MATH 3413 - Physical Mathematics I, Section 001 - Spring 2014
TR 10:30-11:45 a.m., 100 PHSC

Instructor: Nikola Petrov, 802 PHSC, (405)325-4316, npetrov AT math.ou.edu.

Office Hours (tentative): Mon 2:30-3:30 p.m., Thu 1:20-2:30 p.m., or by appointment.

Prerequisites: MATH 2443 (Calculus and Analytic Geometry IV) or concurrent enrollment.

Course catalog description: Complex numbers and functions. Fourier series, solution methods for ordinary differential equations and partial differential equations, Laplace transforms, series solutions, Legendre's equation. Duplicates two hours of 3113. (F)

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Tentative course content:

• Separable equations, linear equations, applications.
• Homogeneous, Bernoulli, exact equations.
• Existence and uniqueness for first order ODEs. Numerical methods.
• Second order nonhomogeneous equations. Variations of parameters. Mass-spring system, resonance.
• First order systems.
• Laplace transform and applications to ODEs. Delta function.
• Power series method. Legendre and Bessel functions.
• Fourier series.
• Heat conduction problem with Dirichlet and Neumann boundary conditions.
• String vibration problems.
• Laplace equation on a rectangle.
• Problems in circular and cylindrical regions.

Text: C. H. Edwards, D. E. Penney. Differential Equations and Boundary Value Problems, 4th ed, Prentice Hall, 2007, ISBN-10: 0131561073, ISBN-13: 978-0131561076.

Homework:

• Homework 1 (problems given on January 14, 16), due January 23 (Thursday).
• Homework 2 (problems given on January 21, 23), due January 30 (Thursday).
• Homework 3 (problems given on January 28, 30), due February 6 (Thursday).
• Homework 4 (problems given on February 4, 6, 13), due February 20 (Thursday).
• Homework 5 (problems given on February 18, 20, 25), due February 27 (Thursday). [Please note the new due date!]
• Homework 6 (problems given on February 27, March 4, 6), due March 11 (Tuesday).
• Homework 7 (problems given on March 11, 25, 27), due April 3 (Thursday).
• Homework 8 (problems given on April 1, 3), due April 10 (Thursday).
• Homework 9 (problems given on April 8, 10), due April 17 (Thursday).
• Homework 10 (problems given on April 15, 17), due April 22 (Tuesday).
• Homework 11 (problems given on April 22, 29), not to be turned in! Solution will be distributed in class on May 1 (Thursday).

Content of the lectures:

