MATH 5163 - Partial Differential Equations, Section 001 - Spring 2014
TR 12:00-1:15 a.m., 809 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: Introduction to Partial Differential Equations (MATH 4163) or permission of instructor.

Course catalog description: First order equations, Cauchy problem for higher order equations, second order equations with constant coefficients, linear hyperbolic equations. (Sp)

Tentative course content:

• PDEs arising in mechanics, electrodynamics, heat propagation, fluid dynamics.
• Laplace's equation, harmonic functions, Dirichlet's Principle, Gauss Mean Value Theorem and its inverse, Maximum Principle, regularity of harmonic functions, Liouville's Theorem.
• Classical Fourier transform (FT) and its inverse, solving PDEs by using FT. Distributions: definition, convergence, derivatives, extension of the FT to distributions, properties, examples.
• Fundamental solutions of Laplace's and Poisson's equations, Green's functions, method of images, inversion.
• Heat equation, solution using the FT, uniqueness, fundamental solutions, non-homogeneous problem, Duhamel's principle.
• Wave equation. Solution in 1D: D'Alembert's formula, uniqueness by the energy method. Solution in 3D: fundamental solution, Kirchhoff's formula, Hadamard's method of descent, Huygens' principle. Fundamental solutions, non-homogeneous problems.
• Banach spaces, Hilbert spaces, weak derivatives, Sobolev spaces. Existence of a weak solution of the Poisson's equation. Embedding theorems for Sobolev spaces.
• Eigenfunction expansions, existence and properties of eigenvalues of certain types of operators, variational principle for the principal eigenvalue.
• (If time permits) First-order PDEs: linear, quasilinear (Burgers' equation), and nonlinear (Hamilton-Jacobi equation).

Text: The main textbook, freely available online for OU students, will be

S. Salsa. Partial Differential Equations in Action. From modelling to theory. Springer, 2008.


We will also use parts of the following books, freely available online for OU students:


S. Salsa, F. Vegni, A. Zaretti, P. Zunino. A Primer on PDEs. Models, methods, simulations. Springer, 2013 (which significantly overlaps with the main textbook);

A. C. King, J. Billingham, S. R. Otto. Differential Equations. Linear, nonlinear, ordinary, partial. Cambridge University Press, 2003.

Homework:

• Homework 1, due Thu, Jan 23.
• Homework 2, due Thu, Jan 30.
• Homework 3, due Thu, Feb 6.
• Homework 4, due Thu, Feb 13.
• Homework 5, due Fri, Feb 21, at 4 p.m. in 802 PHSC.
• Homework 6, due Fri, Feb 28, at 5 p.m. in 802 PHSC.
• Homework 7, due Fri, Mar 6, at 5 p.m. in 802 PHSC.
• Homework 8, due Wed, Apr 9, at 5 p.m. in 802 PHSC.
• Homework 9, due Fri, Apr 18, at 5 p.m. in 802 PHSC.
• Homework 10, due Fri, May 2, at 5 p.m. in 802 PHSC.

Content of the lectures:

