| Description |
1 online resource (xi, 185 pages) : illustrations |
| Series |
Synthesis lectures on mathematics and statistics, 1938-1751 ; #12 |
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Synthesis lectures on mathematics and statistics ; #12.
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| Contents |
1. Partial differential equations in engineering -- 1.1 Introductory comments -- 1.2 Fundamental concepts -- Problems -- 1.3 The heat conduction (or diffusion) equation -- 1.3.1 Rectangular Cartesian coordinates -- 1.3.2 Cylindrical coordinates -- 1.3.3 Spherical coordinates -- The Laplacian operator -- 1.3.4 Boundary conditions -- 1.4 The vibrating string -- 1.4.1 Boundary conditions -- 1.5 Vibrating membrane -- 1.6 Longitudinal displacements of an elastic bar -- Further reading |
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2. The Fourier method: separation of variables -- 2.1 Heat conduction -- 2.1.1 Scales and dimensionless variables -- 2.1.2 Separation of variables -- 2.1.3 Superposition -- 2.1.4 Orthogonality -- 2.1.5 Lessons -- Problems -- 2.1.6 Scales and dimensionless variables -- 2.1.7 Separation of variables -- 2.1.8 Choosing the sign of the separation constant -- 2.1.9 Superposition -- 2.1.10 Orthogonality -- 2.1.11 Lessons -- 2.1.12 Scales and dimensionless variables -- 2.1.13 Getting to one nonhomogeneous condition -- 2.1.14 Separation of variables -- 2.1.15 Choosing the sign of the separation constant -- 2.1.16 Superposition -- 2.1.17 Orthogonality -- 2.1.18 Lessons -- 2.1.19 Scales and dimensionless variables -- 2.1.20 Relocating the nonhomogeneity -- 2.1.21 Separating variables -- 2.1.22 Superposition -- 2.1.23 Orthogonality -- 2.1.24 Lessons -- Problems -- 2.2 Vibrations -- 2.2.1 Scales and dimensionless variables -- 2.2.2 Separation of variables -- 2.2.3 Orthogonality -- 2.2.4 Lessons -- Problems -- Further reading |
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3. Orthogonal sets of functions -- 3.1 Vectors -- 3.1.1 Orthogonality of vectors -- 3.1.2 Orthonormal sets of vectors -- 3.2 Functions -- 3.2.1 Orthonormal sets of functions and Fourier series -- 3.2.2 Best approximation -- 3.2.3 Convergence of Fourier series -- 3.2.4 Examples of Fourier series -- Problems -- 3.3 Sturm-Liouville problems: orthogonal functions -- 3.3.1 Orthogonality of eigenfunctions -- Problems -- Further reading |
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4. Series solutions of ordinary differential equations -- 4.1 General series solutions -- 4.1.1 Definitions -- 4.1.2 Ordinary points and series solutions -- 4.1.3 Lessons: finding series solutions for differential equations with ordinary points -- Problems -- 4.1.4 Regular singular points and the method of frobenius -- 4.1.5 Lessons: finding series solution for differential equations with regular singular points -- 4.1.6 Logarithms and second solutions -- Problems -- 4.2 Bessel functions -- 4.2.1 Solutions of Bessel's equation -- Here are the rules -- 4.2.2 Fourier-Bessel series -- Problems -- 4.3 Legendre functions -- 4.4 Associated Legendre functions -- Problems -- Further reading |
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5. Solutions using Fourier series and integrals -- 5.1 Conduction (or diffusion) problems -- 5.1.1 Time-dependent boundary conditions -- 5.2 Vibrations problems -- Problems -- 5.3 Fourier integrals -- Problem -- Further reading |
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6. Integral transforms: the Laplace transform -- 6.1 The Laplace transform -- 6.2 Some important transforms -- 6.2.1 Exponentials -- 6.2.2 Shifting in the s -domain -- 6.2.3 Shifting in the time domain -- 6.2.4 Sine and cosine -- 6.2.5 Hyperbolic functions -- 6.2.6 Powers of t: tm -- 6.2.7 Heaviside step -- 6.2.8 The Dirac Delta function -- 6.2.9 Transforms of derivatives -- 6.2.10 Laplace transforms of integrals -- 6.2.11 Derivatives of transforms -- 6.3 Linear ordinary differential equations with constant coefficients -- 6.4 Some important theorems -- 6.4.1 Initial value theorem -- 6.4.2 Final value theorem -- 6.4.3 Convolution -- 6.5 Partial fractions -- 6.5.1 Nonrepeating roots -- 6.5.2 Repeated roots -- 6.5.3 Quadratic factors: complex roots -- Problems -- Further reading |
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7. Complex variables and the Laplace inversion integral -- 7.1 Basic properties -- 7.1.1 Limits and differentiation of complex variables: 7.1.1 -- Analytic functions -- Integrals -- 7.1.2 The Cauchy integral formula -- Problems |
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8. Solutions with Laplace transforms -- 8.1 Mechanical vibrations -- Problems -- 8.2 Diffusion or conduction problems -- Problems -- 8.3 Duhamel's theorem -- Problems -- Further reading |
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9. Sturm-Liouville transforms -- 9.1 A preliminary example: Fourier sine transform -- 9.2 Generalization: the Sturm-Liouville transform: theory -- 9.3 The inverse transform -- Problems -- Further reading |
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10. Introduction to perturbation methods -- 10.1 Examples from algebra -- 10.1.1 Regular perturbation -- 10.1.2 Singular perturbation |
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11. Singular perturbation theory of differential equations |
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Appendix A. The roots of certain transcendental equations -- Appendix B. -- Author's biography |
| Summary |
The second edition of this popular book on practical mathematics for engineers includes new and expanded chapters on perturbation methods and theory. This is a book about linear partial differential equations that are common in engineering and the physical sciences. It will be useful to graduate students and advanced undergraduates in all engineering fields as well as students of physics, chemistry, geophysics and other physical sciences and professional engineers who wish to learn about how advanced mathematics can be used in their professions. The reader will learn about applications to heat transfer, fluid flow and mechanical vibrations. The book is written in such a way that solution methods and application to physical problems are emphasized. There are many examples presented in detail and fully explained in their relation to the real world. References to suggested further reading are included. The topics that are covered include classical separation of variables and orthogonal functions, Laplace transforms, complex variables and Sturm-Liouville transforms. This second edition includes two new and revised chapters on perturbation methods, and singular perturbation theory of differential equations |
| Analysis |
Engineering mathematics |
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separation of variables |
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orthogonal functions |
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Laplace transforms |
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complex variables and Sturm-Liouville transforms |
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differential equations |
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perturbation methods |
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perturbation theory |
| Bibliography |
Includes bibliographical references |
| Notes |
Online resource; title from PDF title page (Morgan & Claypool, viewed Mar. 5, 2012) |
| Subject |
Engineering mathematics.
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Differential equations, Partial -- Numerical solutions.
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Differential equations, Linear -- Numerical solutions.
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Fourier analysis.
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Laplace transformation.
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TECHNOLOGY & ENGINEERING -- Engineering (General)
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TECHNOLOGY & ENGINEERING -- Reference.
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Differential equations, Linear -- Numerical solutions
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Differential equations, Partial -- Numerical solutions
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Engineering mathematics
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Fourier analysis
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Laplace transformation
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| Form |
Electronic book
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| ISBN |
9781608457816 |
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1608457818 |
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160845780X |
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9781608457809 |
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9783031024047 |
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3031024044 |