• Lecture 1 (Tue, Jan 14): Differential equations and mathematical models: Ordinary differential equations (ODEs) of nth order, initial conditions (ICs), initial value problem (IVP), IVP=ODE+IC, general solution of an ODE, solution of an IVP, mathematical modeling of natural phenomena, examples from population dynamics: simplest model P'(t)=kP(t), problems with this model (unbounded exponential growth), correction accounting for the limited amount of resources - logistic equation P'=kP(1−P/A) where A is the carrying capacity of the ecosystem [Sec. 1.1]
  Homework: Problems 1.1/6, 15, 23, 31, 35, 38.
  FFT: Problems 1.1/29, 36, 42.
  Remark: The FFT ("Food For Thought") problems are to be solved like regular homework problems, but do not have to be turned in.
• Lecture 2 (Thu, Jan 16): Integrals as general and particular solutions: general solution of an ordinary differential equation (ODE), initial conditions (ICs), initial value problems (IVPs), particular solution of an IVP [Sec. 1.2]
  Slope fields and solution curves: geometric meaning of a first-order differential equation, slope fields (direction fields), solution curves (integral curves); existence and uniqueness of solutions of IVPs: an example of an IVP with no solution, an example of and IVP with infinitely many solutions (growth of the volume of a water droplet in an oversaturated vapor - see Problem 1.3/29) [pages 19-21, 24, 25 of Sec. 1.3]
  Separable equations and applications: separable equations - method of solution, examples; implicit solutions and singular solutions, examples [pages 32-40 of Sec. 1.4]
  Homework: Problems 1.2/5, 6; 1.4/5, 13, 21, 25, 27, 28, 34, additional problem.
  FFT: Problem 1.4/30.
  The complete Homework 1 (problems given on January 14, 16) is due on January 23 (Thursday).
• Lecture 3 (Tue, Jan 21): Linear first-order equations: integrating factor, algorithm for solving linear first-order equations [pages 48-52 of Sec. 1.5]
  Substitution methods and exact equations: substitution method, examples; homogeneous equation: substitution v:=y/x converting it to a separable equations, examples [pages 60-64 of Sec. 1.6]
  Homework: Problems 1.5/17, 23, 27, 29; 1.6/9, 12, 17, 18, 57.
• Lecture 4 (Thu, Jan 23): Substitution methods and exact equations (cont.): Bernoulli equation: substitution v:=y^1−α converting it to a linear equation, examples; second-order ODE with y missing: substitution v:=y' converting it to a first-order equation; second-order ODE with x missing: substitution p:=y' converting it to a first-order equation for the function p(y), examples [pages 64-66, 72, 73 of Sec. 1.6]
  Homework: Problems 1.6/19, 23, 28 [set v(x)=e^y(x)], 43, 44, 48, 54.
  The complete Homework 2 (problems given on January 21, 23) is due on January 30 (Thursday).
• Lecture 5 (Tue, Jan 28): Equilibrium solutions and stability: autonomous first order ODEs, equilibrium solutions, connection between equilibrium solutions and critical points (zeros) of the right-hand side of the ODE, examples; stable and unstable equilibria, representing roughly the time evolution of the system (in the (t,x)-plane); examples of finding the equilibrium solutions and determining their stability [pages 92-95 of Sec. 2.2]
  Second-order linear equations definition of a linear equation, homogeneous and nonhomogeneous linear equations, homogeneous equation associated with a nonhomogeneous equation; a physical example leading to a second-order linear equation: oscillator with resistance force and external driving; Principle of Superposition for homogeneous linear equations; theorem on existence and uniqueness of solutions of 2nd order linear equations; linearly dependent and linearly independent functions; Wronskian of two functions; the Wronskian of two solutions of a second-order linear ODE is either identically zero or never becomes zero; constructing the general solution of a homogeneous second order linear equation as a linear combination of two linearly independent solutions of the equation (i.e, two solutions with nonzero Wronskian); examples [pages 147-150, 152, 154-155 of Sec. 3.1]
  Homework: Problems 1.6/63, 65; 3.1/7, 17, 19, 20, 24, 25, 27, 28; additional problems.
  FFT: Problems 1.3/1, 2, 3.
• Lecture 6 (Thu, Jan 30): Second-order linear equations(cont.): homogeneous linear 2nd-order linear ODEs with constant coefficients, characteristic equation, general solution of a homogeneous linear 2nd-order linear ODEs with constant coefficients in the case of distinct real roots of the characteristic equation (Theorem 5); general solution of a homogeneous linear 2nd-order linear ODEs with constant coefficients in the case of one double real root of the characteristic equation (Theorem 6) [pages 156-158 of Sec. 3.1]
  General solutions of linear equations: general form of an nth-order linear equation and the associated homogeneous equation; Principle of Superposition for homogeneous equations (Theorem 1) [pages 161, 162 of Sec. 3.2]
  Homework: Problems 1.6/29; Ch. 1 Review (p. 78)/31; 3.1/33, 35, 39, 43, 45, 47, 48.
  The complete Homework 3 (problems given on January 28, 30) is due on February 6 (Thursday).
• Lecture 7 (Tue, Feb 4): Homogeneous linear equations with constant coefficients: characteristic equation; notation D:=d/dx, D^k:=d^k/dx^k, polynomial differential operators with constant coefficients L=a_nDⁿ+a_n−1Dⁿ⁻¹+...+a₁D+a₀; writing down the general solution based on the roots of the characteristic equation:
  • case 1 − distinct real roots of the characteristic equation;
  • case 2 − repeated real roots of the characteristic equation: if r₁ is a root of the characteristic equation of multiplicity p, then the corresponding contribution to the general solution is Q_p−1(x)e^r₁x, where Q_p−1(x) is an arbitrary polynomial of degree p−1; examples.
  Complex numbers, algebraic operations with complex numbers; real and imaginary parts of complex numbers, representing the complex numbers as points in the complex plane C; exponent of an imaginary number, Euler's formula e^iθ=cosθ+isinθ; exponent of a general complex number: e^α+iβ=e^α(cosβ+isinβ); sine and cosine functions expressed in terms of complex exponents (derivation from the Euler's formula) [pages 173-178, top of page 181 of Sec. 3.3]
  Homework: Problems 3.1/51, 56; 3.3/2, 3, 5, 10, 26, 39.
• Lecture 8 (Thu, Feb 6): Homogeneous linear equations with constant coefficients (cont.): obtaining the formulas for sin(α+β) and cos(α+β) as a direct consequence from Euler's formula, obtaining all trigonometric formulas as a consequence of these two formulas; complex roots of polynomial equations with real coefficients always come in pairs α+iβ and α−iβ; writing down the general solution based on the roots of the characteristic equation (cont.):
  • case 3 − a pair of complex roots α+iβ and α−iβ of the characteristic equation, each of them with multiplicity p: the corresponding contribution to the general solution of the differential equation is e^αx[Q_p−1(x)cos(βx)+R_p−1(x)sin(βx)], where Q_p−1(x) and R_p−1(x) are arbitrary polynomials (with real coefficients) of degree p−1 [pages 179-182 of Sec. 3.3]
  
