Preface |
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xi | |
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1 | (2) |
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3 | (40) |
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1.1 What are Partial Differential Equations? |
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3 | (3) |
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1.2 PDEs We Can Already Solve |
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6 | (4) |
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1.3 Initial and Boundary Conditions |
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10 | (2) |
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1.4 Linear PDEs--Definitions |
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12 | (4) |
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1.5 Linear PDEs--The Principle of Superposition |
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16 | (3) |
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1.6 Separation of Variables for Linear, Homogeneous PDEs |
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19 | (6) |
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25 | (18) |
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41 | (2) |
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43 | (36) |
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2.1 Second-Order, Linear, Homogeneous PDEs with Constant Coefficients |
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43 | (1) |
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2.2 The Heat Equation and Diffusion |
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44 | (10) |
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2.3 The Wave Equation and the Vibrating String |
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54 | (5) |
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2.4 Initial and Boundary Conditions for the Heat and Wave Equations |
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59 | (7) |
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2.5 Laplace's Equation--The Potential Equation |
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66 | (5) |
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2.6 Using Separation of Variables to Solve the Big Three PDEs |
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71 | (8) |
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77 | (2) |
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79 | (50) |
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79 | (1) |
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3.2 Properties of Sine and Cosine |
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80 | (9) |
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89 | (6) |
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3.4 The Fourier Series, Continued |
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95 | (9) |
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3.5 The Fourier Series--Proof of Pointwise Convergence |
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104 | (13) |
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3.6 Fourier Sine and Cosine Series |
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117 | (7) |
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124 | (5) |
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127 | (2) |
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4 Solving the Big Three PDEs on Finite Domains |
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129 | (34) |
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4.1 Solving the Homogeneous Heat Equation for a Finite Rod |
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129 | (9) |
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4.2 Solving the Homogeneous Wave Equation for a Finite String |
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138 | (9) |
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4.3 Solving the Homogeneous Laplace's Equation on a Rectangular Domain |
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147 | (6) |
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4.4 Nonhomogeneous Problems |
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153 | (10) |
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161 | (2) |
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163 | (50) |
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5.1 First-Order PDEs with Constant Coefficients |
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163 | (11) |
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5.2 First-Order PDEs with Variable Coefficients |
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174 | (6) |
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180 | (12) |
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5.4 Characteristics for Semi-Infinite and Finite String Problems |
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192 | (9) |
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5.5 General Second-Order Linear PDEs and Characteristics |
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201 | (12) |
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211 | (2) |
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213 | (64) |
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6.1 The Laplace Transform for PDEs |
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213 | (7) |
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6.2 Fourier Sine and Cosine Transforms |
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220 | (10) |
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6.3 The Fourier Transform |
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230 | (12) |
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6.4 The Infinite and Semi-Infinite Heat Equations |
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242 | (12) |
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6.5 Distributions, the Dirac Delta Function and Generalized Fourier Transforms |
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254 | (12) |
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6.6 Proof of the Fourier Integral Formula |
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266 | (11) |
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275 | (2) |
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7 Special Functions and Orthogonal Polynomials |
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277 | (52) |
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7.1 The Special Functions and Their Differential Equations |
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277 | (8) |
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7.2 Ordinary Points and Power Series Solutions; Chebyshev, Hermite and Legendre Polynomials |
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285 | (7) |
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7.3 The Method of Frobenius; Laguerre Polynomials |
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292 | (8) |
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7.4 Interlude: The Gamma Function |
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300 | (5) |
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305 | (12) |
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7.6 Recap: A List of Properties of Bessel Functions and Orthogonal Polynomials |
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317 | (12) |
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327 | (2) |
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8 Sturm-Liouville Theory and Generalized Fourier Series |
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329 | (46) |
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8.1 Sturm-Liouville Problems |
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329 | (8) |
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8.2 Regular and Periodic Sturm-Liouville Problems |
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337 | (8) |
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8.3 Singular Sturm-Liouville Problems; Self-Adjoint Problems |
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345 | (9) |
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8.4 The Mean-Square or L2 Norm and Convergence in the Mean |
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354 | (7) |
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8.5 Generalized Fourier Series; Parseval's Equality and Completeness |
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361 | (14) |
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373 | (2) |
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9 PDEs in Higher Dimensions |
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375 | (90) |
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9.1 PDEs in Higher Dimensions: Examples and Derivations |
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375 | (11) |
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9.2 The Heat and Wave Equations on a Rectangle; Multiple Fourier Series |
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386 | (16) |
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9.3 Laplace's Equation in Polar Coordinates: Poisson's Integral Formula |
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402 | (12) |
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9.4 The Wave and Heat Equations in Polar Coordinates |
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414 | (11) |
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9.5 Problems in Spherical Coordinates |
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425 | (14) |
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9.6 The Infinite Wave Equation and Multiple Fourier Transforms |
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439 | (17) |
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9.7 Postlude: Eigenvalues and Eigenfunctions of the Laplace Operator; Green's Identities for the Laplacian |
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456 | (9) |
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463 | (2) |
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10 Nonhomogeneous Problems and Green's Functions |
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465 | (74) |
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10.1 Green's Functions for ODEs |
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465 | (19) |
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10.2 Green's Function and the Dirac Delta Function |
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484 | (16) |
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10.3 Green's Functions for Elliptic PDEs (I): Poisson's Equation in Two Dimensions |
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500 | (16) |
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10.4 Green's Functions for Elliptic PDEs (II): Poisson's Equation in Three Dimensions; the Helmholtz Equation |
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516 | (9) |
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10.5 Green's Functions for Equations of Evolution |
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525 | (14) |
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537 | (2) |
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539 | (40) |
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11.1 Finite Difference Approximations for ODEs |
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539 | (12) |
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11.2 Finite Difference Approximations for PDEs |
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551 | (14) |
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11.3 Spectral Methods and the Finite Element Method |
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565 | (14) |
A Uniform Convergence; Differentiation and Integration of Fourier Series |
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579 | (6) |
B Other Important Theorems |
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585 | (6) |
C Existence and Uniqueness Theorems |
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591 | (10) |
D A Menagerie of PDEs |
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601 | (12) |
E MATLAB Code for Figures and Exercises |
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613 | (14) |
F Answers to Selected Exercises |
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627 | (20) |
References |
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647 | (8) |
Index |
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655 | |