Generation Of Multivariate Hermite Interpolating Polynomials

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ISBN: 9781584885726 | 1584885726
Cover: Hardcover
Copyright: 8/23/2005

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Generation of Multivariate Hermite Interpolating Polynomials advances the study of approximate solutions to partial differential equations by presenting a novel approach that employs Hermite interpolating polynomials and bysupplying algorithms useful in applying this approach.Organized into three sections, the book begins with a thorough examination of constrained numbers, which form the basis for constructing interpolating polynomials. The author develops their geometric representation in coordinate systems in several dimensions and presents generating algorithms for each level number. He then discusses their applications in computing the derivative of the product of functions of several variables and in the construction of expression for n-dimensional natural numbers. Section II focuses on the construction of Hermite interpolating polynomials, from their characterizing properties and generating algorithms to a graphical analysis of their behavior.The final section of the book is dedicated to the application of Hermite interpolating polynomials to linear and nonlinear differential equations in one or several variables. Of particular interest is an example based on the author's thermal analysis of the space shuttle during reentry to the earth's atmosphere, wherein he uses the polynomials developed in the book to solve the heat transfer equations for the heating of the lower surface of the wing.

1 Constrained Numbers

(284)

1 Constrained coordinate system

(28)

1.1 Definition

(16)

1.1.1 Operations with constrained numbers

(2)

1.1.2 Operations with sets of constrained numbers

(5)

1.2 Geometric representation

(3)

1.2.1 Contravariant coordinate representation

(1)

1.2.2 Covariant coordinate representation (dual system)

(2)

1.3 Visualization of the coordinate axes

(1)

1.3.1 Orthogonal representation of the coordinate axes

(1)

1.3.2 Oblique representation of the coordinate axes

(1)

1.4 Geometric location of the constrained numbers

(2)

1.4.1 One-dimensional constrained numbers

(1)

1.4.2 Two-dimensional constrained numbers

(1)

1.5 Two-dimensional representation of η-dimensional coordinate axes

(1)

1.6 Zero-dimensional constrained space

(5)

1.6.1 Definitions

(1)

1.6.2 Geometric representation of the zero-dimensional constrained numbers

(4)

2 Generation of the coordinate system

(82)

2.1 Generation as a cartesian product of two sets

(2)

2.1.1 One-dimensional constrained numbers

(1)

2.1.2 Two-dimensional constrained numbers

(1)

2.1.3 δ-dimensional constrained numbers

(1)

2.2 Generation of the elements of the set Lδlfor the underlying set Z

(5)

2.2.1 Rules for the generation

(3)

2.2.2 Rules for the particular cases

(1)

2.2.3 Algorithm for the generation of the elements

(1)

2.3 Generation of the elements of the set Lγ l for the underlying set Z

(2)

2.3.1 Algorithm for the generation of the elements

(2)

2.4 Generation of the elements of the set Lγl for the underlying set Z+

(6)

2.4.1 Rules for the generation

(3)

2.4.2 Rules for the particular cases

(1)

2.4.3 Algorithm for the generation of the elements of the set Lγl using the coordinate numbers

(2)

2.4.4 Algorithm for the generation of the elements of the set Lγl using sublevels

(1)

2.5 Generation of the elements of the set Lγl for the underlying set Z+

(3)

2.5.1 Algorithm for the generation of the elements of the set L&gamma:l using the coordinate numbers

(1)

2.5.2 Algorithm for the generation of the elements of the set Lγl using sublevels

(1)

2.6 Examples of zero-dimensional constrained space

(4)

2.6.1 Generation of the elements

(1)

2.6.2 Elements of the set Lº0 and Lº(0)

(1)

2.6.3 Elements of the set Lºl

(1)

2.6.4 Elements of the set Lº l

(1)

2.6.5 Geometric representation of the set Lºl

(1)

2.6.6 Geometric representation of the set Lº l

(1)

2.7 Examples of one-dimensional constrained space

(17)

2.7.1 Generation of the elements

(2)

2.7.2 Elements of the set L¹0 and L¹(0)

(1)

2.7.3 Elements of the sets L¹1 and L¹ (1)

(4)

2.7.4 Elements of the set L¹2 and L¹(2)

(5)

2.7.5 Elements of the set L¹3 and L¹(3)

(5)

2.8 Examples of two-dimensional constrained space

(21)