  Homework: Problems 3.3/16, 18, 20, 23, 27, 33, 40, 42.
• Lecture 9 (Tue, Feb 11): Exam 1 [on the material from Sec. 1.1-1.6, 2.2, 3.1, and half of 3.3 covered in Lectures 1-7 and the first half of Lecture 8]
• Lecture 10 (Thu, Feb 13): General solutions of linear equations (cont.): general form of an nth-order linear equation and the associated homogeneous equation; Principle of Superposition for homogeneous equations (Theorem 1); linear independence of a set of n functions, Wronskian of a set of n functions; linear independence of a set of n functions; Wronskian criterion for linear independence of a set of n functions (the functions are linearly independent if and only if their Wronskian is identically zero); the general solution y_c(x) of (H) (of order n) is a linear combination of n linearly independent solutions of (H) (Theorem 4); general solution of (N) is a sum of y_c(x) and a particular solution y_p(x) of (N) (Theorem 5) [only the definitions, the statements of the theorems, and the examlples on pages 161, 164-168, 170 of Section 3.2]
  Nonhomogeneous order n linear equations and undetermined coefficients: denote the nonhomogeneous equation Ly(x)=ƒ(x) by (N), the associated homogeneous equation Ly(x)=0 by (H), and the characteristic equation by (C); two basic rules:
  • (the general solution y(x) of (N)) = (the general solution y_c(x) of (H)) + (a particular solution y_p(x) of (N));
  • if ƒ(x)=ƒ₁(x)+ƒ₂(x), then (gen sol of (N)) = (gen sol of (H)) + (a part sol of Ly(x)=ƒ₁(x)) + (a part sol of Ly(x)=ƒ₂(x)).
  Finding a particular solution of Ly(x) =ƒ(x):
  Case A: ƒ(x)=e^cxP_m(x): if c is a root of the characteristic equation of multiplicity s, then look for a particular solution of (N) of the form y_p(x)=x^se^cxQ_m(x), and find the coefficients of the m^th-degree polynomial Q_m(x) by plugging it in (N) and equating the coefficients of the terms containing the same powers of x [read the handout (the same as the one given in class), read the examples in the Handout that belong to Case A; look at Examples 1, 2, 3, 4, 5, 8 solved on pages 198-206 of Sec. 3.5, and think how the rules from the handout will apply to them]
  Homework: Problems 3.3/43, 51, 53; 3.5/9, 22, 24, 25, 32.
  The complete Homework 4 (problems given on February 4, 6, 13) is due on February 20 (Thursday).
• Lecture 11 (Tue, Feb 18): Nonhomogeneous order n linear equations and undetermined coefficients (cont.):
  Case B: ƒ(x)=e^cx[P_m1(x)cos(dx)+R_m2(x)sin(dx)]: if c+id is a root of the characteristic equation of multiplicity s, then look for a particular solution of (N) of the form y_p(x)=x^se^cx[Q_m(x)cos(dx)+T_m(x)sin(dx)], where Q_m(x) and T_m(x) are polynomials of degree m=max(m₁,m₂), and find the coefficients Q_m(x) and T_m(x) by plugging it in (N) and equating the coefficients of the terms containing the same powers of x [read the handout (the same as the one given in class), read the examples in the Handout that belong to Case A; look at Examples 6, 7, 9, 10 solved on pages 202, 203, 206, 207 of Sec. 3.5, and think how the rules from the handout will apply to them]
  Mechanical vibrations: deriving the ODE governing the motion of a mass under the action of an elastic force F_spring=−kx from a spring (Hooke's law), in presence of a resistance force F_resist=−cx' and an external driving force F_ext(t): mx''+cx'+kx=F_ext(t) or, equivalently, x''+2px'+ω₀²x=ƒ_ext(t); cases of overdamped, critically damped, and underdamped motion; free undamped motion: period T=2π/ω₀ (unit: s), angular frequency ω₀ (unit: s⁻¹), linear frequency ν=1/T=ω₀/(2π) (unit: s⁻¹=Hz, Hertz) [pages 185-188, 190-192 of Sec. 3.4]
  Homework: Problems 3.5/12, 21, 26, 27, 28, 30, 33, 43(a) [in problems 21, 26, 27, 28, 30 explain briefly why you are choosing such form of the particular solution, following the notations from the handout]