• Lecture 1 (Tue, Jan 14): Basic partial differential equations: classification of partial differential equations (PDEs), arbitrariness in the general solution of a PDE of order k for a function of m variables; a detailed derivation of the heat equation cρu_t=∇⋅(k∇u)+Ψ(x,t) (where Ψ(x,t) is the volume density of the power of the heat sources inside the domain), or, for constant k, u_t=α²Δu+ψ(x,t); derivation of the the boundary conditions for the heat equation: Dirichlet condition (the temperature is controlled at the boundary), Neumann condition (the heat flux is controlled at the boundary), Robin condition (convective heat exchange with the surroundings) [Sec. 1.2]
• Lecture 2 (Thu, Jan 16): Other important partial differential equations: diffusion equation, wave equation, Poisson and Laplace equations, Schrodinger equation, Maxwell's equations, Navier-Stokes equations.
  Domains and Green's identities: C^k-domains; Divergence Theorem, Green's identities.
  Properties of harmonic functions: definition of harmonic functions; uniqueness of the solution of Poisson equation with Dirichlet boundary conditions; energy of a vibrating membrane, energy of a membrane in equilibrium in the Earth's gravity field.
• Lecture 3 (Tue, Jan 21): Properties of harmonic functions (cont.): Dirichlet's principle: harmonic functions minimize the energy functional, and the minimizer of the energy functional is a harmonic function; average values over a ball and a sphere: definitions; Mean Value Theorem for harmonic functions; Maximum Principles: statement of the weak and the strong maximum principles (the proof will be given in Lecture 4); if "max" is replaced by "min", then we obtain a "minimum principle" (because if u is harmonic, then −u is also harmonic, and the maxima of u are minima of −u and vice versa); physical interpretation: the maximum or minimum of a steady-state temperature distribution in absence of source in the domain cannot occur inside the domain, but only on the boundary.
• Lecture 4 (Thu, Jan 23): Properties of harmonic functions (cont.): Maximum Principles: proof of the strong maximum principle; uniqueness of the solution of the Poisson equation (proof by using the Maximum Principle); explicit solutions of the Poisson equation in a ball in 2 and 3 dimensions, regularity of these solutions, regularity of harmonic functions in an arbitrary domain; derivative estimate for harmonic functions; Liouvile Theorem.
• Lecture 5 (Tue, Jan 28): Fourier transform − an introduction: L^p spaces; integral transform, integral kernel; Fourier transform (FT) − definition, proof that it maps L²(R) to L^∞(R), examples, elementary properties; convolution, the Convolution Theorem; Inverse FT, the Inversion Theorem, an example of using the Inversion Theorem; Dirac δ-function δ_a − a heuristic definition, FT of δ_a, using the Inversion Theorem to obtain an integral representation of δ; Parseval's Theorem (Plancherel's Theorem) − proof of the theorem and remarks on its physical meaning.
  Using FT to solve PDEs: using FT to solve the heat equation on R.
• Lecture 6 (Thu, Jan 30): Using FT to solve PDEs (cont.): using FT to solve Laplace's equation (i.e., the steady-state heat equation) in the half-plane x∈R, y>0.
  Test functions and mollifiers: support of a function, examples, compactly supported functions, the "standard" compactly supported function η supported on the closed unit ball in Rⁿ, and its rescaled version η_ε supported on the closure of B(0,ε); Young's Theorem; mollifiers, using a mollifier to regularize a "rough" function ƒ by convolving them with η_ε; properties of the convolution ƒ∗η_ε (Lemma 7.1); locally integrable functions [pages 369-372 of Sec. 7.2]
• Lecture 7 (Tue, Feb 4): Test functions and mollifiers (cont.): discussion of some properties of the convolution ƒ∗η_ε (Remarks 7.1 and 7.2, Example 7.2), density of the space of smooth compactly supported functions on Ω in L^p(Ω) for any p∈[1,∞) (Theorem 7.1, without proof) [pages 178, 179 of Sec. 3.2]
