MATH 4163.980 - Introduction to Partial Differential Equations - Fall 2012
Thurday, 12:30-2:40 p.m. (in SCC 4216) or 1:30-4:10 p.m (in SPP 3100)

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

Course catalog description: Prerequisite: 2443, 3113 or 3413. Physical models, classification of equations, Fourier series and boundary value problems, integral transforms, the method of characteristics. (F, Sp, Su)

Text: Richard Haberman, Applied Partial Differential Equations, 4th edition, Pearson/Prentice Hall, 2003. The course will cover parts of chapters 1-10 (and chapter 11, if time permits).

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Homework:

• Homework 1, due August 30 (Thursday).
• Homework 2, due September 6 (Thursday).
• Homework 3, due September 13 (Thursday).
• Homework 4, due September 21 (Friday).
• Homework 5, due September 28 (Friday), at noon.
• Homework 6, due October 5 (Friday), at noon.
• Homework 7, due by 4 p.m. on October 22 (Monday), in 4W138.
• Homework 8-9, due in class on December 6 (Thursday).

Content of the lectures:

• Lecture 1 (Aug 23, 12-2:40 p.m., SCC 4216): ODEs: order-n ODE and its general solution (an n-parameter family of functions, parameterized by the constants C₁,...,C_n), initial conditions (ICs) and initial-value problems (IVPs) for ODEs, boundary conditions (BCs) and boundary-value problems (BVPs) for ODEs; on the importance of being linear.
  PDEs: examples of finding the general solutions of simple PDEs, the general solution of an order-n PDE for a function of d variables, u:R^d→R, has n arbitrary functions of (d-1) variables.
  Vector (linear) spaces: definition, simple implications of the axioms (uniqueness of the zero vector, 0u=0, etc.), basis and dimension of a vector space, an example (Rⁿ).
  Normed vector spaces: definition of a norm, examples of norms - ||u||_∞, ||u||₁, ||u||₂ (Euclidean norm), ||u||_p for p∈[1,∞); shapes of the unit ball in (R²,||•||₂) and in (R²,||•||_∞); convergence of a sequence of vectors v⁽¹⁾, v⁽²⁾, ... in a normed space.
  Inner product (scalar product, dot product) vector spaces: inner product - properties, examples; every inner product defines a norm; orthogonality (by definition, u⊥v if the inner product of u and v is zero); projection of a vector u onto a vector v (more precisely, onto the direction determined by v) - definition and explicit expression.
  Space of polynomials of degree ≤n: definition of addition of polynomials and of multiplication of a polynomial by a number, definition of the space V_n of polynomials of degree ≤n, the dimension of V_n is n+1; remarks about working with polynomials in computer programs; inner product in V_n, weight function w(x).
• Lecture 2 (Aug 30, 1:30-4:10 p.m., SPP 3100): Complex algebra: complex numbers, complex conjugate, Euler's formula.
  Fourier series of 2π-periodic functions: periodic functions of period p, representing a 2π-periodic function in a Fourier series, computing the coefficients c_n, thinking of the functions as vectors in an inner product vector space, expressing the coefficients throught inner products, connection between c_n and c_−n for a real-valued function; The Fourier series in a sin/cos form; convergence of a Fourier series.
  Fourier series of 2L-periodic functions: explicit expressions for the series and its coefficients.
  Fourier series of real-valued functions defined on [0,L]: even and odd functions, extending a real-valued function f defined on [0,L] to R as an even 2π-periodic function, extending a real-valued function f defined on [0,L] to R as an odd 2π-periodic function, Fourier series of the even extension of f, Fourier series of the odd extension of f. Heat equation: set-up of the boundary-value problem (BVP), gradient, normal derivative of a function on a survace, Dirichlet and Neumann boundary conditions (BCs), initial conditions (ICs), example in one spatial dimension.
  Separation of variables: general idea, trying to solve the (1+1)-dimensional heat equation with homogeneous Dirichlet BCs by separation of variables, trying to solve the BVP X''(x)-μX(x)=0, X(0)=0, X(L)=0 for positive or zero value of μ (unsuccessful in both cases).
• Lecture 3 (Sep 6, 1:30-4:10 p.m., SPP 3100): Separation of variables (cont.): successful solution of the BVP X''(x)-μX(x)=0, X(0)=0, X(L)=0 for negative value of μ (μ=-λ²). Finding the set of values λ_n for which the BVP X''(x)+λ²X(x)=0, X(0)=0, X(L)=0 has a non-trivial solutions, and finding the solutions X_n(x)=sin(nπx/L). Determining the functions T_n(t) and the functions u_n(x,t)=X_n(x)T_n(t) that satisfy the PDE and the BCs. Using the Principle of Superposition (since the PDE is linear) to look for a solution u(x,t) of the original BVP as a linear combination of the functions u_n(x,t); determining the coefficients in the linear combination, an example. Discussion about the physical meaning of the solution - the larger n, the faster the term u_n(x,t) decays with time because the heat exchange occurs between points that are closer.
  Heat equation with Neumann BCs in the (1+1)-dimensional case: homogeneous Neumann BCs, u_x(0,t)=0, u_x(L,t)=0, which corresponds to thermally insulated ends; separation of variables, finding the solutions of the BVP X''(x)-μX(x)=0, X'(0)=0, X'(L)=0 and the set of values λ_n for which it has solutions (with μ=-λ²), constructing the functions u_n(x,t) and using the Principle of Superposition to find the solution of the original problem.
  Steady-state temperature distribution in 2 spatial dimensions: derivation of the Laplace's equation, Δu(x,y)=0; meaning of the BCs, an example of separation of variables; a mathematical digression - the Maximum Principle and its physical meaning; solving BVPs for Laplace's equation with more complicated BCs by using the linearity of the equation (hence, Principle of Superposition).
• Lecture 4 (Sep 13, 12-2:40 p.m., SCC 4216): Laplace's equation on a rectangle with Neumann BCs: solving the BVP for the Laplace's equation Δu(x,y)=0 on the rectangle (x,y)∈[0,a]×[0,b] with two Neumann BCs u(0,y)=0, u(a,y)=0, and two Dirichlet BCs u(x,0)=0, u(x,b)=f(x) by separation of variables. A discussion of the physical interpretation of the BVP (interpreted as a BVP describing a steady-state temperature distribution) - Neumann BCs correspond to controlling the heat flux through the wall, while the Dirichlet BCs correspond to controlling the temperature at the wall. A discussion of possible problems that occur if one tries to solve the steady-state heat equation (i.e., the Laplace's equation) with Neumann BCs on all walls (when the net heat flux through the walls may be non-zero, in which case a steady-state is not possible). Read Section 2.4.1 of the book about the heat equation on a rod with thermally insulated ends leading to Neumann BCs; look at the table on page 69 of the book summarizing the solution for the case of Dirichlet and Neumann BCs.
  Laplace's equation in a circular disk: separation of variables in the Laplace's equation in a circular disk - read Section 2.5.2.
  