  FFT: Problem 3.5/43(b).
• Lecture 12 (Thu, Feb 20): Mechanical vibrations (cont.): an alternative way to write the general solution of the free (not forced) damped motion in the underdamped case as Ce^−ptcos(ω₁t−α) instead of e^−pt[C₁cos(ω₁t)+C₂sin(ω₁t)], amplitude C=(C₁²+C₂²)^1/2 [pages 188, 189 of Sec. 3.4]
  Forced oscillations and resonance: eqiation of damped oscillations forced by a periodic external force of the form F_ext(t)=F₀cos(ωt): x''+2px'+ω₀²x=ƒ₀cos(ωt); in the underdamped case the general solution of this equation consists of two parts: a "transient" part x_c(t) which decays with time as e^−pt (this is the general solution of the associated homogeneous equation) and a "persistent" part x₌(t) of the form Acos(ωt)+Bsin(ωt)=Ccos(ωt−α); resonance, applications of resonance in electric circuits to isolate the frequency of a certain radio station [pages 219-221 of Sec. 3.6]
  Laplace transforms (LTs) and inverse transforms: definition of LT, the LT as a unary "machine" ("unary" = "with one input slot"); LT of ƒ(t)=1, ƒ(t)=e^at; linearity of LT, applications of the linearity property [pages 441, 442, 444-446 of Sec. 7.1]
  Homework: Problems 3.6/25, 27; 7.1/7, 13, 15, 16, 17, 23, 27, 28, 30.
• Lecture 13 (Tue, Feb 25): Laplace transforms (LTs) and inverse transforms (cont.): Gamma function and its properties, LT of ƒ(t)=t^a, LT of the unit step function ("Heaviside function") u_a(t)=u(t−a) [pages 443-447 of Sec. 7.1]
  LT of initial value problems: LT of derivatives, solving an IVP by using LT - general idea and an example [pages 452-456 of Sec. 7.2]
  Translation and partial fractions: rules for partial fractions, examples [pages 464, 465 of Sec. 7.3]
  Homework: Problems 7.1/12, 14, 39; 7.2/2, 9.
  FFT: Problems 7.1/35, 38.
  The complete Homework 5 (problems given on February 18, 20, 25) is due on February 27 (Thursday).
• Lecture 14 (Thu, Feb 27): LT of initial value problems (cont.): more examples, transformation of integrals (Theorem 2, with a proof), an example of application; computing the LT of the square-wave function (Problem 7.1/41 = 7.2/34, using the formula for the sum of a geometric series) [pages 455-457, 460 of Sec. 7.2]
  Translation and partial fractions (cont.): Reading assignment: read Theorem 1 (translation on the s-axis) and Examples 1 and 2 [pages 465-467 of Sec. 7.3]
  Derivatives, integrals, and products of transforms: definition of the convolution of two functions, linearity of the convolution with respect to each argument (i.e., (αƒ+g)*h=αƒ*h+g*h), commutativity property of convolution (ƒ*g)=g*ƒ)), the convolution property (Theorem 1 - LT of the convolution of two functions is equal to the product of the LTs of the functions, without proof), using the convolution property to compute the inverse LT of s/(s²+1)² (see also Example 1) [pages 474-475 of Sec. 7.4]
  Homework: Problems 7.2/5, 17, 26, 27(a), 35 (hint: write ƒ(t) as an infinite series of Heaviside functions as in Problem 7.1/40); 7.3/3, 6, 8, 14, 27 (hint for Problem 7.3/6: rewrite F(s) as (s+1−2)/(s+1)³=1/(s+1)²−2/(s+1)³).
  Additional materials: The handout given in class as info for Quiz 3.
• Lecture 15 (Tue, Mar 4): Derivatives, integrals, and products of transforms (cont.): the convolution property (Theorem 1 - LT of the convolution of two functions is equal to the product of the LTs of the functions, with proof); differentiation of transforms, examples of applications [pages 474-477, 479, 480 of Sec. 7.4]
  Periodic and piecewise continuous functions: translation on the t-axis (Theorem 1 on page 475, with proof) - LT of u(t-a)ƒ(t-a) [pages 482-483 of Sec. 7.5]
  Impulses and δ-functions: motivation of the concept of δ-funcion, δ as a limit of "rectangle" functions, definition of a δ-function, integrals involving δ-functions [pages 493-495 of Sec. 7.6]
