MATH 3413 - Physical Mathematics I, Section 002 - Spring 2012
TR 1:30-2:45 p.m., 321 PHSC

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

Office Hours: Mon 2:30-3:30 p.m., Tue 2:45-3:45 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, due Thu, Jan 26.
• Homework 2, due Thu, Feb 2.
• Homework 3, due Thu, Feb 9.
• Homework 4, due Thu, Feb 23.
• Homework 5, due Thu, Mar 1.
• Homework 6, due Thu, Mar 8.
• Homework 7, due Thu, Mar 29.
• Homework 8, due Thu, Apr 5.
• Homework 9, due Thu, Apr 12.
• Homework 10, due Thu, Apr 19.
• Homework 11, due Thu, May 3.

Content of the lectures:

• Lecture 1 (Tue, Jan 17): Differential equations and mathematical models: examples, generalities about the process of mathematical modeling (Sec. 1.1).
  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, examples - population growth, harmonic oscillator (Sec. 1.2).
• Lecture 2 (Thu, Jan 19): Slope fields and solution curves: slope fields (direction fields), solution curves (integral curves) (pages 19-21 of Sec. 1.3).
  Separable equations and applications: separable equations - method of solution, examples; implicit solutions and singular solutions, examples (Sec. 1.4).
  Linear first-order equations: method of solution, integrating factor ρ(x), examples (Sec. 1.5).
• Lecture 3 (Tue, Jan 24): Linear first-order equations (cont.): more examples (Sec. 1.5).
  Substitution methods and exact equations: substitution methods - idea, an example; homogeneous equations - definition, method of solution, examples; Bernoulli equation - definition, idea of solution, an example (Sec. 1.6).
• Lecture 4 (Thu, Jan 26): Substitution methods and exact equations (cont.): implicitly defined solutions F(x,y(x))=0, examples, deriving the ODE satisfied by an implicitly defined solution; exact equations, method of solution, examples, fragility of the "exactness" property (Sec. 1.6).
• Lecture 5 (Tue, Jan 31): Substitution methods and exact equations (cont.): equations with dependent variable (y) missing, generalization of the idea to nth-order equations that do not contain y and its derivatives of orders 1, 2,...,n-2, examples; equations with the independent variable (x) missing, examples (Sec. 1.6).
  Second-order linear equations: definition of a second-order linear equation, classification, examples, homogeneous equation associated to a non-homogeneous equation, Principle of Superposition for homogeneous equations (Theorem 1), existence and uniqueness for linear equations (Theorem 2), linear equations do not have singular solutions (Sec. 3.1).
• Lecture 6 (Thu, Feb 2): Second-order linear equations(cont.): linear independence of two functions, examples; trying to satisfy the initial conditions in an IVP for a second-order ODE and motivaiton of the definition of a Wronskian of two functions, the Wronskian of two linearly dependent functions is identcally zero, Wronskian of two solutions of a homogeneous 2nd-order linear ODE (Theorem 3), general solutions of homogeneous 2nd-order linear ODEs (Theorem 4); 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) (Sec. 3.1).
• Lecture 7 (Tue, Feb 7): Second-order linear equations(cont.): 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); an example of constructing a homogeneous 2nd-order linear ODE from its general solution (Problem 3.1/43) (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); linear independence of a set of n functions, Wronskian of a set of n functions; linear independence of a set of n solutions of a homogeneous linear equation of order n, Wronskian criterion for linear independence of a set of n solutions of a homogeneous linear equation of order n (Theorem 3); general solution of a homogeneous linear equation of order n as a linear combination of n linearly independent solutions (Theorem 4); general solution of a nonhomogeneous linear ODE as a sum of the general solution of the corresponding homogeneous ODE and a particular solution of the nonhomogeneous ODE (Theorem 5) (Sec. 3.2).
• Lecture 8 (Thu, Feb 9): Homogeneous equations with constant coefficients: finding the general solution of a homogeneous linear equation of order n with constant coefficients, Ly(x)=0, where L=a_nDⁿ+a_n-1D^n-1+...+a₁D+a₀: characteristic equation a_nrⁿ+a_n-1r^n-1+...+a₁r+a₀=0, general solutions in the following cases:
  Case 1^- - n distinct real roots r₁, r₂, ..., r_n of the char. eqn.: y(x)=C₁e^r₁x+C₂e^r₂x+...+C_ne^r_nx (Theorem 1).
  Case 1 - repeated roots: each root r of multiplicity s of the char. eqn. contributes a term Q_s−1(x)e^rx, where Q_s−1(x) is a polynomial of degree (s−1) (Theorem 2).
  Complex numbers, Euler's formula.
  Case 2^- - a pair of simple (multiplicity 1) complex roots r=a+ib and r=a−ib of the char. eqn.: y(x)=e^ax[C₁cos(bx)+C₂sin(bx)] (Theorem 3).
  Case 2 - pairs of complex roots r=a+ib and r=a−ib of the char. eqn. of multiplicity s: y(x)=e^ax[P_s−1(x)cos(bx)+Q_s−1(x)sin(bx)], where P_s−1(x) and Q_s−1(x) are polynomials of degree (s−1).
  (handout and Sec. 3.3).
• Lecture 9 (Tue, Feb 14): Exam 1.
• Lecture 10 (Thu, Feb 16): Nonhomogeneous order n linear equations and undetermined coefficients: Denote the nonhomogeneous equation Ly(x)=f(x) by (N), the associated homogeneous equation Ly(x)=0 by (H), and the characteristic equation by (C); then
