MATH 3413 - Physical Mathematics I, Section 001 - Spring 2020
TR 9:00-10:15 a.m., 317 PHSC

Instructor: Nikola Petrov, 1101 PHSC, npetrov AT math.ou.edu.

Office Hours: Mon 2:30-3:30 p.m., Tue 10:30-11:30 a.m., or by appointment, in 1101 PHSC.

First day handout

Text: C. H. Edwards, D. E. Penney, D. T. Calvis. Differential Equations and Boundary Value Problems, 5th edition, Pearson, 2015.
The course will most of chapters 1, 3, 7, 8, and 9 of the book.

Tentative course content:

• Separable equations, linear equations, applications.
• Homogeneous and Bernoulli equations.
• Existence and uniqueness for first order ODEs.
• Second order nonhomogeneous equations. Mass-spring system, resonance.
• Laplace transform and applications to ODEs. Delta function.
• 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.

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, 11), due February 18 (Tuesday).
• Homework 5 (problems given on February 13, 20), due February 27 (Thursday) - will be posted on February 18 and 20.
• Homework 6 (problems given on February 25, 27), due March 5 (Thursday).
• Homework 7 (problems given on March 3, 5), due March 12 (Thursday).
• Homework 8 (problems given on March 10, 12), due on GRADESCOPE by 11:59 p.m. on on March 26 (Thursday).
• Homework 9 (problems given on March 24, 26), due on GRADESCOPE by 11:59 p.m. on on April 4 (Saturday).
  Please note the new due date!
• Homework 10 (problems given on March 31, Apr 2), due on GRADESCOPE by 11:59 p.m. on on April 14 (Tuesday).
• Homework 11 (problems given on Apr 7), due on GRADESCOPE by 11:59 p.m. on on April 16 (Thursday).
  Please note the unusual due date!
• Homework 12 (problems given on Apr 14 and 16), due on GRADESCOPE by NOON on April 22 (Wednesday).
  Please note the VERY unusual due date and TIME!

Content of the lectures:

• Lecture 1 (Tue, Jan 14): Differential equations and mathematical models:
  • ordinary (ODE) and partial (PDE) differential equations, examples;
  • initial value problem (IVP) = ODE + initial conditions (ICs),
  • ODE - general solution - a family of functions (depending on arbitrary constants) that satisfy the ODE,
    IVP - particular solution - one function satisfying the ODE and the ICs;
  • mathematical modeling of natural phenomena:
    • real-life phenomenon,
    • mathematical model,
    • solution of the mathematical model,
    • comparison of the solution with the original phenomenon, if not good - modify the model;
  • 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/M), where M is the carrying capacity of the system;
  • an example from mechanics - one-dimensional motion with constant acceleration.
  [Sec. 1.1]
  Homework 1 (incomplete), due Thu, Jan 23.
  Practice problems: Problems 1.1/5, 14, 22, 32, 42.
  Remark: The Practice problems will not be collected for grading, but nevertheless their completion is essential for your full understanding of the course material. Their solutions will be posted on Canvas. You are expected to know how to solve each of these problems.

