Fourier series and numerical methods for partial differential equations [electronic resource] / Richard Bernatz.
The importance of partial differential equations (PDEs) in modeling phenomena in engineering as well as in the physical, natural, and social sciences is well known by students and practitioners in these fields. Striking a balance between theory and applications, Fourier Series and Numerical Methods...
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Full Text (via Wiley) |
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| Main Author: | |
| Format: | Electronic eBook |
| Language: | English |
| Published: |
Hoboken, N.J. :
Wiley,
©2010.
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| Subjects: |
MARC
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| 100 | 1 | |a Bernatz, Richard, |d 1955- |0 http://id.loc.gov/authorities/names/n2010015268 |1 http://isni.org/isni/0000000120324844. | |
| 245 | 1 | 0 | |a Fourier series and numerical methods for partial differential equations |h [electronic resource] / |c Richard Bernatz. |
| 260 | |a Hoboken, N.J. : |b Wiley, |c ©2010. | ||
| 300 | |a 1 online resource (xiii, 318 pages) : |b illustrations. | ||
| 336 | |a text |b txt |2 rdacontent. | ||
| 337 | |a computer |b c |2 rdamedia. | ||
| 338 | |a online resource |b cr |2 rdacarrier. | ||
| 504 | |a Includes bibliographical references (pages 311-313) and index. | ||
| 505 | 0 | |6 880-01 |a Introduction -- Fourier series -- Sturm -- Liouville problems -- Heat equation -- Heat transfer in 1D -- Heat transfer in 2D and 3D -- Wave equation -- Numerical methods: an overview -- The finite difference method -- Finite element method -- Finite analytic method. | |
| 520 | |a The importance of partial differential equations (PDEs) in modeling phenomena in engineering as well as in the physical, natural, and social sciences is well known by students and practitioners in these fields. Striking a balance between theory and applications, Fourier Series and Numerical Methods for Partial Differential Equations presents an introduction to the analytical and numerical methods that are essential for working with partial differential equations. Combining methodologies from calculus, introductory linear algebra, and ordinary differential equations (ODEs), the book strengthens. | ||
| 546 | |a English. | ||
| 588 | 0 | |a Print version record. | |
| 650 | 0 | |a Fourier series. |0 http://id.loc.gov/authorities/subjects/sh85051090. | |
| 650 | 0 | |a Differential equations, Partial |x Numerical solutions. |0 http://id.loc.gov/authorities/subjects/sh85037915. | |
| 650 | 7 | |a Differential equations, Partial |x Numerical solutions. |2 fast |0 (OCoLC)fst00893488. | |
| 650 | 7 | |a Fourier series. |2 fast |0 (OCoLC)fst00933405. | |
| 776 | 0 | 8 | |i Print version: |a Bernatz, Richard, 1955- |t Fourier series and numerical methods for partial differential equations. |d Hoboken, N.J. : Wiley, ©2010 |z 9780470617960 |w (DLC) 2010007954 |w (OCoLC)526076781. |