  (general solution of (N)) = (general solution of (H)) + (a particular solution of (N)) .
  If f(x)=f₁(x)+f_x(x), then (gen sol of (N)) = (gen sol of (H)) + (a part sol of Ly(x)=f₁(x)) + (a part sol of Ly(x)=f₂(x)) .
  Finding a particular solution of Ly(x) =f(x):
  Case A: f(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;
  Case B: f(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.
  (handout; look at all solved examples on pages 198-207 of Sec. 3.5 and think how the rules from the handout will apply to them).
• Lecture 11 (Tue, Feb 21): Mechanical oscillations: derivation of the equation of (forced damped) motion, equilibrium, Hooke's law, spring constant, undamped/damped, free/forced motion, linearization of the differential equation of the simple pendulum; free undamped motion, amplitude, angular (circular) frequency, phase, initial phase, linear frequency, period; free damped motion (underdamped, critically damped, and overdamped cases); forced undamped motion, resonance, remarks about the case of forced damped motion (Sec. 3.4, 3.6).
• Lecture 12 (Thu, Feb 23): Laplace transforms (LTs) and inverse transforms: definition of LT, LT as a unary "machine", LT of f(t)=1, f(t)=e^at, Gamma function and its properties, LT of f(t)=t^a, linearity of LT, applications of the linearity property, LT of the unit step function u_a(t)=u(t-a) (Sec. 7.1).
  LT of initial value problems: LT of derivatives (pages 452-454 of Sec. 7.2).
• Lecture 13 (Tue, Feb 28): LT of initial value problems (cont.): solving an IVP by using LT - general idea and an example (Sec. 7.2).
  Translation and partial fractions: rules for partial fractions, examples, translation on the s-axis (Sec. 7.3).
• Lecture 14 (Thu, Mar 1): LT of initial value problems (cont.): LT of integrals (Theorem 2, with two proofs), an example (Sec. 7.2).
  Derivatives, integrals, and products of transforms: convolution of two functions, commutativity property of convolution (f*g=g*f), the convolution property (Theorem 1 - LT of the convolution of two functions is equal to the product of the LTs of the functions, with a proof), differentiation of transforms (Theorem 2, with a proof) (Sec. 7.4).
• Lecture 15 (Tue, Mar 6): Derivatives, integrals, and products of transforms (cont.): integration of transofrms (Theorem 3, without proof) (Sec. 3.4).
  Periodic and piecewise continuous functions: translation on the t-axis (Theorem 1 on page 475, with proof) - LT of u(t-a)f(t-a), examples of application, periodic functions, LT of periodic functions (Theorem 2 - statement) (Sec. 7.5).
• Lecture 16 (Thu, Mar 8): Periodic and piecewise continuous functions: LT of periodic functions (Theorem 2 - proof) (Sec. 7.5).
  Impulses and δ-functions: motivation of the concept of δ-funcion, δ_a as a limit of "rectangle" functions, definition of δ-function, integrals involving δ-functions, LT of δ_a: L{δ_a(t)}=e^-as; δ-functions and step functions: δ_a(t)= u'_a(t) - "proof" by looking at the LTs of u'_a and δ_a and using the theorem about LT of derivatives (Sec. 7.6).
• Lecture 17 (Tue, Mar 13): Impulses and δ functions (cont.): transfer function W(s) and weight function w(t) of a system, Duhamel's principle: response x(t)=(w*f)(t) for f(t)-driving of the system, measuring w(t) "experimentally": if the driving is f(t)=δ(t), then the response is x(t)=(w*δ)(t)=w(t); definition of derivatives of δ_a, more examples (Sec. 7.6).
• Lecture 18 (Thu, Mar 15): Exam 2.
• Lecture 19 (Tue, Mar 27): Power series: power series, operations on power series, the power series method, termwise differentiation of power series, identity principle, Examples 1 and 2 (read pages 504-513 of Section 8.1).
  Periodic functions and trigonometric series: vector spaces, inner product (scalar product, dot product), basis in a vector space, orthogonal basis, orthonormal basis, decomposition of vectors with respect of an orthogonal basis: v=Σ_j v_j e_j, where e_j.e_k=0 for j not equal to k; finding the components: v_j=(v.e_j)/(e_j.e_j).
• Lecture 20 (Thu, Mar 29): Periodic functions and trigonometric series (cont.): defining a "dot product" of periodic functions of period p=2π as an integral, checking that the functions 1, cost, sint, cos2t, sin2t, cos3t, sin3t,... are orthogonal to one another, writing a periodic function of period p=2π as a linear combination of the functions 1, cost, sint, cos2t, sin2t, cos3t, sin3t,..., formulae for the Fourier coefficient of a periodic function of period p=2π (Sec. 9.1).
• Lecture 21 (Tue, Apr 3): Periodic functions and trigonometric series (cont.): examples of Fourier series, observation that the Fourier coefficients of a "smoother" function decay faster (the Fourier coefficients of a discontinuous function decay as 1/n, the Fourier coefficients of a continuous function decay as 1/n²) (Sec. 9.1).
  General Fourier series and convergence: formulae for the Fourier coefficient of a periodic function of period p=2L; convergence of a Fourier series (Theorem 1), examples, identities obtained by using Theorem 1 (Sec. 9.2).