• Lecture 5 (Sep 20, 1:30-4:10 p.m., SPP 3100): Occurrence of more complicated Sturm-Liouville (SL) eigenvalue problems: heat flow in a nonuniform rod, circularly symmetric heat flow [Section 5.2].
  SL eigenvalue problems: classification, regular SL eigenvalue problems. Main theoretical results about SL eigenvalue problems: real eigenvalues λ_n; infinite number of eigenvalues with a smallest one, the eigenvalues λ_n tend to ∞ as n tends to ∞; uniqueness (up to a multiplicative constant) of the eigenfunctions φ_n corresponding to the eigenvalue λ_n, number of zeros of φ_n completeness of the set of eigenfunctions; orthogonality of eigenfunctions corresponding to different eigenvalues. Examples, using the orthogonality of the eigenfunctions to determine the coefficients in the expansion of a function in the basis of eigenfuncitons [pages 161-166 of Section 5.3].
  Self-adjoint operators and SL eigenvalue problems: Linear operators occurring in SL eigenvalue problems, Lagrange's identity, Green's formula, self-adjointness of the linear operators occurring in SL eigenvalue problems with homogeneous BCs; proof of the orthogonality of φ_n and φ_k for n≠k, proof that the eigenvalues λ_n are real [pages 174-179 of Section 5.5].
  BCs of the third kind: physical meaning of the BC for the case of the heat equation in one spatial dimension: u_x(0,t)=-hu(0,t) describes the convective heat exchange at the end of the rod; deriving an transcendental equation for the eigenvalues λ_k and graphical analysis of the behavior of λ_k in the physical case h>0; a brief discussion of the non-physical case h<0 [pages 198-203 of Section 5.8].
• Lecture 6 (Sep 27, 1:30-4:10 p.m., SPP 3100): Vibrating rectangular membrane: separating variables in the wave equation in a rectangular membrane with fixed ends, double Fourier series, frequencies of vibration of the membrane [Section 7.3].
  General properties of the solutions of the Helmnoltz equation: brief summary (without derivations) of the properties of the solutions of the Helmholtz equation ΔS+λS=0 [pages 289-291 of Section 7.4].
  Vibrating circular membrane and Bessel functions: separating variables in the wave equation in a circular membrane with fixed ends, periodicity condition for the angular function Θ(θ), expression the radial function R(r) in terms of the solutions of the Bessel's differential equation, J_m(x) and Y_m(x), eliminating Y_m(x) because of the boundedness condition at r=0, expressing the radial functions as R_mn(r)=J_m(β_mnx/a), where β_mn is the mth zero of J_m, ordered as follows: 0<β_m1<β_m2<β_m3<... [Section 7.7].
• Lecture 7 (Oct 4, 12-2:40 p.m., SCC 4216): Laplace's equation in a circular cylinder: separation of variables in Laplaces equation in a circular cylinder with cold top and bottom and an arbitrary Dirichlet boundary conditions on the side wall; discretization of the constants because of the periodicity in θ and because of the zero boundary conditions at the top and the bottom (which implies the conditions Z(0)=0, Z(b)=0); modified Bessel functions of the first kind, I_m, modified Bessel functions of the second kind (Macdonald functions), K_m; behavior of I_m(x) and K_m(x) near x=0⁺ and discarding K_m(x) because it tends to ∞ when x tends to 0; determining the coefficients in the series expansion of the solution u(r,θ,z) of the BVP [read Sections 7.9.1, 7.9.2, 7.9.4, and skim through Section 7.9.5].