2.8.1 Generation of the elements

(2)

2.8.2 Elements of the set L²0 and L²(0)

(1)

2.8.3 Elements of the sets L²1 and L²(1)

(4)

2.8.4 Elements of the set L²2 and L²(2)

(8)

2.8.5 Elements of the set L²3 and L²(3)

(6)

2.9 Examples of three-dimensional constrained space

(15)

2.9.1 Generation of the elements

(3)

2.9.2 Elements of the set L³0 and L³(0)

(1)

2.9.3 Elements of the sets L³1 and L³(1)

(3)

2.9.4 Elements of the set L³2 and L³(2)

(3)

2.9.5 Elements of the set L³3 and L³(3)

101

(5)

2.10 Properties of sets of constrained numbers

106

(7)

3 Natural coordinates

113

(22)

3.1 One-dimensional natural coordinate numbers

113

(4)

3.1.1 Location of a point

113

(1)

3.1.2 Relation between the one-dimensional natural coordinate numbers and the one-dimensional cartesian coordinate numbers

114

(2)

3.1.3 Relation between the one-dimensional natural coordinate numbers and the two-dimensional cartesian coordinate numbers

116

(1)

3.2 Two-dimensional natural coordinate numbers

117

(6)

3.2.1 Location of a point

117

(3)

3.2.2 Relation between the two-dimensional natural coordinate numbers and the two-dimensional cartesian coordinate numbers

120

(1)

3.2.3 Coordinates orthogonal to the sides of the triangle

121

(2)

3.3 Three-dimensional natural coordinate numbers

123

(5)

3.3.1 Location of a point

123

(2)

3.3.2 Relation between the three-dimensional natural coordinate numbers and the three-dimensional cartesian coordinate numbers

125

(1)

3.3.3 Coordinates orthogonal to the faces of the tetrahedron

126

(2)

3.4 δ-dimensional natural coordinate numbers

128

(7)

3.4.1 Location of a point

128

(1)

3.4.2 Relation between the δ-dimensional natural coordinate numbers and the δ-dimensional cartesian coordinate numbers

128

(2)

3.4.3 Coordinates orthogonal to the faces of the δ-hedron

130

(5)

4 Computation of the number of elements

135

(50)

4.1 Zero-dimensional constrained space

135

(4)

4.1.1 Underlying set Z

135

(2)

4.1.2 Underlying set Z+

137

(1)

4.1.3 Summary for the zero-dimensional space

138

(1)

4.2 One-dimensional constrained space

139

(11)

4.2.1 Underlying set Z

139

(3)

4.2.2 Underlying set Z+

142

(7)

4.2.3 Summary for the one-dimensional numbers

149

(1)

4.3 Two-dimensional space

150

(13)

4.3.1 Underlying set Z

150

(5)

4.3.2 Underlying set Z+

155

(7)

4.3.3 Summary for the two-dimensional numbers

162

(1)

4.4 Three-dimensional constrained space

163

(7)

4.4.1 Underlying set Z

163

(3)

4.4.2 Underlying set Z+

166

(3)

4.4.3 Summary for the three-dimensional numbers

169

(1)

4.5 δ-dimensional constrained space

170

(1)

4.5.1 Underlying set Z+

170

(1)

4.6 Summary

171

(4)

4.6.1 If the underlying set is Z

171

(2)

4.6.2 If the underlying set is Z+

173

(2)

4.7 Mathematical tools

175

(10)

4.7.1 Sum of the terms of an arithmetic progression

175

(1)

4.7.2 Sum of squares

175

(2)

4.7.3 Sum of cubes

177

(1)

4.7.4 Selected combinatorics properties

178

(6)

4.7.5 Pochhammer's symbol

184

(1)

5 An ordering relation

185

(32)

5.1 Definition of order

185

(6)

5.1.1 Possible definitions

185

(2)

5.1.2 Adopted definition

187

(4)

5.2 Integer numbers as the underlying set

191

(13)

5.2.1 Zero-dimensional constrained space

191

(1)

5.2.2 One-dimensional constrained space

191

(4)

5.2.3 Two-dimensional constrained space

195

(5)

5.2.4 Three-dimensional constrained space

200

(4)

5.3 Nonnegative integer numbers as the underlying set

204

(6)

5.3.1 Zero-dimensional constrained space

204

(1)