  Homework: Problems 7.4/5, 7, 8, 17, 25, 36; 7.5/2, 5, 13.
  Additional materials: The handout given in class as info for Quiz 4.
• Lecture 16 (Thu, Mar 6): Impulses and δ-functions (cont.): δ-function δ_a "concentrated at a" defined by the rule that integral of δ_a(x) times a smooth function φ(x) from −∞ to ∞ is equal to φ(a); derivatives of δ_a(x): integral of δ_a⁽ⁿ⁾(x) times a smooth function φ(x) is equal to (−1)ⁿφ⁽ⁿ⁾(a); proof that the derivative of the Heaviside (unit step) function u_a(x) is δ_a(x); solving linear constant-coefficient ODEs with right-hand side (i.e., driving force) δ_a(t) by Laplace transform; transfer function W(s)=1/(As²+Bs+C)⁻¹ and weight function w(t) (the inverse LT of W(s)) of the initial value problem Ax''(t)+Bx'(t)+Cx(t)=ƒ(t), x(0)=0, x'(0)=0; expressing the solution of this initial value problem as a convolution: x=w*ƒ, Duhamel's principle; determining the weight function by using a delta-function input because in this case x(t)=(w*δ)(t)=w(t) [pages 495-501 of Sec. 7.6]
  Homework: Problems 7.6/5, 11, 13, 14, 15.
  The complete Homework 6 (problems given on February 27, March 4, 6) is due on March 11 (Tuesday).
• Lecture 17 (Tue, Mar 11): Impulses and δ-functions (cont.): a brief recap of what we know about delta-function, Heaviside function, Laplace transforms, and using these concepts to solve IVPs for ODEs; proving (again) that the derivative of u_a(t) is δ_a(t) by comparing the Laplace transforms of of u'_a and δ_a; periodic functions, Laplace transform of a periodic function [pages 496 of Sec. 7.6, 487-491 of Sec. 7.5]
  Homework: Problems 7.4/29 (hint); 7.5/25.
  FFT: Problem 7.6/22.
• Lecture 18 (Thu, Mar 13): Exam 2 [on the material from Sec. 3.3-3.6, 7.1-7.6 covered in Lectures 7, 8, 10-16]
• Lecture 19 (Tue, Mar 25): Introduction to partial differential equations: solving elementary partial differential equations by consecutive integration; the general solution of a differential equation of order n for a function of d variables contains n arbitrary functions, each of which is a function of (d−1) variables.
  Vector spaces: definition of a vector space (linear space); basis in a vector space; dimension of a vector space; components of a vector in a certain basis; polynomials of order no greater than n form a vector space V_n of dimension (n+1); inner product vector spaces; norm ||u||=⟨u,u⟩^1/2; orthogonal basis v₁,...,v_n of a vector space - such that ⟨v_i,v_j⟩=0 if i≠j; orthonormal basis of a vector space - such that ⟨v_i,v_j⟩=δ_ij.
  Homework: No homework is assigned with this lecture.
• Lecture 20 (Thu, Mar 27): Vector spaces (cont.): inner product in the space V_n of polynomials, weight function, examples.
  Periodic functions and trigonometric series: definition of a periodic function of period p; Fourier series of a 2π-periodic function (i.e., a periodic function of period 2π); the functions {1/2, cos(t), sin(t), cos(2t), sin(2t), cos(3t), sin(3t), ...} as a basis in the (infinite-dimensional) vector space of 2π-periodic functions [pages 580-581, 584 of Sec. 9.1; page 590 of Sec. 9.2]
  Homework: Click here to download the assgned homework.
  The complete Homework 7 (problems assigned on March 11 and 27) is due on April 3 (Thursday).
• Lecture 21 (Tue, Apr 1): Periodic functions and trigonometric series (cont.): determining the components of a vector in an orthogonal or orthonormal basis (in an inner product vector space) by taking the dot product with a basis vector; inner product in the space of 2π-periodic functions; orthogonality of the basis {1/2, cos(t), sin(t), cos(2t), sin(2t), cos(3t), sin(3t), ...}; expression for the coefficients a_j and b_j of a 2π-periodic function; examples; convergence of the Fourier series of a function ƒ at a points where the function is continuous and at a point where the function has a finite jump; using the theorem on convergece of Fourier series to find sum of a series; Fourier series of a 2L-periodic function, expression for the coefficients a_j and b_j of a 2L-periodic function [pages 582-586 of Sec. 9.1; pages 589-594 of Sec. 9.2]
