| Description |
1 online resource (694 pages) |
| Contents |
Cover; Title Page ; Copyright; Contents ; Preface ; PART I: THEORY ; Chapter 1: Introduction ; 1.1 Partial Differential Equations ; 11.2 Classification of Second-order Linear Pdes ; 1.3 Side Conditions ; 1.3.1 Boundary Conditions on an Interval ; 1.4 Linear Pdes ; 1.4.1 Principle of Superposition ; 1.5 Steady-state and Equilibrium Solutions ; 1.6 First Example for Separation of Variables ; 1.7 Derivation of the Diffusion Equation ; 1.7.1 Boundary Conditions ; 1.8 Derivation of the Heat Equation ; 1.9 Derivation of the Wave Equation ; 1.10 Examples of Laplace''s Equation ; 1.11 Summary |
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1.11.1 Problems and Notes Chapter 2: Fourier Series ; 2.1 Piecewise Continuous Functions ; 2.2 Even, Odd, and Periodic Functions ; 2.3 Orthogonal Functions ; 2.4 Fourier Series ; 2.4.1 Fourier Sine and Cosine Series ; 2.5 Convergence of Fourier Series ; 2.5.1 Gibbs'' Phenomenon ; 2.6 Operations on Fourier Series ; 2.7 Mean Square Error ; 2.8 Complex Fourier Series ; 2.9 Summary ; 2.9.1 Problems and Notes ; Chapter 3: Separation of Variables ; 3.1 Homogeneous Equations ; 3.1.1 General Linear Homogeneous Equations ; 3.1.2 Limitations of the Method of Separation of Variables |
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3.2 Nonhomogeneous Equations 3.2.1 Method of Eigenfunction Expansions ; 3.3 Summary ; 3.3.1 Problems and Notes ; Chapter 4: Sturm Liouville Theory ; 4.1 Formulation ; 4.2 Properties of Sturm-liouville Problems ; 4.3 Eigenfunction Expansions ; 4.4 Rayleigh Quotient ; 4.5 Summary ; 4.5.1 Problems and Notes ; Chapter 5: Heat, Wave, and Laplace Equations ; 5.1 One-dimensional Heat Equation ; 5.2 Two-dimensional Heat Equation ; 5.3 One-dimensional Wave Equation ; 5.3.1 d'' Alembert''s Solution ; 5.4 Laplace''s Equation ; 5.4.1 Potential in a Rectangle ; 5.5 Maximum Principle |
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5.6 Two-dimensional Wave Equation 5.7 Eigenfunctions in Two Dimensions ; 5.8 Summary ; 5.8.1 Problems and Notes ; Chapter 6: Polar Coordinates ; 6.1 Interior Dirichlet Problem for a Disk ; 6.1.1 Poisson Integral Formula ; 6.2 Vibrating Circular Membrane ; 6.3 Bessel''s Equation ; 6.3.1 Series Solutions of Odes ; 6.4 Bessel Functions ; 6.4.1 Properties of Bessel Functions ; 6.4.2 Integral Representation of Bessel Functions ; 6.5 Fourier-bessel Series ; 6.6 Solution to the Vibrating Membrane Problem ; 6.7 Summary ; 6.7.1 Problems and Notes ; Chapter 7: Spherical Coordinates |
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7.1 Spherical Coordinates 7.1.1 Derivation of the Laplacian ; 7.2 Legendre''s Equation ; 7.3 Legendre Functions ; 7.3.1 Legendre Polynomials ; 7.3.2 Fourier-legendre Series ; 7.3.3 Legendre Functions of the Second Kind ; 7.3.4 Associated Legendre Functions ; 7.4 Spherical Bessel Functions ; 7.5 Interior Dirichlet Problem for a Sphere ; 7.6 Summary ; 7.6.1 Problems and Notes ; Chapter 8: Fourier Transforms ; 8.1 Fourier Integrals ; 8.1.1 Fourier Integral Representation ; 8.1.2 Examples ; 8.1.3 Fourier Sine and Cosine Integral Representations ; 8.1.4 Proof of Fourier''s Theorem |
| Summary |
Uniquely provides fully solved problems for linear partial differential equations and boundary value problems Partial Differential Equations: Theory and Completely Solved Problems utilizes real-world physical models alongside essential theoretical concepts. With extensive examples, the book guides readers through the use of Partial Differential Equations (PDEs) for successfully solving and modeling phenomena in engineering, biology, and the applied sciences. The book focuses exclusively on linear PDEs and how they can be solved using the separation of variables technique. The authors begin |
| Bibliography |
Includes bibliographical references and index |
| Notes |
Print version record |
| Subject |
Differential equations, Partial.
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MATHEMATICS -- Calculus.
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MATHEMATICS -- Mathematical Analysis.
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Differential equations, Partial
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| Form |
Electronic book
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| Author |
Leonard, I. Ed., 1938- author
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Van Roessel, Henry, 1956- author
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| LC no. |
2012017382 |
| ISBN |
9781118441466 |
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111844146X |
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1322078173 |
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9781322078175 |
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1118063309 |
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9781118063309 |
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