5.3.2 One-dimensional constrained space

205

(2)

5.3.3 Two-dimensional space

207

(2)

5.3.4 Generalization for the constrained numbers

209

(1)

5.4 Properties of the ordering function

210

(1)

5.5 Summary

211

(6)

5.5.1 Summary for the underlying set is Z

211

(3)

5.5.2 Summary for the underlying set is Z+

214

(3)

6 Application to symbolic computation of derivatives

217

(68)

6.1 Polynomial notation

217

(7)

6.1.1 Examples

217

(2)

6.1.2 Notation for a complete polynomial

219

(1)

6.1.3 Examples

220

(3)

6.1.4 Notation for the terms in the level l of complete polynomials

223

(1)

6.2 Partial derivative notation

224

(1)

6.3 Derivative of order upsilon of a polynomial

225

(5)

6.3.1 Polynomials whose powers are elements in the set Z+

226

(2)

6.3.2 Polynomial whose powers are elements in the set Z

228

(1)

6.3.3 Algorithm to obtain the derivative of a polynomial whose powers are elements in the set Z+

229

(1)

6.3.4 Algorithm to obtain the derivative of a polynomial whose powers are elements in the set Z

229

(1)

6.4 Partial integration notation

230

(3)

6.4.1 Integration of order upsilon of a polynomial

231

(2)

6.5 Elements characterizing a polynomial

233

(3)

6.6 Order of derivative

236

(1)

6.7 Derivative of the product of functions in one variable

236

(10)

6.7.1 Derivative of the product of two functions using constrained multiplication

236

(2)

6.7.2 Derivative of the product of t functions using constrained multiplication

238

(1)

6.7.3 Derivative of the product of two functions in one variable

239

(2)

6.7.4 Derivative of the product of three functions in one variable

241

(2)

6.7.5 Derivative of the product of t functions in one variable

243

(3)

6.8 Derivative of a product of functions of several variables

246

(17)

6.8.1 Derivative of a product of two functions of two variables

246

(7)

6.8.2 Derivative of the product of t functions of d variables

253

(10)

6.9 Expansion of the power of a sum of functions

263

(2)

6.9.1 Expansion of the power of a sum of two functions

263

(1)

6.9.2 Expansion of the power of a sum of t functions

264

(1)

6.10 Computation of the derivatives of the form 1/(x-β)s

265

(17)

6.10.1 One function and one variable

265

(2)

6.10.2 Product of t functions of the same variable x

267

(5)

6.10.3 Computation of the derivatives of the product of d functions, which are product of kj functions of the same variable xj

272

(8)

6.10.4 Summary

280

(2)

6.11 Computation of the derivatives of the form (coxo + c1x1 + c2)

282

(5)

6.11.1 Derivative of one function

282

(1)

6.11.2 Derivative of product of t functions

283

(2)

II Hermite Interpolating Polynomials

285

(250)

7 Multivariate Hermite Interpolating Polynomial

287

(48)

7.1 Definition

287

(3)

7.2 Polynomial construction

290

(17)

7.2.1 The polynomial Pe(x)

290

(12)

7.2.2 The polynomial QE,N(x)

302

(1)

7.2.3 The polynomial ƒe,n(α,x)

303

(4)

7.3 Computation of the coefficients of ƒe,n(α,x)

307

(8)

7.3.1 Hypersurface region

307

(1)

7.3.2 Construction of the system of equations

308

(1)

7.3.3 Computation of the term [1/P,(x)](upsilon)

308

(1)

7.3.4 Computation of [ƒe,n(α,x)](upsilon)

309

(2)

7.3.5 Solution of the system of equations

311

(4)

7.4 Properties of the Hermite Interpolating Polynomials

315

(20)

7.4.1 Computation of the coefficients of ƒh,n(g,x) in terms of the coefficients of fe,n(α,x) if the reference nodes e and h differ only for one coordinate number

315

(5)

7.4.2 Computation of the coefficients of ƒh,n(g,x) in terms of the coefficients of fe,n(α,x) if the reference nodes e and h differ for several coordinate number

320

(1)

7.4.3 Relation between the polynomials Ψ(n)e,n(x)|x=α and Φ(n)e,n(x)|x=αif Ψ(n)e,n(x)|x=α=ρΦ(n)e,n(x)|x=α

321

(2)