• 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;
  • examples.
  [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 and IVP with infinitely many solutions: growth of the volume of a water droplet in an oversaturated vapor (see Problem 1.3/29).
  [Sec. 1.3]
  Homework 1 (complete), due Thu, Jan 23.
  The complete Homework 1 (problems given on January 14, 16) is due on January 23 (Thursday).
  Practice problems: Problems 1.2/2, 9, 18, 30, 40, 1.3/12, 14, 16.
• Lecture 3 (Tue, Jan 21):
  Separable equations and applications:
  • separable equations - definition;
  • method of solution of separable equations, examples;
  • implicit solutions, examples;
  • singular solutions, examples;
  • applicaitons of separable equations:
    • Newton's law of cooling and heating;
    • Torricelli's law for a liquid leaking out of a small hole in a tank.
  [Sec. 1.4]
  Linear first-order equations:
  • integrating factor;
  • algorithm for solving linear first-order equations;
  • examples.
  [pages 45-48 of Sec. 1.5]
  Homework 2 (incomplete), due Thu, Jan 30.
  Practice problems: Problems 1.4/2, 5, 17, 22; 1.5/4, 9, 20, 25.
• Lecture 4 (Thu, Jan 23):
  Linear first-order equations (cont.):
  • an example: (x²+1)y'(x)+y(x)=1, the equation written in the "standard" form: y'(x)+y(x)/(x²+1)=1/(x²+1), integrating factor ρ(x)=e^arctan(x), general solution y(x)=1+Ce^−arctan(x), particular solution with IC y(3^1/2)=4: y(x)=1+4e^{(π/3)−arctan(x)};
  • remark: the equation in the previous example is also separable: y'=(1−y)/(x²+1);
  • an example: xy'(x)+y(x)=e^−x2, the equation written in the "standard" form: y'(x)+(1/x)y(x)=e^−x2/x, integrating factor ρ(x)=x, general solution y(x)=1/x[(π^1/2/2)erf(x)+C], where erf(x) is the error function ((2/π^1/2) times the definite integral from 0 to x of e^−t2dt; obvious properties of erf(x): erf(0)=0, (d/dx)erf(x)=e^−x2, erf(x) is an odd function.
  [pages 45-50 of Sec. 1.5]
  Substitution methods:
  • general idea of the method;
  • example: y'=(x+y+3)², substitution: v=x+y+3, then y=v−x−3, y'=v'−1, the ODE becomes v'=1+v², separable equation with general solution v(x)=tan(x+C), so y(x)=tan(x+C)−x−3;
  • example (Problem 1.6/17): y'=(4x+y)^1/2, v=4x+y, y=v−4x, y'=v'−4, v'=4+v², y(x)=2tan(2x+C)−4x.
  • homogeneous equations: y'(x)=F(y/x), substitution: v=y/x, so y(x)=xv(x), y'(x)=v(x)+xv'(x), and the ODE becomes v(x)+xv'(x)=F(v).
  [pages 57-60 of Sec. 1.6]
  Homework 2 (complete), due Thu, Jan 30.
  The complete Homework 2 (problems given on January 21, 23) is due on January 30 (Thursday).
  Practice problems: Problems 1.6/2, 5, 20, 27.
• Lecture 5 (Thu, Jan 28):
  Substitution methods (cont.):
  • Bernoulli equation: substitution v:=y^1−α converting it to a linear 1st order equation;
  • an example: y'−y/x=y²/x;
  • second-order ODE with y missing: substitution v:=y' converting it to a first-order equation;
  • an example: y''−y'/x=(y')²/x (after the substitution it becomes the Bernoulli equation above);
  • y⁽⁴⁾+y'''=1 (substitution v:=y''', find v(x), then integrate three times to obtain y(x));
  • second-order ODE with x missing: substitution p:=y' converts it to a first-order equation for the function p(y);
  • an example: yy''=(y')².
  [pages 61, 62, 67-69 of Sec. 1.6]
  Homework 3 (incomplete), due Thu, Jan 30.
  Practice problems: Problems 1.6/48, 54.
• Lecture 6 (Thu, Jan 30):
  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;
  • constructing the general solution of a homogeneous second order linear equation as a linear combination of two linearly independent solutions of the equation;
  • 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).
  [Sec. 3.1; skip the text on page 142 about the Wronskian of solutions]
  Homework 3 (complete), due Thu, Feb 6.
  The complete Homework 3 (problems given on January 28, 30) is due on February 6 (Thursday).