  Fourier sine and cosine series: extending a function defined on [0,L] to an odd or an even function of period 2L; Fourier sine (for the odd extension) or Fourier cosine (for the even extension) of a function defined on [0,L] [pages 597-600 of Sec. 9.3]
  Homework: Click here to download the assigned homework.
• Lecture 22 (Thu, Apr 3): Fourier sine and cosine series (cont.): termwise integration and differentiation of Fourier series (only read the statements of Theorems 1 and 2, and Example 3) [pages 601, 605-606 of Sec. 9.3]
  Application of Fourier series: idea of finding a periodic particular solution (the steady periodic solution) of a non-homogeneous linear constant-coefficient ODE by (1) expanding the right-hand side (i.e., the driving term) in a Fourier series, (2) solving the corresponding system of (infinitely many in general) ODEs with only one sin or cos term in the right-hand side, and (3) writing the periodic particular solution as a superposition of the solutions of the solutions of the ODEs from step (2) [pages 609-611 of Sec. 9.4]
  Heat conduction and separation of variables: a detailed derivation of the heat equation in R³ by using the Conservation of Energy Law and the Divergence Theorem.
  Homework: Click here to download the assigned homework.
  The complete Homework 8 (problems assigned on April 1, 3) is due on April 10 (Thursday).
• Lecture 23 (Tue, Apr 8): Heat conduction and separation of variables (cont.): finding the solutions u_n(x,t) by separation of variables in the case of Dirichlet BCs u(0,t)=0, u(L,t)=0; superposition of solutions u_n(x,t) each of which satisfies the PDE and the BCs, adjusting the coefficients in the superposition of functions u_n(x,t) in order to satisfy the IC u(x,0)=u₀(x) [pages 618-621 of Sec. 9.5]
  Homework: No homework is assigned with this lecture.
• Lecture 24 (Thu Apr 10): Heat conduction and separation of variables (cont.): recap of the main ideas of the method of separation of variables; separation of variables in the case of Neumann BCs u_x(0,t)=0, u_x(L,t)=0, examples [pages 622-626 of Sec. 9.5]
  Homework: Click here to download the assigned homework.
  The complete Homework 9 (problems assigned on April 10) is due on April 17 (Thursday).
• Lecture 25 (Tue, Apr 15): Vibrating strings and the one-dimensional wave equation: physical meaning of the wave equation and the boundary and initial conditions for it; representation of a solution of the wave equation in the form u(x,t)=φ(x-ct)+ψ(x+ct) (d'Alembert's formula), physical meaning - waves moving to the right and to the left with speed c; separation of variables in the wave equation in the case of homogeneous Dirichlet BCs u(0,t)=0, u(L,t)=0, imposing the ICs u(x,0)=g(x), u_t(x,0)=h(x); concepts related to a vibration that is periodic in time and space: speed c (unit m/s), wavelength λ (unit m), period T (unit s), (linear) frequency ν=1/T (unit s⁻¹=Hertz), angular frequency ω=2π/T (unit s⁻¹), basic relation λ=cT; discussion of concepts related to the solution of the homogeneous Dirichlet BVP for the wave equation in one spatial dimension for x∈[0,L]: allowed wavelengths λ_n=2L/n, allowed periods T_n=λ_n/c=2L/(nc), allowed frequencies ν_n=1/T_n=nc/(2L); illustrations with guitar strings; flageolets - supressing some harmonics by touching the string at certain positions, see the Wikipedia article Harmonic [Sec. 9.6]
  Homework: Click here to download the assgned homework.