7.4.4 Relation between [1/(n!Pe(x))](upsilon) and [1/n!Pe(x)](μ) if μ is a permutation of upsilon

323

(1)

7.4.5 Equality of the coefficients of the terms of the polynomial ƒe,n(a,x) that are in the same level and which indices are permutations

324

(8)

7.4.6 Computation of the polynomials related to one node in terms of the polynomials related to a symmetric node

332

(3)

8 Generation of the Hermite Interpolating Polynomials

335

(4)

8.1 Polynomials of minimum degree

335

(2)

8.2 The degree of ƒe,n(a,x) is kept constant

337

(2)

8.2.1 The degree of ƒe,n(a,x) is η

337

(2)

9 Hermite Interpolating Polynomials: the classical and present approaches

339

(16)

9.1 Classical approach, one variable

339

(2)

9.2 Computation of a one-variable polynomial

341

(12)

9.2.1 Generating expression

341

(1)

9.2.2 Generation of the polynomials related to the reference node a = -1

342

(1)

9.2.3 Construction of the polynomial Φa,0 (x)

342

(4)

9.2.4 Construction of the polynomial Φa,1(x)

346

(3)

9.2.5 Construction of the polynomial Φa,2 (x)

349

(2)

9.2.6 Generation of the polynomials related to the reference node b = 1

351

(1)

9.2.7 Polynomial Φb,0 (x)

351

(1)

9.2.8 Polynomial Φb,1 (x)

352

(1)

9.2.9 Polynomial Φb,2 (x)

352

(1)

9.3 Comparison between the two techniques

353

(2)

9.3.1 Classical approach

353

(1)

9.3.2 The present approach

353

(2)

10 Normalized symmetric square domain

355

(78)

10.1 Computation of a two-variable polynomial

355

(1)

10.2 Generation of the polynomial Φa0,00(x0, x1) related to the reference node a0 = (-1,-1)

355

(21)

10.2.1 Elements common to all polynomials related to the node a0 = (-1,-1)

356

(2)

10.2.2 Generation of the polynomial Φa0,00(x0,x1)

358

(1)

10.2.3 Computation of the coefficients of the polynomial ƒa0,00(x)

359

(17)

10.3 Generation of the polynomial Φa0,10 (x0,x1) related to the reference node a0 = (-1,-1)

376

(12)

10.3.1 Expression for the polynomial φa0,10(x0,x1)

376

(1)

10.3.2 Computation of the polynomial ƒa0,n(x)

376

(9)

10.3.3 Polynomial Φa0,10 (x0, x1)

385

(1)

10.3.4 Summary of the polynomials Φa0,n, related to the reference node a0 = (-1,-1)

386

(2)

10.4 Generation of the polynomials Φa1,n related to the reference node (1,-1)

388

(3)

10.4.1 Generation of the polynomials Φa1,n

388

(1)

10.4.2 Summary of the polynomials Φa1,n related to the reference node a 1 = (1,-1)

389

(2)

10.5 Generation of the polynomials Φa2,n, related to the reference node a2=(1,1)

391

(4)

10.5.1 Generation of the polynomials

391

(2)

10.5.2 Summary of the polynomials Φa2,n related to the reference node a2 = (1,1)

393

(2)

10.6 Generation of the polynomials Φa3,n related to the reference node a3 = (-1,1)

395

(3)

10.6.1 Generation of the polynomials

395

(1)

10.6.2 Summary of the polynomials Φa3,n related to the reference node a3 = (-1,1)

396

(2)

10.7 Generation of the polynomial Φa0,00(x0,x1)) related to the reference node a0 = (-1,-1) with q = 6

398

(35)

10.7.1 Computation of the coefficients of the polynomial ƒa0,00(x)

398

(29)

10.7.2 Summary of the polynomials Φa0,n related to the reference node ao = (-1,-1) and q = 6

427

(6)

11 Rectangular nonsymmetric domain

433

(34)

11.1 Generation of a two-variable polynomial in a rectangular non-symmetric domain

433

(1)

11.2 Generation of the polynomial Φa3,00(x0,x1) related to the reference node a3 = (2,4)

433

(25)

11.2.1 Information about the polynomial

433

(2)

11.2.2 Common factors related to the reference node a3 = (2,4)

435

(1)

11.2.3 Expression for the polynomial Qa3,n(x)

436

(1)