  Distributions: multiindex notations in Rⁿ: α=(α₁,...,α_n) where α_j∈{0,1,2,...}, length |α| of the multiindex defined as the sum of its components α_j, differential operator D^α=∂^|α|/∂x₁^α₁...∂x_n^α_n, product of components x^α=(x₁)^α₁...(x_n)^α_n; the space D(Ω) of test functions as the set of compactly supported smooth functions with an appropriately defined meaning of convergence; examples; the set D'(Ω) of distributions as continuous linear functionals on D(Ω), examples of distributions coming from L² functions or from locally integrable functions; linear combinations of distributions; continuous embeddings of L^p(Ω) into the space of locally integrable functions on Ω and of the space of locally integrable functions on Ω into the space of distributions D'(Ω), proof of the continuity of the second embedding [pages 373-375 of Sec. 7.3]
• Lecture 8 (Thu, Feb 7): Distributions (cont.): Holder inequality; example: the locally integrable function 1(x)≡1 defines a distribution; example: Dirac comb [pages 375-376 of Sec. 7.3]
  Calculus: motivation of the definition of a derivative of a distribution, definition of a derivative of a distribution F∈D'(Ω) for Ω⊆Rⁿ: ⟨D^αF,φ⟩:=(−1)^|α|⟨F,D^αφ⟩, where φ∈D(Ω) is an arbitrary test function; examples; proof that the derivative of the Heaviside function H is the Dirac function δ (and H_a'=δ_a) [pages 377-379 of Sec. 7.4]
• Lecture 9 (Tue, Feb 11): Calculus (cont.): continuity of differentiation in D'(Ω) (Proposition 7.3); vector-valued test functions φ∈D(Ω,Rⁿ) and vector valued distributions F∈D'(Ω,Rⁿ); pairing between elements of D'(Ω,Rⁿ) and elements of D(Ω,Rⁿ); gradient of ƒ∈D'(Ω) (with Ω⊆Rⁿ), ⟨∇ƒ,φ⟩=−⟨ƒ,∇⋅φ⟩; divergence of F∈D'(Ω,Rⁿ), ⟨∇⋅F,φ⟩=−⟨F,∇φ⟩; Laplacian of ƒ∈D'(Ω), ⟨Δƒ,φ⟩=⟨ƒ,Δφ⟩; derivation of the fundamental solution of the Laplacian operator in R³: −Δu=δ [pages 379-382 of Sec. 7.4]
  Multiplication, composition, division, convolution: impossibility of defining multiplication of distirbutions; multiplying a distribution by a C^∞ function [pages 382-383 of Sec. 7.5]
• Lecture 10 (Thu, Feb 13): Multiplication, composition, division, convolution (cont.): more multiindex notations: α!=α₁!...α_n!, β≤α if β_j≤α_j for all j=1,...,n, α−β=(α₁−β₁,...,α_n−β_n) defined for β≤α, α "choose" β is defined as α!/[β!(α−β)!]; Leibniz rule for the derivative of a product of a smooth function and a distribution, examples; problems with defining a composition of a distribution with a smooth function, defining a composition of a distribution and a 1-to-1 smooth function with a smooth inverse, example: delta function of a function of x with simple zeros; dividung a distribution by a smooth function that never vanishes, dividing a distribution by a smooth function with zeros (possibly of higher multiplicities) (Proposition 7.4), principal value of an integral, distribution P.v.(1/x)∈D'(R), the general solution of the equation xG=1 is G=P.v.(1/x)+Cδ (where C is an arbitrary constant); problems with defining the convolution of two distributions, support of a distribution, distributions with compact support, definition of the convolution of a distribution with compact support and a smooth function, the convolution of a distribution with compact support and a C^∞ function is a C^∞ function (Proposition 7.5), the delta function is the identity with respect to convolution [pages 383-388 of Sec. 7.5]
• Lecture 11 (Tue, Feb 18): Tempered distributions: need for using another space of test functions in order to define Fourier transform (FT) for the corresponding distributions (footnote on page 388); space S(Rⁿ) of functions rapidly vanishing at ∞ (rapidly decreasing), examples; definition onf convergence of a sequence in S(Rⁿ); D(Rⁿ)⊂S(Rⁿ); if (v_k)⊂D(Rⁿ) converges to v in D(Rⁿ), then (v_k)⊂S(Rⁿ) and v_k→v in S(Rⁿ); space S'(Rⁿ) of tempered distributions (giving the definition in stages); D(Rⁿ)⊂S(Rⁿ)⊂L^p(Rⁿ)⊂S'(Rⁿ)⊂D'(Rⁿ); completeness of S'(Rⁿ); definition of convergence of a sequence of tempered distributions; example: the Dirac comb as an element of S'(Rⁿ); the convolution of a tempered distribution and a rapidly decreasing function is a tempered distribution which coincides with a C^∞ function [Sec. 7.6.1]