  Practice problems: Problems 3.1/2, 11, 34, 35, 40, 46.
  Suggested reading: examples of 1st order equations that can be solved in different ways, their solutions, a Mathematica notebooks first-order-ODEs-solved-examples-using-mathematica.nb and the same notebook in pdf format illustrating how to use Mathematica.
• Lecture 7 (Tue, Feb 4):
  Homogeneous linear equations with constant coefficients:
  • characteristic equation;
  • writing down the general solution based on the roots of the characteristic equation:
    • case 1 − distinct real roots of the characteristic equation: each root r_j contributes a term c_je^r_jx; examples;
    • 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 P_p−1(x)e^r₁x, where P_p−1(x) is an arbitrary polynomial of degree p−1; examples;
    • 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; examples.
  [Sec. 3.3; skip the material about the complex-valued functions and Euler's formula]
  Homework 4 (incomplete), due Tue, Feb 18.
  Practice problems: Problems 3.3/12, 15, 18, 29, 34, 40.
• Lecture 8 (Thu, Feb 6): Lecture canceled due to weather.
• Lecture 9 (Tue, Feb 11):
  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 y_p(x) in (N) and equating the coefficients of the terms containing the same powers of x;
    • 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 y_p(x) in (N) and equating the coefficients of the terms containing the same powers of x.
  [pages 184-192 of Sec. 3.5]
  Homework 4 (complete), due Tue, Feb 18.
  The complete Homework 4 (problems given on February 4, 11) is due on February 18 (Tuesday).
  Practice problems: Problems 3.5/2, 10, 22, 28, 31, 38.
  More practice: read the solved problems in this handout as well as Examples 1-10 on pages 185-192 of the book, and apply the rules for looking for a particular solutions.
• Lecture 10 (Thu, Feb 13):
  Laplace transforms and inverse transforms:
  • functions, functionals, transforms;
  • definition of Laplace transform (LT);
  • example: LT of ƒ(t)=1;
  • example: LT of ƒ(t)=e^at;
  • integral transforms in general, integral kernel K(t,s);
  • unit step function ("Heaviside function") u(t), shifted Heaviside function u_a(t)=u(t−a);
  • integral kernel of the LT: K(t,s)=e^−stu(t);
  • linearity of LT (with proof);
  • Gamma function and its properties;
  • LT of ƒ(t)=t^a,
  • inverse Laplace transform.
  [pages 437-442 of Sec. 7.1]
  Homework 5: will be posted later.
  Practice problems: Problems 7.1/1, 4, 7, 13, 17, 25, 30.
• Lecture 11 (Tue, Feb 18):
  Exam 1, on the material from Sections 1.1-1.6, 3.1, 3.3, and 3.5, covered in Lectures 1-9]
• Lecture 12 (Thu, Feb 20):
  Laplace transform of initial value problems:
  • LT of derivatives;
  • solving an IVP by using LT - general idea and examples;
  • LT of integrals;
  • using the formula for LT of integrals to find the inverse LT of expressions of the form sF(s) (Example 6).
  [Sec. 7.2]
  Homework 5 (complete), due Thu, Feb 27.
  The complete Homework 5 (problems given on February 13, 20) is due on February 27 (Thursday).
  Practice problems: Problems 7.2/3, 7, 10, 19, 22.
  Reading assignment (mandatory): Examples 4 and 5 from Sec. 7.2.
• Lecture 13 (Tue, Feb 25):
  Translation and partial fractions:
  • rules for partial fractions: linear factors and quadratic factors;
  • translation on the s-axis (Theorem 1, with proof);
  • using LT and inverse LT to solve IVPs: Examples 1, 3, and 4.
  [pages 458-462 of Sec. 7.3]
  Homework 6 (incomplete), due Thu, Mar 5.
  Practice problems: Problems 7.3/3, 6, 14, 19, 28, 33, 37.
  Reading assignment (mandatory): Examples 5 and 6 from Sec. 7.3.
• Lecture 14 (Thu, Feb 27):
  Derivatives, integrals, and products of transforms:
  • definition of the convolution of two functions;
  • linearity of the convolution with respect to each argument: (αƒ+g)*h=αƒ*h+g*h, α=constant;
  • commutativity property of convolution: ƒ*g=g*ƒ (with proof);