| 856 | 4 | 0 | |u https://colorado.idm.oclc.org/login?url=https://onlinelibrary.wiley.com/doi/book/10.1002/9780470651384 |z Full Text (via Wiley) |
| 880 | 0 | 0 | |6 505-01/(S |g Machine generated contents note: |g 1. |t Introduction -- |g 1.1. |t Terminology and Notation -- |g 1.2. |t Classification -- |g 1.3. |t Canonical Forms -- |g 1.4. |t Common PDEs -- |g 1.5. |t Cauchy-Kowalevski Theorem -- |g 1.6. |t Initial Boundary Value Problems -- |g 1.7. |t Solution Techniques -- |g 1.8. |t Separation of Variables -- |t Exercises -- |g 2. |t Fourier Series -- |g 2.1. |t Vector Spaces -- |g 2.1.1. |t Subspaces -- |g 2.1.2. |t Basis and Dimension -- |g 2.1.3. |t Inner Products -- |g 2.2. |t Integral as an Inner Product -- |g 2.2.1. |t Piecewise Continuous Functions -- |g 2.2.2. |t Inner Product on Cp (a, b) -- |g 2.3. |t Principle of Superposition -- |g 2.3.1. |t Finite Case -- |g 2.3.2. |t Infinite Case -- |g 2.3.3. |t Hubert Spaces -- |g 2.4. |t General Fourier Series -- |g 2.5. |t Fourier Sine Series on (0, c) -- |g 2.5.1. |t Odd, Periodic Extensions -- |g 2.6. |t Fourier Cosine Series on (0, c) -- |g 2.6.1. |t Even, Periodic Extensions -- |g 2.7. |t Fourier Series on ( -c, c) -- |g 2.7.1. |t 2c-Periodic Extensions -- |g 2.8. |t Best Approximation -- |g 2.9. |t Bessel's Inequality -- |g 2.10. |t Piecewise Smooth Functions -- |g 2.11. |t Fourier Series Convergence -- |g 2.11.1. |t Alternate Form -- |g 2.11.2. |t Riemann-Lebesgue Lemma -- |g 2.11.3. |t Dirichlet Kernel Lemma -- |g 2.11.4. |t Fourier Theorem -- |g 2.12. |t 2c-Periodic Functions -- |g 2.13. |t Concluding Remarks -- |t Exercises -- |g 3. |t Sturm-Liouville Problems -- |g 3.1. |t Basic Examples -- |g 3.2. |t Regular Sturm-Liouville Problems -- |g 3.3. |t Properties -- |g 3.3.1. |t Eigenfunction Orthogonality -- |g 3.3.2. |t Real Eigenvalues -- |g 3.3.3. |t Eigenfunction Uniqueness -- |g 3.3.4. |t Non-negative Eigenvalues -- |g 3.4. |t Examples -- |g 3.4.1. |t Neumann Boundary Conditions on [0, c] -- |g 3.4.2. |t Robin and Neumann BCs -- |g 3.4.3. |t Periodic Boundary Conditions -- |g 3.5. |t Bessel's Equation -- |g 3.6. |t Legendre's Equation -- |t Exercises -- |g 4. |t Heat Equation -- |g 4.1. |t Heat Equation in 1D -- |g 4.2. |t Boundary Conditions -- |g 4.3. |t Heat Equation in 2D -- |g 4.4. |t Heat Equation in 3D -- |g 4.5. |t Polar-Cylindrical Coordinates -- |g 4.6. |t Spherical Coordinates -- |t Exercises -- |g 5. |t Heat Transfer in 1D -- |g 5.1. |t Homogeneous IBVP -- |g 5.1.1. |t Example: Insulated Ends -- |g 5.2. |t Semihomogeneous PDE -- |g 5.2.1. |t Variation of Parameters -- |g 5.2.2. |t Example: Semihomogeneous IBVP -- |g 5.3. |t Nonhomogeneous Boundary Conditions -- |g 5.3.1. |t Example: Nonhomogeneous Boundary Condition -- |g 5.3.2. |t Example: Time-Dependent Boundary Condition -- |g 5.3.3. |t Laplace Transforms -- |g 5.3.4. |t Duhamel's Theorem -- |g 5.4. |t Spherical Coordinate Example -- |t Exercises -- |g 6. |t Heat Transfer in 2D and 3D -- |g 6.1. |t Homogeneous 2D IBVP -- |g 6.1.1. |t Example: