer-Verlag, New York. fox l.(1964), An introduction to Numerical Linear Algebra, Clarendon Press, Oxford.
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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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Page 305 and 306: REFERENCES adams r.(1975), Sobolev
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