11.2.4 Expression for the polynomial Φa3,n(x)

436

(1)

11.2.5 Properties of the polynomial Φa3,00 (x0,x1)

436

(1)

11.2.6 Construction of the constant term n!Pa3(α)

437

(1)

11.2.7 Construction of the polynomial Qa3,a(x)

437

(1)

11.2.8 The polynomial Φa3,00 (x0,x1)

437

(1)

11.2.9 Computation of the coefficients of the polynomial ƒa3,00 (x)

437

(21)

11.3 Summary of the polynomials

458

(9)

11.3.1 Polynomials Φe,i related to the node a1 = (6, 12)

458

(2)

11.3.2 Polynomials Φe,i related to the node &alhpa;2 = (2, 12)

460

(2)

11.3.3 Polynomials Φe,n related to the node a3 = (2,4)

462

(2)

11.3.4 Polynomials Φe,i related to the node a4 = (6,4)

464

(3)

12 Generic domains

467

(14)

12.1 Expression for the polynomial

467

(2)

12.1.1 Information about the polynomial

467

(1)

12.1.2 Properties of the polynomial Φa3,00(x0,x1)

468

(1)

12.2 Expression for the polynomial Φe,n(x)

469

(5)

12.2.1 Common factors related to the reference node a3 = (-5,2)

472

(1)

12.2.2 Expression for the polynomial Qa3,n(x)

472

(1)

12.2.3 Construction of the constant term n!Pa3(α)

472

(1)

12.2.4 Construction of the polynomial Qa3,n(x)

472

(1)

12.2.5 The polynomial Φa3,00 (x0,x1)

473

(1)

12.2.6 Computation of the coefficients of the polynomial ƒa3,00 (a,x)

473

(1)

12.3 Triangular domain

474

(7)

12.3.1 Information about the polynomial

474

(1)

12.3.2 Common factors related to the reference node ao = square root of (0, square root of 3/2)

475

(1)

12.3.3 Expression for the polynomial Qa,0n,(x)

476

(1)

12.3.4 Expression for the polynomial Φa0,n(x)

476

(1)

12.3.5 Properties of the polynomial Φa0,00(x0,x1)

476

(1)

12.3.6 Construction of the constant term n!Pa0 (α)

477

(1)

12.3.7 Construction of the polynomial Qa0,n(x)

477

(1)

12.3.8 The polynomial Φa0,00(x0,x1)

477

(1)

12.3.9 Computation of the coefficients of the polynomial ƒa0,00 (x)

477

(4)

13 Extensions of the constrained numbers

481

(6)

13.1 Comparison to the divided difference method

481

(2)

13.1.1 Comparison

481

(1)

13.1.2 Example

482

(1)

13.2 Generalization to the set of positive real numbers

483

(2)

13.3 Constrained coordinate system with matrices

485

(2)

14 Field of the complex numbers

487

(12)

14.1 Generalization to the field of the complex numbers

487

(1)

14.2 Generation of the polynomial Φao,o(x) related to the reference node ao = -1 -i

488

(9)

14.2.1 Elements common to all polynomials related to the node ao = (-1 -i)

488

(2)

14.2.2 Generation of the polynomial Φa0,0(x) related to the reference node ao = 1 -i

490

(1)

14.2.3 Computation of the coefficients of the polynomial ƒa0,00(x)

491

(5)

14.2.4 Polynomials related to the node -1 -i

496

(1)

14.3 Polynomial Φa1,0(x) related to the reference node a1 = 1 -i

497

(1)

14.4 Polynomial Φa2,0(x) related to the reference node a2 = -1 + i

498

(1)

14.5 Polynomial Φa3,0(x) related to the reference node a3 = 1 + i

498

(1)

15 Analysis of the behavior of the Hermite Interpolating Polynomials

499

(36)

15.1 One dimensional Hermite Interpolating Polynomials

499

(2)

15.1.1 The set of functions

499

(2)

15.2 Graphical representation of the Hermite Interpolating Polynomials Φ(μ)a,n (x) for η = 4

501

(4)

15.2.1 Graphic of Φa,0(x) and its derivatives

501

(1)

15.2.2 Graphic of Φa,1(x) and its derivatives

502

(1)

15.2.3 Graphic of the polynomial Φa,2 (x) and its derivatives

502

(2)