  • the convolution property (Theorem 1 - LT of the convolution of two functions is equal to the product of the LTs of the functions);
  • using the convolution property to compute the inverse LT of 2/[(s−1)(s²+4)];
  • differentiation of transforms (Theorem 2);
  • integration of transforms (Theorem 3);
  • using Theorem 2 to compute LT of t²sin(kt);
  • using Theorem 3 to compute the inverse LT of arctan(1/s) (motivated by the observation that the derivative of arctan is a rational function);
  • using Theorem 2 to solve the initial value problem tx''(t)+(t−1)x'(t)+x(t)=0, x(0)=0, x'(0)=0.
  [pages 467-471 of Sec. 7.4]
  Homework 6 (complete), due Thu, Mar 5.
  The complete Homework 6 (problems given on February 25, 27) is due on March 5 (Thursday).
  Practice problems: Problems 7.4/5, 10, 16, 22, 24, 31.
  Reading assignment (mandatory): Read all examples from Sec. 7.4.
  Reading assignment (optional): The proof of Theorem 1 on pages 471-472 of Sec. 7.4.
• Lecture 15 (Tue, Mar 3):
  Periodic and piecewise continuous functions:
  • translation on the t-axis: the LT of u(t-a)ƒ(t-a) is e^−asF(s) (Theorem 1, with proof);
  • Examples 2 and 3;
  • definition of a periodic function of period p;
  • LT of a periodic function (Theorem 2, with proof);
  • Example 6;
  • example of a harmonic oscillator driven by a periodic piecewise-constant function.
  [Sec. 7.5]
  Homework 7 (incomplete), due Thu, Mar 12.
  Practice problems: Problems 7.5/3, 8, 12, 16, 26, 28, 31.
  Reading assignment (mandatory): Read all examples from Sec. 7.5.
• Lecture 16 (Thu, Mar 5):
  Impulses and δ-functions:
  • δ_a(t) as a limit of "rectangle" functions;
  • definition of δ_a;
  • notations: δ(t):=δ₀(t), δ(t−a):=δ_a, justification of the latter notation;
  • Laplace transform of δ_a; solving linear constant-coefficient ODEs with right-hand side (i.e., driving force) δ_a(t) by Laplace transform;
  • transfer function W(s)=(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 w(t) by using a delta-function input because in this case x(t)=(w*δ)(t)=w(t)
  [Sec. 7.6; skip the text about δ_a as the derivative of u_a]
  Homework 7 (complete), due Thu, Mar 12.
  The complete Homework 7 (problems given on March 3, 5) is due on March 12 (Thursday).
  Practice problems: Problems 7.6 / 2, 6, 8, 19b.
  Additional materials: A handout on the properties of LT (given in class).
• Lecture 17 (Tue, Mar 10):
  Impulses and δ-functions (cont.):
  • more about δ_a(t):
    • δ*w=w (a self-quiz question: δ_a*w?);
    • a proof that u_a'=δ_a (based on applying the integration by parts formula);
    • food for thought: if v_a(t)=0 for t<0 and v_a(t)=t for t>0, then show that v_a'=u_a;
  • more discussion of the Duhamel's principle;
  • exercise: showing that the inverse LT of 1/(s²−6s+25) is (1/4)e^3tsin(4t);
  • exercise: showing that the inverse LT of s/(s²−6s+25) is e^3t[cos(4t)+(3/4)sin(4t)].
  [pages 488-492 of Sec. 7.6]
  Homework 8 (incomplete), due on GRADESCOPE by 11:59 p.m. on Thu, Mar 26.
• Lecture 18 (Thu, Mar 12):
  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);
  • determining the Fourier coefficients of a 2π-periodic function;
  • relation between the parity (i.e., the property of being even/odd) of a function with its Fourier series;
  • examples.
  [Sec. 9.1]
  Homework 8 (complete), due on GRADESCOPE by 11:59 p.m. on on March 26 (Thursday).
  The complete Homework 8 (problems given on March 10, 12)
  is due on GRADESCOPE by 11:59 p.m. on on March 26 (Thursday).
  Practice problems: Problems 9.1/1, 3, 9, 11, 14, 20, 26.
• Lecture 19 (Tue, Mar 24):
  Periodic functions and trigonometric series:
  • definition of a periodic function of period p=2L;
  • Fourier series of a 2L-periodic function (i.e., a periodic function of period 2L);
  • expression for the coefficients a_n and b_n of a 2L-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 (Theorem 1);
