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280 Polynomial Approximation of Differential Equations Figure 12.4.1 - Plot of J1(x), x ∈ [0,18]. The proof of the orthogonal<strong>it</strong>y relation (which somehow justifies the interest for the Bessel zeroes): (12.4.4) 1 is left to the reader. 0 x Jk(z (k) i (k) x) Jk(z j x) dx = 0, if i = j, k ≥ 1,
13 AN EXAMPLE IN TWO DIMENSIONS To illustrate the construction of polynomial approximations of the solution of a partial differential equation, we examine a simple boundary-value problem in two dimensions. 13.1 Poisson’s equation Let Ω be the open square ] − 1,1[×] − 1,1[ in R 2 . We denote by ∂Ω the boundary of Ω. Given the function f : Ω → R, we want to find the solution U : ¯ Ω → R of the boundary-value problem ⎧ 2 ∂ U ⎪⎨ −∆U := − ∂x (13.1.1) ⎪⎩ 2 + ∂2U ∂y2 U ≡ 0 on ∂Ω, = f in Ω. The partial differential equation in (13.1.1) is known as Poisson’s equation. The symbol ∆ is called the Laplace operator. There are many applications of equation (13.1.1) in the field of physics. For example, if Ω is a plane region of dielectric material surrounded by a grounded conductor, and f is the charge dens<strong>it</strong>y in Ω, then the solution U of (13.1.1) is proportional to the electrostatic potential on Ω.
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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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