  Heat flow with sources and nonhomogeneous BCs: Case 1: no sources and time-independent BCs - find the equilibrium temperature u_∞(x) and reformulate the problem for the "displacement from equilibrium" function v(x,t):=u(x,t)-u_∞(x) which satisfies a BVP with no sources and zero BCs (and a modified IC). Case 2: time-independent sources Q(x) and time-independent BCs - find the equilibrium temperature u_∞(x) which satisfies the ODE α²u''_∞(x)+Q(x)=0, introducing the "displacement from equilibrium" function v(x,t):=u(x,t)-u_∞(x) which satisfies a BVP with no sources and zero BCs (and a modified IC). Case 3: time-dependent sources Q(x,t) and time-dependent BCs - find any "reference temperature distribution" r(x,t) that satisfies the time-dependent BCs, then the "displacement from equilibrium" function v(x,t):=u(x,t)-r(x,t) satisfies the PDE with (modified) time-dependent sources, homogeneous BCs, and modified ICs [Section 8.2].
  Method of eigenfunction expansion with homogeneous BCs: in Case 3 above write the associated homogeneous BVP (i.e., the problem occurring for v(x,t) but with zero sources in the PDE), separating variables in the associated homogeneous BVP to obtain a basis of eigenfunctions X_n(x), looking for a solution u(x,t) of the original BVP for v(x,t) (with time-dependent source in the PDE) as a superposition of functions of the form T_n(t)X_n(x), writing an ODE for T_n(t) (coming from the PDE with the source expanded in the basis X_n(x)) and ICs for T_n(t) (coming from the expansion of the IC for v(x,t) in the basis X_n(x)); solving the initial-value problems for T_n(t) [Section 8.3].
• Lecture 8 (Oct 11, 1:30-4:10 p.m., SPP 3100): Fourier transform: integral transforms, integral kernel, Fourier transform (FT), inverse FT, examples - FT of the Dirac delta function δ_a(x), inverse FT of a Gaussian G(ω)=e^-α2ω2 (with α a positive constant) [pages 450-454 of Section 10.3].
  FTs and PDEs on the real line R: FT of the derivatives u_t(x,t), u_x(x,t), and u_xx(x,t) of a function u(x,t) with respect to the variable x, expressing the solution of the heat equation on R as an integral involving the initial condition f(x)=u(x,0). Influence function for the heat equation on R: explicit expression and physical interpretation. Example - the heat equation on R with initial condition equal to the Heaviside function Θ(x): expressing the solution in terms of the error function, similarity solution (reducing the heat equation to an ODE for the similarity variable z=x/t^1/2) [pages 459-466 of Section 10.4].
• Lecture 9 (Oct 18, 1:30-4:10 p.m., SPP 3100): FTs and PDEs on the real line R (cont.): definition and properties of convolution of two functions, the inverse FT of the product of the FTs F(ω) and G(ω) of the functions f(x) and g(x) is equal to 1/2π times the convolution of the two functions, application to the heat equation, Parseval's identity (Plancherel's formula) - expressions for the "energy" of a function as an integral over x and as an integral of the spectral energy density over ω [pages 466-469 of Section 10.4].
  Fourier sine and cosine transforms: the heat equation on semi-infinite intervals: solving the heat equation for x∈[0,∞) by separation of variables, obtaining the solution in the form of an integral over ω∈[0,∞) of an expression containing sin(ωx); definition of Fourier sine and cosine transforms; Fourier sine and cosine transform of derivatives of functions; a complete solution of an initial-value problem for the heat equation on the half-line by using Fourier sine and cosine transforms [Section 10.5].
  Lord Kelvin's calculation of the age of the Earth: a link to the paper "Kelvin and the age of the Earth" by F. Richter published in The Journal of Geology, Vol. 94, No. 3, May 1986, pp. 395-401.
• Lecture 9a (Oct 25, 12-2:40 p.m., SCC 4216): Midterm Exam.