• Lecture 26 (Thu, Apr 17): Vibrating strings and the one-dimensional wave equation (cont.): solution of the initial boundary problem for the wave equation on the interval [0,L) with zero Neumann boundary conditions; sound waves in a pipe for the cases of: (1) both ends open, (2) both ends closed, (3) one end open and the other end closed. [Sec. 9.6]
  Homework: Click here to download the assgned homework.
  The complete Homework 10 (problems assigned on April 15 and 17) is due on April 22 (Tuesday).
• Lecture 27 (Tue, Apr 22) Steady-state temperature and Laplace equation: physical problems leading to Poisson's equation Δu(x)=ψ(x) and Laplace's equation Δu(x)=0 - steady-state temperature with time-independent heat sources in the domain and time-independent boundary conditions; boundary value problems for 2-dimensional Laplace's equation in a rectangular domain (x,y)∈[0,a]×[0,b]; separation of variables in Laplace's equation in the case of Dirichlet BCs Δu=0, u(x,0)=0, u(x,b)=g(x), u(0,y)=0, u(a,y)=0; solving Laplace's equation with BCs u(x,0)=0, u(x,b)=0, u(0,y)=0, u(a,y)=h(y) by analogy; read the case of BCs u(x,0)=ƒ(x), u(x,b)=0, u(0,y)=0, u(a,y)=0 from the book (Example 1); think about the case with BCs u(x,0)=0, u(x,b)=0, u(0,y)=f(y), u(a,y)=0 (analogous to Example 1); the solution of the BVP Δu=0, u(x,0)=f₁(x), u(x,b)=f₂(x), u(0,y)=g₁(y), u(a,y)=g₁(y) as a superposition of the solutions of four BVPs each of which has nonzero temperature on one side only [pages 643-649 of Sec. 9.7]
  Homework: Click here to download the assgned homework.
• Lecture 28 (Thu, Apr 24): Exam 3 [on the material from Sec. 9.1-9.5 covered in Lectures 19-26]
• Lecture 29 (Tue, Apr 29): Steady-state temperature and Laplace equation (cont.): attempting to solve the Neumann BVP Δu(x,y)=0, u_x(0,y)=0, u_x(π,y)=0, u_y(x,0)=0, u_y(x,π)=5 and discovering that the method of separation of variables does not yield a solution; a more general discussion of the Neumann BVP Δu(x,y)=0, u_x(0,y)=0, u_x(π,y)=0, u_y(x,0)=0, u_y(x,π)=ƒ(x) and discovering that solution exists only if the zeroth term in the cosine-Fourier expansion of f(x) is equal to zero (or, equivalently, that the integral of the function ƒ(x) from x=0 to x=π is 0), physical explanation of this condition as a condition for zero net amount of heat entering the domain through the "upper" wall if the other three walls are thermally insulated [Sec. 9.7]
  Homework: Reading assignment: Read "The Dirichlet problem for a circular disk" on pages 649-651 of Sec. 9.7. Here are the important steps in the derivation: (a) separating variables as usual - looking in a solution in the form R(r)Θ(θ); (b) the function Θ(θ) must be periodic of period 2π, which implies that the constant of separation of variables must take a discret set of values; (c) solving the equation for the functions R₀(r) and R_n(r); (d) eliminating the term with ln(r) in R₀(r) and the term with r⁻ⁿ in R_n(r) because they "explode" as r→0⁺; (e) finding the coefficients in the series expansion.
• Lecture 30 (Thu, May 1): Separation of variables in cylindrical geometry: remarks about the physical problems leading to the heat equation and the Laplace equation in cylindrical domains; setting up the general IBVP for the temperature in a cylinder of radius a and height h; solving the heat equation in an infinite cylinder if the temperature distribution depends only on the radial coordinate r and the time t - derivation of the equations for the radial function R(r) and the temporal function T(t), Bessel equation, Bessel functions J_n(ξ) and Neumann functions Y_n(ξ), n=0,1,2,...; the discretization of the constant in the separation of variables comes from zeros of the equation J₀(ξ)=0; removing the Neumann function Y₀(ξ) because it tends to −∞ as ξ→0⁺; using the orthogonality relation for Bessel functions to determine the constants in the series expansion of u(r,t) from the initial condition [Sec. 10.4]
• Final exam: Monday, May 5, 8:00-10:00 a.m.