  FT in S'(Rⁿ): direct and inverse FTs on L^p(Rⁿ) functions; continuity of the direct and the inverse FT; weak Parseval identity; definition of a FT of a tempered distribution; properties of the FT on S'(Rⁿ) (please read them from the book); examples: FT of 1, FT of the coordinate function x_j [Sec. 7.6.2]
• Lecture 12 (Thu, Feb 20): FT on L²(Rⁿ): u∈L²(Rⁿ) if and only if its FT is in L²(Rⁿ); strong Parseval identity (i.e., the FT is an isometry in L²(Rⁿ), up to a factor); an example of application of the strong Parseval identity [Sec. 7.6.3]
  Laplace and Poisson equations: fundamental solution of a linear differential operator; derivation of the fundamental solution Φ(x) of the Poisson's equation in R³ by using FT [recall that we verified by a direct integration that Φ(x)=1/(4π|x|) is a fundamental solution of Poisson's equation in R³ in Lecture 9]; applying a Green formula to the case when one of the functions is Φ(x)=1/(4π|x|) [Sec. 3.5.1]
• Lecture 13 (Tue, Feb 25): Laplace and Poisson equations (cont.): definition of the Green function of the operator −Δ in a domain Ω⊂Rⁿ, general properties: positivity, symmetry; derivation of the Green function of −Δ in the upper half-space {x∈R³:x₃>0} - the method of electrostatic images; derivation of the Green function of −Δ in a ball B_R(0)⊂R³ - an image charge located at the inversion of x∈B_R(0); Green representation formula for the solution of the Dirichlet BVP −Δu(x)=ƒ(x) for x∈Ω⊂Rⁿ, u(x)=g(x) for x∈∂Ω, single and double layer potentials, Poisson kernel; solution of the Dirichlet BVP in the upper half-space of R³ [pages 133-136 of Sec. 3.5.2 and 3.5.3]
• Lecture 14 (Thu, Feb 27): Laplace and Poisson equations (cont.): Poisson kernel and Poisson formula for the solution of the Dirichlet BVP for the Poisson equation in a ball of radius R in R³; the Neumann BVP for the Poisson equation in bounded domain - compatibility condition between the equation and the boundary condition, and how to take care of this condition in finding Neumann function (i.e., the Green function in this case) [pages 137, 138 of Sec. 3.5.3 and 3.5.4]
• Lecture 15 (Tue, Mar 4): The heat/diffusion equation: discussion of the physical meaning of the heat and diffusion equations and the Dirichlet, Neumann, and Robin boundary conditions; the linearity of the homogeneous equation u_t−α²Δu=0 implies the Principle of Superposition of solutions [pages 13-18 of Sec. 2.1]
  Uniqueness: integral method of proving the uniqueness of the solution of the heat equation for the three boundary conditions [pages 30, 31 of Sec. 2.2]
  The fundamental solution: time reversal in the heat equation; invariance of the heat equation with respect of space and time translations; parabolic dilations - constructing a dimensionless expression x/(αt^1/2) that does not change under the transformations x→ax, t→a²t with respect to which the homogeneous heat equation is invariant [pages 34-35 of Sec. 2.3]
• Lecture 16 (Thu, Mar 6): The fundamental solution (cont.): dilation and conservation of mass - if we perform a dilation x→ax, t→a²t, u→cu, then conservation of mass (for the diffusion equation) implies that c=aⁿ (where n is the dimension of the space Rⁿ); looking for the fundamental solution of the heat equation in R×(0,∞) in the form Γ(x,t)=A/(αt^1/2)v(ζ), where A is a dimensionless constant, ζ=x/(αt^1/2), and v is an unknown function of one variable - this substitution leads to an ODE for v(ζ) which together with some appropriate conditions leads to the fundamental solution Γ(x,t)=1/(2α(πt)^1/2)exp{−x²/(4α²t)} in R×(0,∞) (this calculation was done in Problem 1 of Homework 5); obtaining the fundamental solution of the heat equation in R×(0,∞) by Fourier transform with respect to the spatial variable x (done in Lecture 5); generalization - looking for the fundamental solution of the heat equation in Rⁿ×(0,∞) in the form Γ(x,t)=A/(αt^1/2)v(ζ), where A is a dimensionless constant, ζ=|x|/(αt^1/2), and v is an unknown function of one variable - this substitution leads to an ODE for v(ζ) which yields the fundamental solution Γ(x,t)=1/(4α²πt)^n/2exp{−|x|²/(4α²t)} in Rⁿ×(0,∞) [pages 35-38, 42, 43 of Sec. 2.3]
• Lecture 17 (Tue, Mar 11): Exam 1
• Lecture 18 (Thu, Mar 13): The global Cauchy problem: recall that the fundamental solution on (x,t)∈Rⁿ×(0,∞),
  
