3150 PDE Course Spring 2014 Hybrid Book, Edwards-Penney and Haberman (abbreviation EPH), or Haberman 4E or 5E (abbreviation H) Reference to Asmar's textbook can be any edition. Chapter references are to EPH 12,13,14,15,16. These chapters are identical to Haberman H1,H2,H3,H4,H10. The PDE course runs January 6 to April 23, 2014 CHAPTER 12 Heat Equation 717 WEEK 1, Jan 6, 8 ====== 12.1 Introduction 717 12.2 Derivation of the Conduction of Heat in a One-Dimensional Rod 718 Thermal energy density, heat energy, conservation of heat energy, heat flux, heat sources, conservation of heat energy in a thin slice, conservation of heat energy in the limit (exact), temperature and specific heat, thermal energy, Fourier's law of heat conduction, heat equation, initial conditions, diffusion of a chemical pollutant, Fick's law of diffusion. 12.3 Boundary Conditions 726 Prescribed temperature at the ends, insulated boundary, prescribed heat flux, Newton's law of cooling, convection coefficient. 12.4 Equilibrium Temperature Distribution 728 Prescribed temperature, equilibrium, steady-state, temperature distribution, insulated boundaries, total thermal energy, averages. WEEK 2, Jan 13, 15 ====== 12.5 Derivation of the Heat Equation in Two or Three Dimensions 733 Heat energy, heat energy conservation law, heat flux, normal vectors, integral equation for heat conservation, divergence theorem, heat flow equation, Fourier's law of heat conduction, Heat equation, Laplace equation, diffusion equation, initial boundary value problem, insulated boundary, steady-state, Poisson's equation, Laplace's equation, potential equation, Green's theorem, polar and cylindrical coordinates, spherical coordinates. 12.6 Appendix to 1.5: Review of Gradient and a Derivation of Fourier's Law of Heat Conduction 741 Directional derivative, heat gradient, level surface theorem. Independent Reading assignments. The lecture provides a brief intro to the topics. =============================== 4.6 Edwards-Penney, Orthogonal Vectors 269 Inner product, inner product space, orthogonal vectors, CSB inequality, triangle inequality, parallelogram law 4.10 Edwards-Penney, Inner Product Spaces 299 Inner product on a function space, CSB and triangle inequalities in an abstract inner product space, orthogonal projection onto a subspace with orthogonal basis, inner product definitions with various sets, trig polynomials, Fourier coefficients, approximation of time signals by trig polynomial projections 13.3 Haberman, Appendix, Orthogonality of Functions 763 Mostly remarks here, requiring reading of Edwards-Penney sections 4.6 to 4.10. CHAPTER 13 Method of Separation of Variables 744 WEEK 3, Jan 20, 22 ====== Monday Holiday, Martin Luther King. No classes. 13.1 Introduction 744 13.2 Linearity 745 Linear operator, linear equation, linear homogeneous and nonhomogeneous, superposition. WEEK 4, Jan 27, 29 ====== 13.3 Heat Equation with Zero Temperatures at Finite Ends 747 Classical heat BVP, separation of variables, product solution, separation constant, linear ordinary DE with boundary conditions, linear time invariant system, linear ordinary time dependent system, eigenvalue problem of Sturm-Liouville type, eigenvalues and eigenfunctions, the three cases for lambda positive, zero and negative, eigenfunction summary for the two-point BVP, product solutions and superposition, initial value problems, principle of superposition, extended principle, orthogonality of sines, abstract orthogonal set in an inner product space, orthogonal set of functions, welding rod example with constant initial temperature and ice-pack ends, steady-state approximation by series truncation, peak amplitude, exponential decay. 13.4 Worked Examples with the Heat Equation (Other Boundary Value Problems) 765 Rod with insulated ends, orthogonality of cosines, insulated circular ring, perfect thermal contact, periodic BVP for u'' + lambda u = 0, orthogonality relations for the sine-cosine system, summary for the heat problem with three types of BC. WEEK 5, Feb 3, 5 ====== 13.5 Laplace's Equation: Solutions and Qualitative Properties 775 Steady-state heat conduction on a rectangle, equilibrium temperature, superposition with boundary conditions, hyperbolic functions, Fourier coefficient formulas for the rectangle problem, circular disk BVP formulation, periodic BVP for u'' + lambda u =0, Cauchy-Euler ODE, Fourier coefficient formulas for the disk problem, fluid flow outside a cylinder, mass conservation, velocity and stream functions, circulation, pressure, drag, lift, Bernoulli condition, Mean Value Theorem for Laplace's equation, well-posed problems, uniqueness, maximum principles, compatibility condition (61) Exam Review, for weeks 1-4 CHAPTER 14 Fourier Series 791 WEEK 6, Feb 10, 12 ====== 14.1 Introduction 791 14.2 Statement of the Convergence Theorem 793 14.3 Fourier Cosine and Sine Series 796 WEEK 7, Feb 17 ,19 ====== Monday Holiday, President's Day. No classes. EXAM 1, for weeks 1-6 WEEK 8, Feb 24, 26 ====== 14.4 Term-by-Term Differentiation of Fourier Series 813 14.5 Term-By-Term Integration of Fourier Series 822 14.6 Complex Form of Fourier Series 826 CHAPTER 15 Wave Equation: Vibrating Strings and Membranes 829 15.1 Introduction 829 15.2 Derivation of a Vertically Vibrating String 829 WEEK 9, Mar 3, 5 ====== 15.3 Boundary Conditions 832 15.4 Vibrating String with Fixed Ends 835 WEEK 10, Mar 8-16 is Spring Break. No classes. ======= WEEK 11, Mar 17, 19 ======= 15.5 Vibrating Membrane 840 15.6 Reflection and Refraction of Electromagnetic (Light) and Acoustic (Sound) Waves 843 Solved examples, wave equation. CHAPTER 16 Infinite Domain Problems: Fourier Transform 848 WEEK 12, Mar 24, 26 ======= 16.1 Introduction 848 16.2 Heat Equation on an Infinite Domain 848 16.3 Fourier Transform Pair 852 WEEK 13, Mar 31, Apr 2 ======= 16.4 Fourier Transform and the Heat Equation 859 16.5 Fourier Sine and Cosine Transforms: The Heat Equation on Semi-Infinite Intervals 870 WEEK 14, Apr 7, 9 ======= 16.6 Worked Examples Using Transforms 878, part I. Exam review and solved examples, weeks 7-12. EXAM 2, for weeks 7-12 WEEK 15, Apr 14, 16 ======= 16.6 Worked Examples Using Transforms 878, Part II. 16.7 Scattering and Inverse Scattering 897 Additional transform examples: Welding torch, Shannon's theorem on signals WEEK 16, Apr 21, 23 ======= Exam review for the final exam, weeks 1-13 Additional worked examples for heat, wave, Laplace and Poisson equations. Textbook Linear Algebra & Differential Equations with Introductory Partial Differential Equations and Fourier Series, ISBN-13: 978-1-269-42557-5. This text is a hybrid of the three texts: Differential Equations and Linear Algebra 3rd Edition, by Edwards and Penney; Applied Partial Differential Equations with Fourier Series and Boundary Value Problems, 5th edition, by Haberman; Elementary Linear Algebra, by Edwards and Penney. This version of the text is required for the 4th semester in the new engineering math sequence, Math 3140 and the PDE course Math 3150.