  • using the theorem on convergence of Fourier series to find sum of a series;
  • examples.
  [Sec. 9.2]
  Homework 9 (incomplete), due on GRADESCOPE by 11:59 p.m. on Sat, Apr 4. Please note the new due date!
  Practice problems: Problems 9.2/3, 7, 9, 13, 17.
  Just for fun: a very interesting video with a proof of the formula for the sum of the reciprocals of the natural numbers (the so-called Basel problem) based on physical and geometric principles.
• Lecture 20 (Thu, Mar 26)
  Fourier sine and cosine series:
  • even and odd functions;
  • parity of a product and ratio of even/odd functions;
  • Fourier series of even/odd functions;
  • extending a function defined on (0,L) to an odd or an even function on (−L,L) (of period 2L);
  • Fourier sine (for the odd extension) or Fourier cosine (for the even extension) of a function defined on (0,L);
  • relation between the regularity of a function and the rate of decay of its Fourier coefficients.
  [pages 580-584 of Sec. 9.3]
  Homework 9 (complete), due on GRADESCOPE by 11:59 p.m. on Sat, Apr 4. Please note the new due date!
  The complete Homework 9 (problems given on March 24, 26)
  is due on GRADESCOPE by 11:59 p.m. on on April 2 (Thursday). Please note the new due date!
  Practice problems: Problems 9.3/1, 5, 7.
• Lecture 21 (Tue, Mar 31)
  Fourier sine and cosine series:
  • using Fourier sine and cosine expansion for solving boundary value problems (BVPs) for linear constant-coefficient ODEs, like an example: solving the BVP ax''+bx'+cx=ƒ(t), x(0)=A, x(L)=B;
  • an example: solving the BVP ax''+4x=4t x(0)=0, x(1)=0 by using Fourier sine series
  [pages 586-587 of Sec. 9.3]
  Heat conduction and separation of variables:
  • a derivation of the heat equation in one spatial dimension;
  • Example 1: constructing a solution of the heat equation by finding several functions satisfying the PDE and the BCs, and constructing a superposition of them that satisfies the IC
  [pages 597-601 of Sec. 9.5]
  Homework 10 (incomplete), due date TBA.
  Practice problems: Problems 9.3/12, 14.
• Lecture 22 (Thu, Apr 2)
  Heat conduction and separation of variables (cont.):
  • finding functions u_n(x,t)=F_n(x)G_n(t) by separation of variables that satisfy the heat equation and the BCs u(0,t)=0, u(L,t)=0;
  • adjusting the coefficients in the superposition of functions u_n(x,t) in order to satisfy the IC u(x,0)=ƒ(x);
  • recap of the main ideas of the method of separation of variables;
  • examples.
  [pages 601-605 of Sec. 9.5]
  Homework 10 (complete), due on GRADESCOPE by 11:59 p.m. on Tue, Apr 14.
  The complete Homework 10 (problems given on March 31, April 2)
  is due on GRADESCOPE by 11:59 p.m. on on April 14 (Tuesday).
  Practice problems: Problems 9.5/1, 3, 10, 13.
  Reading assignment (optional): the Appendix, "A comparison of the Laplace and Fourier transforms" (page 8 of Lecture 22 from the Lecture notes uploaded on Canvas), in which you can find a brief discussion of the similarities and differences between the Laplace transform and the Fourier transform.
• Lecture 23 (Tue, Apr 7):
  Heat conduction and separation of variables (cont.):
  • finding functions u_n(x,t)=F_n(x)G_n(t) by separation of variables that satisfy the heat equation and the BCs u_x(0,t)=0, u_x(L,t)=0;
  • adjusting the coefficients in the superposition of functions u_n(x,t) in order to satisfy the IC u(x,0)=ƒ(x);
  • Example 3.
  [pages 605-608 of Sec. 9.5]
  Homework 11 (complete), due on GRADESCOPE by 11:59 p.m. on Thu, Apr 16.
  PLEASE NOTE THE UNUSUAL DUE DATE!
  Practice problems: Problems 9.5/5, 11, 14.
• Lecture 24 (Thu, Apr 9):
  Exam 2, on the material from Sections 7.1-7.6, covered in Lectures 10, 12-17]
• Lecture 25 (Tue, Apr 14):
  Vibrating strings and the one-dimensional wave equation:
  • derivation of the wave equation describing waves on a string;