• Lecture 10 (Nov 1, 1:30-4:10 p.m., SPP 3100): Worked examples using transforms: one-dimensional wave equation on R, u_tt(x,t)=c²u_xx(x,t), with zero initial velocity by using Fourier transform in x - the solution is a superposition of waves moving to the left and to the right with speed c: u(x,t)=(1/2)[f(x-ct)+f(x+ct)]; Laplace's equation in a semi-infinite strip - representing the solution as a superposition of functions u₁ (vanishing at the side walls) and u₂ (vanishing at the bottom), solving the problem for u₁ by separation of variables, solving the problem for u₂ by performing Fourier sine transform with respect to x; Laplace's equation in a half-plane - performing Fourier transform in x, using the Convolution Theorem to represent the solution as an integral, an example [Sections 10.6.1-10.6.3].
  Green's functions for BVPs for ODEs: steady-state heat equation in one spatial dimension with time-independent sources in the domain, and homogeneous Dirichlet BCs - reduction to the BVP u''(x)=fx, u(0)=0, u(L)=0; more general problem: Lu(x)=[p(x)u'(x)]'+q(x)u(x)=0; method of variation of parameters: u(x)=v₁(x)u₁(x)+v₂(x)u₂(x), where Lu₁=0 and Lu₂=0, deriving a linear system for v₁'(x) and v₂'(x), Wronskian of two functions, obtaining the solution; an example [pages 385-389 of Section 9.3].
• Lecture 11 (Nov 8, 12-2:40 p.m., SCC 4216): Green's functions for BVPs for ODEs (cont.): method of eigenfunction expansion for Green's functions - idea and an example; Dirac's δ-function and Green's functions - definition of Dirac's δ-function, derivatives of δ-function, properties of δ-function, generalized derivatives of discontinuous functions (equal to the regular derivatives when these exist plus j_kδ(x-x_k) for any "jump" of size j_k at x_k), interpreting the Green's function G(x,ξ) as the response at x due to a concentrated source δ(x-ξ) at ξ, using Green's formula to write an expression for the solution of a BVP as an integral over ξ of the product of the source f(ξ) and the Green's function G(x,ξ), Maxwell's reciprocity, finding the Green's function by using the jump conditions at ξ and the fact that δ(x-ξ) is zero except when x=ξ, physical meaning of the Green's function for the differential operator L=d²/dx² with homogeneous Dirichlet BCs; nonhomogeneous BCs - using Green's formula to obtain the expression for the solution of the BVP using the same Green's function as for the case of homogeneous BCs [Sections 9.3.3-9.3.5].
  Green's function for Poisson's equation: Poisson's equation and Laplace's equation, Divergence Theorem (Gauss Law) in 3 and 2 dimensions, derivation of the Green's formula for the Laplacian, mutidimensional δ-functions, Green's functions in several dimensions, using Green's formula for the Laplacian to obtain an expression for the solution of the BVP as an integral of the product of the source function and the Green's function; Green's functions by eigenfunction expansions - derivation of an expression for the Green's function when λ=0 is not an eigenvalue of the differential operator [pages 416-419 of Section 9.5].
• Lecture 12 (Nov 15, 1:30-4:10 p.m., SPP 3100): Green's function for Poisson's equation (cont.): using Green's functions for problems with nonhomogeneous BCs - an example for the Poisson's equation, using Greene's formula to derive an expression taking into account the sources in the domain as well as the BCs, physical interpretation of the "boundary contribution" as the field due to an infinitely thin layer of dipols at the boundary; infinite-space Green's functions - deriving the Green's function G(x,ξ) of the Poisson's equation in R² (by solving an ODE for the radial part and using the Divergence Theorem for an infinitesimal sphere centered at ξ; obtaining Green's functions for bounded domains by using the infinite-space Green's functions by the method of images - examples of the Green's function of the Poisson's equation in the half-plane and in the first quadrant [Sections 9.5.6-8].
  