Homogeneous IBVP -- |g 6.2. |t Semihomogeneous 2D IBVP -- |g 6.2.1. |t Example: Internal Source or Sink -- |g 6.3. |t Nonhomogeneous 2D IBVP -- |g 6.4. |t 2D BVP: Laplace and Poisson Equations -- |g 6.4.1. |t Dirichlet Problems -- |g 6.4.2. |t Dirichlet Example -- |g 6.4.3. |t Neumann Problems -- |g 6.4.4. |t Neumann Example -- |g 6.4.5. |t Dirichlet, Neumann BC Example -- |g 6.4.6. |t Poisson Problems -- |g 6.5. |t Nonhomogeneous 2D Example -- |g 6.6. |t Time-Dependent BCs -- |g 6.7. |t Homogeneous 3D IBVP -- |t Exercises -- |g 7. |t Wave Equation -- |g 7.1. |t Wave Equation in 1D -- |g 7.1.1. |t d'Alembert's Solution -- |g 7.1.2. |t Homogeneous IBVP: Series Solution -- |g 7.1.3. |t Semihomogeneous IBVP -- |g 7.1.4. |t Nonhomogeneous IBVP -- |g 7.1.5. |t Homogeneous IBVP in Polar Coordinates -- |g 7.2. |t Wave Equation in 2D -- |g 7.2.1. |t 2D Homogeneous Solution -- |t Exercises -- |g 8. |t Numerical Methods: an Overview -- |g 8.1. |t Grid Generation -- |g 8.1.1. |t Adaptive Grids -- |g 8.1.2. |t Multilevel Methods -- |g 8.2. |t Numerical Methods -- |g 8.2.1. |t Finite Difference Method -- |g 8.2.2. |t Finite Element Method -- |g 8.2.3. |t Finite Analytic Method -- |g 8.3. |t Consistency and Convergence -- |g 9. |t Finite Difference Method -- |g 9.1. |t Discretization -- |g 9.2. |t Finite Difference Formulas -- |g 9.2.1. |t First Partials -- |g 9.2.2. |t Second Partials -- |g 9.3. |t ID Heat Equation -- |g 9.3.1. |t Explicit Formulation -- |g 9.3.2. |t Implicit Formulation -- |g 9.4. |t Crank-Nicolson Method -- |g 9.5. |t Error and Stability -- |g 9.5.1. |t Error Types -- |g 9.5.2. |t Stability -- |g 9.6. |t Convergence in Practice -- |g 9.7. |t 1D Wave Equation -- |g 9.7.1. |t Implicit Formulation -- |g 9.7.2. |t Initial Conditions -- |g 9.8. |t 2D Heat Equation in Cartesian Coordinates -- |g 9.9. |t Two-Dimensional Wave Equation -- |g 9.10. |t 2D Heat Equation in Polar Coordinates -- |t Exercises -- |g 10. |t Finite Element Method -- |g 10.1. |t General Framework -- |g 10.2. |t 1D Elliptical Example -- |g 10.2.1. |t Reformulations -- |g 10.2.2. |t Equivalence in Forms -- |g 10.2.3. |t Finite Element Solution -- |g 10.3. |t 2D Elliptical Example -- |g 10.3.1. |t Weak Formulation -- |g 10.3.2. |t Finite Element Approximation -- |g 10.4. |t Error Analysis -- |g 10.5. |t 1D Parabolic Example -- |g 10.5.1. |t Weak Formulation -- |g 10.5.2. |t Method of Lines -- |g 10.5.3. |t Backward Euler's Method -- |t Exercises -- |g 11. |t Finite Analytic Method -- |g 11.1. |t 1D Transport Equation -- |g 11.1.1. |t Finite Analytic Solution -- |g 11.1.2. |t FA and FD Coefficient Comparison -- |g 11.1.3. |t Hybrid Finite Analytic Solution -- |g 11.2. |t 2D Transport Equation -- |g 11.2.1. |t FA Solution on Uniform Grids -- |g 11.2.2. |t Poisson Equation -- |g 11.3. |t Convergence and Accuracy -- |t Exercises -- |g Appendix |t A FA 1D Case -- |g Appendix B |t FA 2D Case -- |g B.1. |t Case θ = 1 -- |g B.2. |t Case θ = Bx + Ay. |
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