15.2.4 Graphic of Φa,3(x) and its derivatives

504

(1)

15.2.5 Graphic of the polynomial Φa,4(x) and its derivatives

504

(1)

15.3 Graphic of polynomials Φ(μ)a,n(x), n = 0, 1, 2, 3, 4

505

(4)

15.3.1 Graphic of polynomials Φ(μ)a,n(x), n = 0, 1, 2, 3, 4

505

(1)

15.3.2 Graphic of polynomials Φ(¹)a,n(x), n = 0, 1, 2, 3, 4

506

(1)

15.3.3 Graphic of the polynomials Φ(²)a,n(x),n = 0, 1, 2, 3, 4

506

(1)

15.3.4 Graphic of the polynomials &PHi;(³)a,n(x),n = 0, 1, 2, 3, 4

507

(1)

15.3.5 Graphic of the polynomials Φ(4)a,n(x),n = 0, 1, 2, 3, 4

508

(1)

15.3.6 Graphic of the polynomials Φa,i,(x),i = 0, 3,..., 27

508

(1)

15.3.7 Graphic of the first derivatives of the polynomials Φ(0)a,i(x),i = 3, 6,..., 27

509

(1)

15.3.8 Graphic of the second derivatives of the polynomials Φ(0)a,i(x),i = 3, 6,..., 27

509

(1)

15.4 Graphical representation of the Hermite Interpolating Polynomials Φ(&mu)a,n(x) and Φ(μ)b,n(x) for n = 0, 1, 2, 3, 4

509

(8)

15.4.1 Graphic of the polynomials Φ(º)a,n(x) and Φ(0)b,n(x) for n = 0, 1, 2, 3, 4

510

(1)

15.4.2 Graphic of the polynomials Φ(¹)a,n(x) and Φ(¹)b,n(x) for n = 0, 1, 2, 3, 4

511

(2)

15.4.3 Graphic of the polynomials Φ(²)a,n(x) and Φ(²)b,n(x) for n = 0, 1, 2, 3, 4

513

(1)

15.4.4 Graphic of the polynomials Φ(³)a,n(x) and Φ(³)b,n(x) for n = 0, 1, 2, 3, 4

514

(2)

15.4.5 Graphic of the polynomials Φ(4)a,n(x) and Φ(4)b,n(x) for n = 0, 1, 2, 3, 4

516

(1)

15.5 One-dimensional modified Hermite Interpolating Polynomials

517

(4)

15.5.1 The set of functions

517

(4)

15.6 Analysis of the error for selected degrees of ƒa,n (a, x)

521

(2)

15.6.1 Approximation to a line

521

(2)

15.7 Selected examples of Hermite Interpolating Polynomials

523

(14)

15.7.1 One-dimensional Hermite Interpolating Polynomial

523

(1)

15.7.2 Two-dimensional Hermite Interpolating Polynomial

523

(1)

15.7.3 Three-dimensional Hermite Interpolating Polynomial

524

(1)

15.7.4 Summary of the polynomials Φ(n0,n1,n2)e,n related to the node ae = (-1, -1, -1)

524

(3)

15.7.5 Summary of the polynomials Φ(n0,nn,n2)e,n related to the node ae = (-1, -1, -1) and η = 4

527

(5)

15.7.6 Four-dimensional Hermite Interpolating Polynomial

532

(1)

15.7.7 Summary of the polynomials Φ(n0,n1,n2,n3)e,n related to the node ae = (-1, -1, -1, -1) and η = 3

532

(3)

III Selected applications

535

(124)

16 Construction of the approximate solution

537

(14)

16.1 One-dimensional problems

537

(3)

16.1.1 Solution expanded in terms of the Hermite Interpolating Polynomials

537

(1)

16.1.2 Coordinate transformation

538

(2)

16.1.3 One-dimensional boundary conditions

540

(1)

16.2 Higher-dimensional problems

540

(8)

16.2.1 Analysis of the boundary conditions

540

(2)

16.2.2 Representation of a two-variable polynomial on any basis

542

(1)

16.2.3 Representation of a three-variable polynomial on any basis

543

(1)

16.2.4 Example of a three-variable polynomial on a Chebyshev basis

544

(1)

16.2.5 Example of the two-dimensional sine function

545

(1)

16.2.6 The polynomial approximation

546

(1)

