MATH 3413 - Physical Mathematics I, Section 001 - Fall 2010
MWF 2:30-3:20 p.m., 122 PHSC

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

Office Hours: Wed 3:30-4:30 p.m., Fri 10:00-11:00 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 (Solutions are posted after the due date in Bizzell Library)

• Homework 1, due Mon, Sep 6.
• Homework 2, due Wed, Sep 15.
• Homework 3, due in class on Mon, Sep 20. Please note the unusual date!
• Homework 4, due Wed, Sep 29.
• Homework 5, due Wed, Oct 6.
• Homework 6, due Fri, Oct 15.
• Homework 7, due in class on Fri, Oct 22.
• Homework 8, due Mon, Nov 8.
• Homework 9, due Mon, Nov 15.
• Homework 10, due Mon, Nov 29.
• Homework 11, due Fri, Dec 10.
  A solution of Additional problem 1 from Homework 11.

Content of the lectures:

• Lecture 1 (Mon, Aug 23): 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 (Wed, Aug 25): 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).
• Lecture 3 (Fri, Aug 27): Separable equations and applications (cont.): applications - natural growth and decay, radioactive decay, half-life of an isotope, ¹⁴C dating (Sec. 1.4).
  Linear first-order equations: method of solution, integrating factor ρ(x) (Sec. 1.5).
• Lecture 4 (Mon, Aug 30): Linear first-order equations (cont.): examples (Sec. 1.5).
  Substitution methods and exact equations: substitution methods - idea, examples (Sec. 1.6).
• Lecture 5 (Wed, Sep 1): Substitution methods and exact equations (cont.): homogeneous equations - definition, method of solution, examples; Bernoulli equation - definition, idea of solution, examples; implicitly defined solutions F(x,y(x))=0, examples, deriving the ODE satisfied by an implicitly defined solution (Sec. 1.6).
• Lecture 6 (Fri, Sep 3): Substitution methods and exact equations (cont.): exact equations, method of solution, examples, fragility of the "exactness" property; equations with dependent variable (y) missing, generalization of the idea, examples (Sec. 1.6).
• Lecture 7 (Wed, Sep 8): Substitution methods and exact equations (cont.): equations with the independent variable (x) missing, examples (Sec. 1.6).
  Second-order linear equations: a general definition of an n^th order linear equation, homogeneous and non-homogeneous equations, Principle of Superposition for homogeneous linear equations (Sec. 3.1).
• Lecture 8 (Fri, Sep 10): Remarks about exact equations (Sec. 1.6).
  Second-order linear equations: linear independence of two functions, Wronskian of two functions (Sec. 3.1).
• Lecture 9 (Mon, Sep 13): Second-order linear equations (cont.): 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; general solution of a homogeneous linear equation of order n as a linear combination of n linearly independent solutions, 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 (Sec. 3.1, 3.2).
  Homogeneous equations with constant coefficients: characteristic equation (Sec. 3.3).
• Lecture 10 (Wed, Sep 15): Homogeneous equations with constant coefficients (cont.): 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;
  Case 2 - 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) (Sec. 3.3).
• Lecture 11 (Fri, Sep 17): Homogeneous equations with constant coefficients (cont.): complex numbers, Euler's formula; finding the general solution of a homogeneous linear equation of order n with constant coefficients, Ly(x)=0: characteristic equation, general solutions in the following cases:
  Case 3 - 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)]
  Case 4 - 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) (Sec. 3.3).
• Lecture 12 (Mon, Sep 20): Nonhomogeneous order n linear equations and undetermined coefficients: (general solution of nonhomogeneous equation) = (general solution of homogeneous equation) + (particular solution of nonhomogeneous equation); finding a particular solution of Ly(x) =f(x) in the case 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 the nonhomogeneous equation 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 the nonhomogeneous equation (Sec. 3.5).
• Lecture 13 (Wed, Sep 22): Exam 1.
• Lecture 14 (Fri, Sep 24): Nonhomogeneous order n linear equations with constant coefficients (cont.): finding a particular solution of Ly(x)=f(x) in the case 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 the nonhomogeneous equation 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 the nonhomogeneous equation (Sec. 3.5).
• Lecture 15 (Mon, Sep 27): 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 (three cases); forced damped motion (Sec. 3.4, 3.6).
• Lecture 16 (Wed, Sep 29): Mechanical oscillations (cont.): forced damped motion in the underdamped case (Sec. 3.4, 3.6).
• Lecture 17 (Mon, Oct 4): 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).
• Lecture 18 (Wed, Oct 6): Laplace transforms (LTs) and inverse transforms (cont.): existence and uniqueness of LT and inverse LT (very briefly) (Sec. 7.1).
  LT of initial value problems: LT of derivatives, solving an IVP by using LT (Sec. 7.2).
  Translation and partial fractions: translation on the s-axis (Sec. 7.3).
• Lecture 19 (Fri, Oct 8): Translation and partial fractions (cont.): rules for partial fractions, examples (Sec. 7.3).
• Lecture 20 (Mon, Oct 11): Derivatives, integrals, and products of transforms: convolution of two functions, commutativity property of convolution (f*g=g*f), the convolution property (LT of the convolution of two functions is equal to the product of the LTs of the functions), examples of application, differentiation of transforms (Sec. 7.4).
• Lecture 21 (Wed, Oct 13): Derivatives, integrals, and products of transforms (cont.): integration of transofrms, examples. (Sec. 3.4).
  Periodic and piecewise continuous functions: translation on the t-axis (Theorem 1 on page 475) - LT of u(t-a)f(t-a), proof of the theorem, examples of application (Sec. 7.5).
