Neta B. - Partial Differential Equations, (MA3132 lecture notes).pdf

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PARTIAL DIFFERENTIAL EQUATIONS
MA 3132 LECTURE NOTES
B. Neta Department of Mathematics
Naval Postgraduate School
Code MA/Nd
Monterey, California 93943
October 10, 2002
1996 - Professor Beny Neta
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Contents
1 Introduction and Applications 1
1.1 BasicConceptsandDefinitions ......................... 1
1.2 Applications.................................... 4
1.3 Conduction of Heat in a Rod .......................... 5
1.4 Boundary Conditions ............................... 7
1.5 AVibratingString ................................ 10
1.6 Boundary Conditions ............................... 11
1.7 DiffusioninThreeDimensions.......................... 13
2 Classification and Characteristics 15
2.1 PhysicalClassification .............................. 15
2.2 ClassificationofLinearSecondOrderPDEs .................. 15
2.3 CanonicalForms ................................. 19
2.3.1 Hyperbolic................................. 19
2.3.2 Parabolic ................................. 22
2.3.3 Elliptic . . . ................................ 24
2.4 EquationswithConstantCoecients ...................... 28
2.4.1 Hyperbolic................................. 28
2.4.2 Parabolic ................................. 29
2.4.3 Elliptic . . . ................................ 29
2.5 LinearSystems .................................. 32
2.6 GeneralSolution ................................. 33
3 Method of Characteristics 37
3.1 AdvectionEquation(firstorderwaveequation) ................ 37
3.1.1 NumericalSolution............................ 42
3.2 Quasilinear Equations .............................. 44
3.2.1 The Case S =0 ,c = c ( u ) ........................ 45
3.2.2 GraphicalSolution ............................ 46
3.2.3 Fan-likeCharacteristics.......................... 49
3.2.4 ShockWaves ............................... 50
3.3 SecondOrderWaveEquation .......................... 58
3.3.1 Infinite Domain .............................. 58
3.3.2 Semi-infinite String ............................ 62
3.3.3 Semi Infinite String with a Free End .................. 65
3.3.4 FiniteString................................ 68
3.3.5 ParallelogramRule ............................ 70
4 Separation of Variables-Homogeneous Equations 73
4.1 Parabolicequationinonedimension ...................... 73
4.2 Other Homogeneous Boundary Conditions ................... 77
4.3 EigenvaluesandEigenfunctions ......................... 83
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5 Fourier Series 85
5.1 Introduction.................................... 85
5.2 Orthogonality . . . ................................ 86
5.3 ComputationofCoe cients ........................... 88
5.4 RelationshiptoLeastSquares .......................... 96
5.5 Convergence.................................... 97
5.6 FourierCosineandSineSeries.......................... 99
5.7 TermbyTermDifferentiation .......................... 106
5.8 TermbyTermIntegration ............................ 108
5.9 FullsolutionofSeveralProblems ........................ 110
6 Sturm-Liouville Eigenvalue Problem 120
6.1 Introduction.................................... 120
6.2 Boundary Conditions of the Third Kind .................... 127
6.3 ProofofTheoremandGeneralizations ..................... 131
6.4 LinearizedShallowWaterEquations ...................... 137
6.5 EigenvaluesofPerturbedProblems ....................... 140
7 PDEs in Higher Dimensions 147
7.1 Introduction.................................... 147
7.2 HeatFlowinaRectangularDomain ...................... 148
7.3 VibrationsofarectangularMembrane ..................... 151
7.4 HelmholtzEquation................................ 155
7.5 VibratingCircularMembrane .......................... 158
7.6 Laplace’sEquationinaCircularCylinder ................... 164
7.7 Laplace’sequationinasphere.......................... 170
8 Separation of Variables-Nonhomogeneous Problems 179
8.1 Inhomogeneous Boundary Conditions . ..................... 179
8.2 MethodofEigenfunctionExpansions ...................... 182
8.3 ForcedVibrations................................. 186
8.3.1 PeriodicForcing.............................. 187
8.4 Poisson’sEquation ................................ 190
8.4.1 Homogeneous Boundary Conditions ................... 190
8.4.2 Inhomogeneous Boundary Conditions .................. 192
9 Fourier Transform Solutions of PDEs 195
9.1 Motivation..................................... 195
9.2 FourierTransformpair .............................. 196
9.3 HeatEquation................................... 200
9.4 FourierTransformofDerivatives......................... 203
9.5 FourierSineandCosineTransforms ...................... 207
9.6 FourierTransformin2Dimensions ....................... 211
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10 Green’s Functions 217
10.1Introduction.................................... 217
10.2OneDimensionalHeatEquation......................... 217
10.3 Green’s Function for Sturm-Liouville Problems ................. 221
10.4DiracDeltaFunction............................... 227
10.5 Nonhomogeneous Boundary Conditions ..................... 230
10.6FredholmAlternativeAndModifiedGreen’sFunctions ............ 232
10.7Green’sFunctionForPoisson’sEquation .................... 238
10.8 Wave Equation on Infinite Domains . . ..................... 244
10.9 Heat Equation on Infinite Domains . . ..................... 251
10.10Green’sFunctionfortheWaveEquationonaCube .............. 256
11 Laplace Transform 266
11.1Introduction.................................... 266
11.2SolutionofWaveEquation............................ 271
12 Finite Differences 277
12.1TaylorSeries.................................... 277
12.2FiniteDifferences................................. 278
12.3IrregularMesh................................... 280
12.4ThomasAlgorithm ................................ 281
12.5MethodsforApproximatingPDEs........................ 282
12.5.1 Undeterminedcoecients ........................ 282
12.5.2 IntegralMethod.............................. 283
12.6 Eigenpairs of a Certain Tridiagonal Matrix ................... 284
13 Finite Differences 286
13.1Introduction.................................... 286
13.2DifferenceRepresentationsofPDEs....................... 287
13.3HeatEquationinOneDimension ........................ 291
13.3.1 Implicitmethod.............................. 293
13.3.2 DuFortFrankelmethod ......................... 293
13.3.3 Crank-Nicholsonmethod......................... 294
13.3.4 Theta ( θ )method............................. 296
13.3.5 Anexample ................................ 296
13.4TwoDimensionalHeatEquation ........................ 301
13.4.1 Explicit .................................. 301
13.4.2 CrankNicholson ............................. 302
13.4.3 AlternatingDirectionImplicit...................... 302
13.5Laplace’sEquation ................................ 303
13.5.1 Iterativesolution ............................. 306
13.6VectorandMatrixNorms ............................ 307
13.7 Matrix Method for Stability ........................... 311
13.8 Derivative Boundary Conditions ......................... 312
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13.9HyperbolicEquations............................... 313
13.9.1 Stability . . ................................ 313
13.9.2 EulerExplicitMethod .......................... 316
13.9.3 UpstreamDifferencing .......................... 316
13.10InviscidBurgers’Equation............................ 320
13.10.1LaxMethod................................ 321
13.10.2LaxWendroffMethod .......................... 322
13.11ViscousBurgers’Equation............................ 324
13.11.1FTCSmethod............................... 326
13.11.2LaxWendroffmethod .......................... 328
14 Numerical Solution of Nonlinear Equations 330
14.1Introduction.................................... 330
14.2BracketingMethods................................ 330
14.3FixedPointMethods............................... 332
14.4Example...................................... 334
14.5Appendix ..................................... 336
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