  • physical interpretation of the boundary conditions y(0,t)=0, y(L,t)=0, and the initial conditions y(x,0)=ƒ(x,0) (initial position) y_t(x,0)=g(x,0) (initial velocity);
  • representing the solution y(L,t) of the wave equation, y_tt=a²y_xx, with BCs y(0,t)=0, y(L,t)=0, and ICs y(x,0)=ƒ(x) y_t(x,0)=g(x) as y(L,t)=y_A(L,t)+y_B(L,t),
    where y_A(L,t) satisfies the PDE, the BCs, and the ICs y(x,0)=ƒ(x) y_t(x,0)=0,
    while y_B(L,t) satisfies the PDE, the BCs, and the ICs y(x,0)=0, y_t(x,0)=g(x);
  • finding the functions y_A(L,t) and y_B(L,t) by separation of variables.
  [pages 611-615, 619, 620 of Sec. 9.6]
• Lecture 26 (Thu, Apr 16):
  Vibrating strings and the one-dimensional wave equation (cont.):
  • several examples (including Examples 1, 2, and 4 from the book).
  [pages 615, 616, 620 of Sec. 9.6]
Homework 12 (complete), due on GRADESCOPE by NOON on Wed, Apr 22.
PLEASE NOTE THE VERY UNUSUAL DUE DATE AND TIME!
Practice problems: Problems 9.6/1, 2, 4, 8, 10.
• Lecture 27 (Tue, Apr 21):
  Steady-state temperature and Laplace equation:
  • physical problems leading to Laplace's equation Δu(x)=0 - steady-state temperature with time-independent boundary conditions;
  • boundary value problems for 2-dimensional Laplace's equation in a rectangular domain (x,y)∈[0,a]×[0,b];
  • solving Laplace's equation with BCs u(x,0)=ƒ(x), u(x,b)=0, u(0,y)=0, u(a,y)=0 by separation of variables (Example 1).
  [pages 625-628 of Sec. 9.7]
  Vibrating strings and the one-dimensional wave equation (cont.):
  • concepts related to a vibration that is periodic in time and space:
    • speed a (unit m/s),
    • wavelength λ (unit m),
    • period τ (unit s),
    • (linear) frequency ν=1/τ (unit s⁻¹=Hertz),
    • angular frequency ω=2π/τ (unit s⁻¹),
    • basic relation λ=aτ;
  • concepts related to the waves in a guitar string of length L (with fixed ends):
    • harmonics (see the Wikipedia article Harmonic),
    • allowed angular frequencies ω_n=nπa/L,
    • allowed linear frequencies ν_n=ω_n/(2π)=na/(2L),
    • allowed wavelengths λ_n=2L/n,
    • allowed periods τ_n=1/ν_n=2L/(na);
  • dependence of the fundamental frequency ν₁ on the physical parameters length L (unit: m), tension T (unit: N), (linear) density ρ (unit: kg/m):
    ν₁=a/(2L)=(T/ρ)^1/2/(2L)
  • illustrations with guitar strings;
  • flageolets - supressing some harmonics by touching the string at certain positions (see the Wikipedia article Flageolet).
• Lecture 28 (Thu, Apr 23):
  Exam 2, on the material from Sections 9.1-9.3, 9.5, 9.6, covered in Lectures 18-23, 25, 26]
• Lecture 29 (Tue, Apr 28):
  Steady-state temperature and Laplace equation (cont.):
  • example: steady-state temperature in a semi-infinite strip with zero temperatures on the infinitely long walls and a given tempearture on the finite wall;
  • example: steady-state temperature in a disk - separation of variables in polar coordinates;
  • example: steady-state temperature in a rectangle with BCs u_x(0,y)=0, u_x(a,y)=0, u_y(x,0)=0, u_y(x,b)=5 - the problem has no solution;
  • physical reasons for non-existence of solution of the last problem: the BCs mean that three of the walls are thermally insulated, while there is heat flux going into the domain, so that steady state is physically impossible (the temperature in the rectangle will increase without bound due to the heat influx);
  • modification of the previous example: steady-state temperature in a rectangle with BCs u_x(0,y)=0, u_x(a,y)=0, u_y(x,0)=0, u_y(x,b)=ƒ(x) - solution exists if and only if the integral of ƒ(x) for x from 0 to a is zero, which means that the net heat entering the domain is zero, hence steady-state solution exists.
  [pages 629-632 of Sec. 9.7]
• Lecture 30 (Thu, Apr 30):
  A quick review of some topics for the final.
• Final Exam:
  Friday, May 8, 8:00-10:00 a.m., cumulative.

Good to know:

• The greek_alphabet.
• Some useful notations.