• Lecture 22 (Fri, Oct 15): Periodic and piecewise continuous functions (cont.): periodic functions, LT of periodic functions (Sec. 7.5).
• Lecture 23 (Mon, Oct 18): 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 24 (Wed, Oct 20): 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 (Sec. 7.6).
• Lecture 25 (Fri, Oct 22): Impulses and δ functions (cont.): definition of derivatives of δ_a, more examples (Sec. 7.6).
• Lecture 26 (Mon, Oct 25): Exam 2.
• Lecture 27 (Wed, Oct 27): Impulses and δ functions (cont.): measuring the weight function w(t) and the transfer function W(s) experimentally, remarks about some practical issues (Sec. 7.6).
• Lecture 28 (Fri, Oct 29): 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).
• Lecture 29 (Mon, Nov 1): Periodic functions and trigonometric series: periodic functions, period; idea of Fourier series - expansion of a (periodis) function in terms of sines and cosines; vector spaces, inner product (scalar product, dot product), basis in a vector space, orthogonal 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), 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 30 (Wed, Nov 3): General Fourier series and convergence: formulae for the Fourier coefficient of a periodic function of period p=2L (Sec. 9.2).
• Lecture 31 (Fri, Nov 5): General Fourier series and convergence (cont.): convergence of a Fourier series (Theorem 1), examples, identities obtained by using Theorem 1 (Sec. 9.2).
  Fourier sine and cosine series: extending a function defined on (0,L) to a periodic function of period 2L as an odd function (Fourier sine series), extending a function defined on (0,L) to a periodic function of period 2L as an odd function (Fourier sine series) (Sec. 9.3).
• Lecture 32 (Mon, Nov 8): Applications of Fourier series: using Fourier series to solve linear constant-coefficient ODEs with periodic driving term (i.e., right-hand side f(t)), resonance, secular term (Sec. 9.4).
  Heat conduction and separation of variables: detailed derivation of the heat equation based on conservation of energy (Sec. 9.5).
• Lecture 33 (Wed, Nov 10): Heat conduction and separation of variables (cont.): finishing the derivation of the heat equation, boundary and initial conditions; idea of separation of variables (Sec. 9.5).
• Lecture 34 (Fri, Nov 12): 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) (Sec. 9.5).
• Lecture 35 (Mon, Nov 15): 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; separation of variables in the case of Neumann BCs u_x(0,t)=0, u_x(L,t)=0, examples (Sec. 9.5).
• Lecture 36 (Wed, Nov 17): Heat conduction and separation of variables (cont.): recap of the main ideas of the method of separation of variables, more examples (Sec. 9.5).
• Lecture 37 (Fri, Nov 19): 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 Dirichlet BCs u(0,t)=0, u(L,t)=0, imposing the ICs u(x,0)=f(x), u_t(x,0)=g(x); 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 (Sec. 9.6).
• Lecture 38 (Mon, Nov 22): 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; 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 (Sec. 9.7).
• Lecture 39 (Mon, Nov 29): Recap of Fourier series and PDEs.
• Lecture 40 (Wed, Dec 1): Exam 3.
• Lecture 41 (Fri, Dec 3): Steady-state temperature and Laplace equation (cont.): more on the two-dimensional Laplace equation on a rectangle with Dirichlet or Neumann BCs; Laplace's equation on a semi-infinite strip (Example 2); an example of a problem without solution - Laplace's equation on a rectangle with certain Neumann BCs: Δu=0, u_y(x,0)=0, u_y(x,b)=0, u_x(0,y)=0, u_x(a,y)=5, physical reason - steady-state temperature distribution is impossible if heat comes in through one of the walls of the rectangle and the other three walls are thermally insulated (Sec. 9.7).
• Lecture 42 (Mon, Dec 6): Stationary temperature distribution and Laplace equation (cont.): solution of Laplace's equation in a circular domain - periodic BCs for Θ(θ), consequences for the function R(r) from the requirement of boundedness of the solution u(r,θ) (Sec. 9.7). (Sec. ).
• Lecture 43 (Wed, Dec 8): Separation of variables in cylindrical geometry: 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) (Sec. 10.4).
• Lecture 44 (Fri, Dec 10): Separation of variables in cylindrical geometry (cont.): 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: Homework assignments will be set regularly throughout the semester. Each homework will consist of several problems, of which some pseudo-randomly chosen problems will be graded. Your lowest homework grade will be dropped. All hand-in assignments will carry a specific due date and must be submitted in class on the due date. If you cannot come to class, you can turn in your homework in my office no later than 5 p.m. on the same day (if I am not in my office, you can slip it in under the door). No late homeworks will be accepted.

Shortly after a homework assignment's due date, solutions to the problems from that assignment will be placed on restricted reserve in Bizzell Library. All homework assignments will be posted on this page one week before the assignment is due.

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).

Academic Calendar for Fall 2010.

Course schedule for Fall 2010.

Policy on W/I Grades : Through October 1 (Friday), you can withdraw from the course with an automatic "W". In addition, from October 4 (Monday) to December 10 (Friday), you may withdraw and receive a "W" or "F" according to your standing in the class. Dropping after November 1 (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 A Student's Guide to Academic Integrity. For information on your rights to appeal charges of academic misconduct consult the Rights and Responsibilities Under 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.