  Fourier sine and cosine series: even and odd functions; extending a function defined on (0,L) to a periodic function of period 2L as an even function, extending a function defined on (0,L) to a periodic function of period 2L as an odd function (Sec. 9.3).
• Lecture 22 (Thu, Apr 5): Fourier sine and cosine series (cont.): Fourier series of a function defined on (0,L) obtained through extending it to R as an even or as an odd periodic function of period 2L; termwise differention and integration of Fourier series (Sec. 9.3).
  Application of Fourier series: Examples 1 and 3 of Sect. 9.3.
  Heat conduction and separation of variables: derivation of the heat equation from the law of conservation of energy (pages 615-617 of Sec. 9.5).
• Lecture 23 (Tue, Apr 10): Heat conduction and separation of variables (cont.): finishing the derivation of the heat equation, boundary and initial conditions; 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) (Sec. 9.5).
• Lecture 24 (Thu, Apr 12): 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 (Sec. 9.5).
  Vibrating strings and the one-dimensional wave equation: physical meaning of the wave equation and the boundary and initial conditions for it, 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)=f(x), u_t(x,0)=g(x) (Sec. 9.6).
• Lecture 25 (Tue, Apr 17): Vibrating strings and the one-dimensional wave equation: representation of a solution of the wave equation in the form u(x,t)=φ(x-ct)+ψ(x+ct), physical meaning - waves moving to the right and to the left with speed c; concepts related to a vibration that is periodic in 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 (Sec. 9.6).
  Steady-state temperature and Laplace equation: physical problems leading to Laplace's equation (sources of heat and Poisson's equation Δu(x)=ψ(x)), 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)=f(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)=f(y) by analogy (Sec. 9.7).
• Lecture 26 (Thu, Apr 19): Steady-state temperature and Laplace equation (cont.): read the case of BCs u(x,0)=f(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; attempting to solve the Neumann BVP Δu(x,y)=0, u_x(0,y)=0, u_x(a,y)=0, u_y(x,0)=0, u_y(x,b)=f(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 f(x) from x=0 to x=a 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).
• Lecture 27 (Tue, Apr 24): Exam 3.
• Lecture 28 (Thu, Apr 26): A digression on functions: definition of addition of functions and multiplicaiton of a function and a number, functions as elements of an infinite dimensional linear space, Fourier basis in the space of continuous 2π-periodic functions.
  A digression on general solutions of differential equations: the general solution of an ODE of order n has n arbitrary constants C₁, ..., C_n (which can be thought of as functions of zero variables); the general solution of a PDE of order n for a function u(x₁,...,x_d) of d variables has n arbitrary functions φ₁(x₁,...,x_d-1), ..., φ_n(x₁,...,x_d-1) of d−1 variables; obtaining the general solution of simple PDEs, like, u_xx(x,y)=sin(x), u_xz(x,y,z)=xy², etc.
  Steady-state temperature and Laplace equation (cont.): separation of variables in Laplace's equation u(r,θ)=R(r)Θ(θ) in a disk r≤a in polar coordinates, discrete values of the constant in the separation of variables because of the 2π-periodicity condition on the angular function Θ(θ) (Sec. 9.7).
• Lecture 29 (Tue, May 1): Stationary temperature distribution and Laplace equation (cont.): complete solution of Laplace's equation in a circular domain - explicit expressions for the functions Θ_n(θ) and R_n(r), determining the coefficients in the expansion from the Fourier expansion of the temperature at the boundary: f(θ)=u(a,θ); remarks on solving Laplace equation in an annulus r∈[a,b], θ∈[0,2π) (Sec. 9.7).
  Separation of variables in cylindrical geometry: remarks about the physical problems leading to the heat equation and the Laplace equation in cylindrical domains (Sec. 10.4).
• Lecture 30 (Thu, May 3): Separation of variables in cylindrical geometry (cont): 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 and Neumann functions, discretizing the separation constant coming from zeros of the equation J₀(ξ)=0, orthogonality relations for Bessel functions, determining the constants in the series expansion of u(r,t) from the initial condition by using an orthogonality relation (Sec. 10.4).

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. 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 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.

Academic Calendar for Spring 2012.

Course schedule for Spring 2012.

Policy on W/I Grades : Through February 24 (Friday), you can withdraw from the course with an automatic "W". In addition, from February 27 (Monday) to May 4 (Friday), you may withdraw and receive a "W" or "F" according to your standing in the class. Dropping after April 2 (Monday) requires a petition to the Dean. (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 Misconduct Code. Students are also bound by the provisions of the OU Student Code.

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.