Grading: Your grade will be determined by your performance on the following coursework:

Homework: It is absolutely essential to solve the assigned homework problems! Homework assignments will be given regularly throughout the semester and will be posted on this web-site. The homework will be due at the start of class on the due date. Each homework will consist of several problems, of which some pseudo-randomly chosen problems will be graded. Your lowest homework grade will be dropped. Your homework should have your name clearly written on it, and should be stapled. No late homeworks will be accepted!

Attendance: You are required to attend class on those days when an examination is being given; attendance during other class periods is also strongly encouraged. When you are in class, do your best to follow, take notes, feel free to ask questions at any time. Using computers, phones, iPads, and other electronic equipment in class is not allowed (unless you are using them to take notes). You are fully responsible for the material covered in each class, whether or not you attend. Make-ups for missed exams will be given only if there is a compelling reason for the absence, which I know about beforehand and can document independently of your testimony (for example, via a note or a phone call from a doctor or a parent). You should come to class on time; if you miss a quiz because you came late, you won't be able to make up for it.

Useful links: the academic calendar.

Policy on W/I Grades : From January 13 to January 27, you can withdraw from the course without record of grade. From January 28 to March 28, you can withdraw from the course with an automatic "W". From March 31 to May 2 you may petition to the Dean to withdraw and receive a "W" or "F" grade according to your standing in the class. (Such petitions are not often granted. Furthermore, even if the petition is granted, I will give you a grade of "Withdrawn Failing" if you are indeed failing at the time of your petition.) Please check the dates in the Academic Calendar!

The grade of "I" (Incomplete) is not intended to serve as a benign substitute for the grade of "F". I only give the "I" grade if a student has completed the majority of the work in the course (for example everything except the final exam), the coursework cannot be completed because of compelling and verifiable problems beyond the student's control, and the student expresses a clear intention of making up the missed work as soon as possible.

Academic Misconduct: All cases of suspected academic misconduct will be referred to the Dean of the College of Arts and Sciences for prosecution under the University's Academic Misconduct Code. The penalties can be quite severe. Don't do it!
For details on the University's policies concerning academic integrity see the Student's Guide to Academic Integrity at the Academic Integrity web-site. For information on your rights to appeal charges of academic misconduct consult the Academic Integrity Code. Please check out the web-site of the OU Student Conduct Office.

Students With Disabilities: The University of Oklahoma is committed to providing reasonable accommodation for all students with disabilities. Students with disabilities who require accommodations in this course are requested to speak with the instructor as early in the semester as possible. Students with disabilities must be registered with the Office of Disability Services prior to receiving accommodations in this course. The Office of Disability Services is located in Goddard Health Center, Suite 166: phone 405-325-3852 or TDD only 405-325-4173.

Good to know:

• The greek_alphabet.
• Some useful notations.