• Lecture 13 (Nov 29, 1:30-4:10 p.m., SPP 3100): Green's function for Poisson's equation (cont.): Green's functions for bounded domains by using the infinite-space Green's functions by the method of images - examples of the Green's function of the Poisson's equation in an infinite strip. Green's function of the Poisson's equation in a circle by using inversion transformation [Section 9.5.9].
  Green's function for the wave equation: definition of the Green's function, causality principle, time-translation property; using Greene's identity from Sturm-Liouville theory and Greene's formula from vector calculus to derive integral identities related to the wave operator; reciprocity (without derivation) [Sections 12.2.1-12.2.3].
• Lecture 14 (Dec 6, 12-2:40 p.m., SCC 4216): Green's function for the wave equation (cont.): using the Green's function to derive an expression for the solution of the heat equation; alternate differential equation for the Green's function; invinite-space Green's function for the one-dimensional wave equation, d'Alembert solution; infinite-space Green's function for the three-dimensional wave equation (without derivation), idea of deriving the infinite-space Green's function for the three-dimensional wave equation from the Green's function for the three-dimensional wave equation by using the method of descent [Sections 11.2.4-11.2.9].

Attendance: You are required to attend class on those days when an examination is being given; attendance during other class periods is also expected. 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).

Homework: It is absolutely essential to solve a large number of problems on a regular basis!
Homework will be assigned regularly and will be posted on the 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. Please write the problems in the same order in which they are given in the assignment. No late homework will be accepted!
You are allowed (and encouraged) to work in small groups. However, each of you will need to prepare individual solutions written in your own words - this is the only way to achieve real understanding!

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Useful links: the academic calendar, the class schedules.

Policy on W/I Grades : Until August 31, there is no record of a grade for dropped courses. From September 4 through September 28, you may withdraw and receive a "W" grade, no matter what scores you have so far achieved. If you drop the course after September 28, you will receive either a "W" or an "F" depending on your grade at time of your withdrawal. A withdrawal on or after October 29 would require a petition to the Dean (such petitions are not often granted). Avoidance of a low grade is not sufficient reason to obtain permission to withdraw after October 29. The grade of "I" is a special-purpose grade given when a specific task needs to be completed to finish the coursework, it is not intended to serve as a benign substitute for the grade of "F". Please check the dates in the academic calendar!

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Good to know:

• The greek_alphabet.
• Some useful notations.