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276 Polynomial Approximation of Differential Equations (12.3.5) ⎧ ⎪⎨ ⎪⎩ − q(k) n − q (k−1) n h p (k) n − p (k−1) n h = − ζ 2 [p(k) = − ζ 2 [q(k) n + p (k−1) n n + q (k−1) n ] ′′ − ǫ 4 [Θ(k) n + Θ (k−1) n ] ′′ − ǫ 4 [Θ(k) n + Θ (k−1) n where Θ (k) n := [p (k) n ] 2 + [q (k) n ] 2 , 0 ≤ k ≤ m. The polynomials p (m) n ][p (k) ][q (k) n + p (k−1) n n + q (k−1) n ] and q (m) n expected to be the approximations of V (·,T) and W(·,T) respectively. Using the notations of section 9.9, (12.3.5) is equivalent to (12.3.6) ⎡ ⎣ Mn −µIn + ǫ ⎡ ⎣ 2ζ ∆(k) n where ∆ (k) n := diag (Θ (k) ⎤⎡ µIn ⎦ Mn 0 0 ∆ (k) n n +Θ (k−1) n ⎣ ¯p(k) n ¯q (k) n ⎤⎡ ⎦ ⎤ ⎡ ⎦ = −⎣ Mn ⎤⎡ −µIn ⎦ ⎣ ¯p(k) n + ¯p (k−1) n ¯q (k) n + ¯q (k−1) n )(η (n) 1 ), · · · ,(Θ (k) µIn ⎤ Mn ⎣ ¯p(k−1) n ¯q (k−1) n ⎦ 1 ≤ k ≤ m, n +Θ (k−1) n ⎤ ⎦ ] are )(η (n) n−1 ) and µ := 2/ζh. The inverse of the matrix on the left-hand side of (12.3.6) can be evaluated once and for all with the help of (9.9.4). For any 1 ≤ k ≤ m, we can solve the nonlinear equation (12.3.6) iteratively. One approach is to construct a sequence of vectors according to the prescription (12.3.7) where ∆ (k,j) n Θ (k,j) n ⎡ ⎣ ¯p(k,j) n ¯q (k,j) n + ǫ 2ζ ⎡ ⎤ ⎦ := ⎡ ⎣ ∆(k,j−1) n ⎣ Mn := diag (Θ (k,j) n −µIn 0 0 ∆ (k,j−1) n ⎤−1 ⎧ µIn ⎨ ⎦ ⎩ Mn − ⎡ ⎣ Mn ⎤⎡ −µIn ⎦ µIn Mn ⎤⎡ := [p (k,j) n ] 2 + [q (k,j) n ] 2 , j ∈ N, and ¯p (k,0) n ⎦ ⎣ ¯p(k,j−1) n ¯q (k,j−1) n + ¯p (k−1) n + ¯q (k−1) n + Θ (k−1) n )(η (n) 1 ), · · · ,(Θ (k,j) n := ¯p (k−1) n , ¯q (k,0) n ⎤⎫ ⎬ ⎦ ⎭ ⎣ ¯p(k−1) n ¯q (k−1) n j ≥ 1, ⎤ ⎦ + Θ (k−1) n )(η (n) n−1 ) with := ¯q (k−1) n .
Examples 277 Therefore, one obtains (12.3.8) lim ⎡ ⎣ j→+∞ ¯p(k,j) n ¯q (k,j) n ⎤ ⎦ = ⎡ ⎣ ¯p(k) n ¯q (k) n ⎤ ⎦, 1 ≤ k ≤ m. The scheme considered here was suggested in delfour, fortin and payre(1981) for finite-difference approximations. It is second-order accurate, i.e., the error in the variable t decays as h 2 . In addition, it preserves certain physical quantities. For example, using formula (3.5.1) for w ≡ 1, we can prove the discrete counterpart of (12.3.2), i.e., that the sum n i=0 Θ(k) n (η (n) i ) ˜w (n) i does not change with k. Figure 12.3.1 - Evolution of an approximate solution of equation (12.3.1).
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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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Page 313 and 314: References 301 pavoni d.(1988), Sin
Page 315: References 303 vainikko g.m.(1964),
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