16.2.7 Computation of the coefficients of the expansion of the solution

547

(1)

16.3 Residual analysis

548

(1)

16.3.1 Residual analysis of the governing equation

548

(1)

16.3.2 Residual analysis of the solution

548

(1)

16.3.3 Residual analysis in the domain

549

(1)

16.4 Comments about the computer program

549

(2)

17 One-dimensional two-point boundary value problems

551

(52)

17.1 Linear differential equation

551

(9)

17.1.1 Approximate solution in the domain [-0.5,0.5]

551

(4)

17.1.2 Approximate solution in the domain [-1,1]

555

(4)

17.1.3 Error analysis

559

(1)

17.2 Nonlinear differential equation

560

(19)

17.2.1 Modeling viscoelastic flows: case I

560

(9)

17.2.2 Modeling viscoelastic flows: case II

569

(9)

17.2.3 Analysis of the solution in the interval [-1,1]

578

(1)

17.3 Flow of non-Newtonian fluids: Powell-Eyring model

579

(7)

17.3.1 Prandtl-Eyring model: case I

579

(4)

17.3.2 Powell-Eyring model: case II

583

(3)

17.4 Two-point boundary value problem

586

(17)

17.4.1 Singular perturbation: problem I

586

(10)

17.4.2 Singular perturbation: problem II

596

(4)

17.4.3 Singular perturbation: problem III

600

(3)

18 Application to problems with several variables

603

(14)

18.1 Application to heat problems

603

(5)

18.1.1 One-dimension heat flow equation

603

(2)

18.1.2 Two-dimension heat flow equation

605

(1)

18.1.3 Two-dimension heat flow equation with lateral heat loss

606

(2)

18.2 Application to the wave equation

608

(5)

18.2.1 Homogeneous one-dimension wave equation

608

(1)

18.2.2 Inhomogeneous one-dimension wave equation

609

(1)

18.2.3 Homogeneous two-dimension wave equation

610

(2)

18.2.4 Inhomogeneous two-dimension wave equation

612

(1)

18.3 Nonlinear partial differential equation

613

(1)

18.4 Inhomogeneous advection equation

614

(3)

19 Thermal analysis of the surface of the space shuttle

617

(42)

19.1 Physical model

617

(6)

19.1.1 Thermal properties

617

(1)

19.1.2 Material modeling

618

(3)

19.1.3 Properties of the remaining layers

621

(1)

19.1.4 Heat generated at the external surface

622

(1)

19.1.5 Geometry

622

(1)

19.1.6 Temperatures

623

(1)

19.2 Mathematical model

623

(5)

19.2.1 Nonlinear governing differential equations

623

(1)

19.2.2 Linear governing differential equations

624

(1)

19.2.3 Summary of governing equations

624

(1)

19.2.4 Boundary conditions

625

(3)

19.3 Construction of the approximate solution

628

(8)

19.3.1 Types of solutions

628

(2)

19.3.2 Second-order differential equation formulation

630

(5)

19.3.3 First-order differential equation formulation

635

(1)

19.3.4 Computation of the residual of the differential equation

636

(1)

19.4 Numerical computation

636

(4)

19.5 Analysis of the solutions

640

(9)

19.5.1 Governing equation residual analysis

640

(1)

19.5.2 Two-temperature boundary condition approximate solution

641

(2)

19.5.3 Heat-flux and temperature boundary condition

643

(2)

19.5.4 First-order differential governing equation

645

(4)

19.6 Temperature distribution

649

(2)

19.7 Coefficients for the approximate solution

651

(5)

19.7.1 Power series, case 1 degree 15

651

(1)

19.7.2 Chebyshev polynomial, case 1 degree 15

652

(1)

19.7.3 Power series, case 7 degree 15

652

(1)

19.7.4 Chebyshev polynomial, case 7 degree 15

653

(1)

19.7.5 Power series, case 8 degree 15

653

(1)

19.7.6 Chebyshev polynomial, case 8 degree 15

654

(1)

19.7.7 Power series, case 9 degree 15

654

(1)

19.7.8 Chebyshev polynomial, case 9 degree 15

655

(1)

19.8 Conclusion

656

(3)

References

659

(4)

Index

663

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Generation Of Multivariate Hermite Interpolating Polynomials

Summary

Table of Contents

Supplemental Materials