  satisfies the initial-value problem on (x,t)∈Rⁿ×(0,∞)
  
  and notice that it also satisfies on (x,t)∈Rⁿ×(0,∞)
  
  because of translational invariance, for given y∈Rⁿ and s∈(0,∞), the solution on (x,t)∈Rⁿ×(s,∞) of the initial value problem
  
  or, equivalently,
  
  is
  
  by the Principle of Superposition, the solution of the initial-value problem on (x,t)∈Rⁿ×(0,∞)
  
  is ("Duhamel's principle")
  
  using physics arguments to solve PDEs: advection-diffusion-reaction equation describing the evolution of the concentration u(x,t) of a pollutant in a channel in which the water is moving in positive direction with velocity c and the pollutant is decaying exponentially in time:
  
  if y=x−ct is the coordinate in the coordinate system "comoving" with the water, and u(x,t)=v(x−ct,t), where v(y,t) is the concentration of the pollutant in the comoving coordinate system, then v(y,t) satisfies
  
  since if v(y,t) satisfies does not depend on y, then u will decay exponentially with time, we set v(y,t)=e^−γtw(y,t), and the new unknown function w(y,t) satisfies the diffusion/heat equation
  
  so by Duhamel's principle
  
  [pages 68, 69, 71-73 of Sec. 2.8; pages 54-57 of Sec. 2.5, Problem 1 of Homework 3]
• Lecture 19 (Tue, Mar 25): Waves and vibrations - general concepts: important particular cases: travelling waves, harmonic waves; plane waves, spherical waves; important concepts: amplitude, wave number, wavelength, angular frequency, period, linear frequency, phase speed, (group speed) [pages 221-223 of Sec. 5.1]
  Transversal waves in a string: the equation describing transversal waves in a string (without derivation); energy in the waves in a string, conservation of energy [pages 226, 228, 229 of Sec. 5.2]
  The one-dimensional wave equation: equation, initial conditions, and different boundary conditions (Dirichlet, Neumann, Robin) for one-dimensional waves; principle of superposition for the homogeneous wave equation [pages 229-231 of Sec. 5.3]
  The d'Alembert formula: changing the variables (x,t) to η=x−ct and ζ=x+ct; general solution of the 1-D wave equation on R: u(x,t)=φ(x−ct)+ψ(x+ct); relating the functions φ and ψ with the initial conditions - d'Alembert formula; range of influence, domain of dependence, speed of propagation of the signal [pages 236-238 of Sec. 5.4]
• Lecture 20 (Thu, Mar 27): The d'Alembert formula (cont.): remark about factorizing the wave operator: ∂_tt−c²∂_xx=(∂_t+c∂_x)(∂_t−c∂_x), uni- and bidirectional wave equations (unidirectional of first order, bidirectional of second order in time); demonstrating the well-posedness of the 1-D wave equation on a finite time interval from the d'Alembert formula for the solution; the fundamental solution K(x,y,t) of the 1-D wave equation - a derivation from the d'Alembert formula; derivation of the d'Alembert formula by using the expression for K(x,y,t); non-homogeneous equation - Duhamel's method, physical interpretation [pages 238, 244-247 of Sec. 5.4]
• Lecture 21 (Tue, Apr 1): The d'Alembert formula (cont.): formal derivation of the expression for the solution of the nonhomogeneous 1-D wave equation with nonzero initial conditions by using the fundamental solution K(x,y,t).
  The Cauchy problem: fundamental solution K(x,t) of the 3-D wave equation - derivation by using Fourier transform in the spatial variable, delta-function concentrated on a sphere and its Fourier transform; using the fundamental solution to derive solution for zero initial displacement u(x,0) and non-zero initial velocity u_t(x,0); using the fundamental solution to derive solution for non-zero initial displacement u(x,0) and zero initial velocity u_t(x,0); Kirchhoff formula; Huygens principle [pages 274, 275, 277-279 of Sec. 5.9]
• Lecture 22 (Thu, Apr 3): The Cauchy problem (cont.): nonhomogeneous problem, Duhamel's principle, retardation effects due to the finite speed of propagation of the waves; Hadamard's method of descent for deriving the solution of the wave equation in two spatial dimensions by using the solution in three spatial dimension given by Kirchhoff's formula, fundamental solution of the wave equation in two spatial dimensions, Huygens principle does not hold in two spatial dimensions; physical interpretation of Hadamard's method of descent - the fundamental solution of the wave equation in two spatial dimensions can be obtained as the solution of the 3-dimensional wave equation with a source term spread along the x₃-axis, i.e., equal to δ(x₁)δ(x₂)δ(t) - therefore, one can obtain the fundamental solution K⁽²⁾(x₁,x₂,t) by integrating the fundamental solution K⁽³⁾(x₁,x₂,x₃,t) with respect to x₃ (from −∞ to ∞) [pages 279-282 of Sec. 5.9]
• Lecture 23 (Tue, Apr 8): Norms and Banach spaces: norm, normed linear space; distance, metric space; convergent sequence in a metric (or normed) space, Cauchy sequence; convergent implies Cauchy; complete normed space, Banach space; continuous operator; continuity of the norm; space C(A) of continuous functions (for A⊂Rⁿ compact), sup (max) norm on C(A), completeness of C(A) with the sup norm (i.e., C(A) with the sup norm is a Banach space); L²(A) norm on C(A), this normed space is not complete; C^k(A) wiht the maximum norm of order k is a Banach space; sets of measure zero, functions equal almost everywhere (a.e.) in an open set Ω, equivalence classes of functions that differen only on subsets of measure zero, essentially bounded functions, essential supremum, spaces L^p(A) for p∈[1,∞) and L^∞(A); Holder inequality [pages 307-311 of Sec. 6.2]
  Hilbert spaces: inner product, inner product linear space; norm induced by an inner product; Schwarz inequlity; Hilbert space; isomorphism between two Hilbert spaces; example: Rⁿ with the Euclidean inner product or with a more general inner product defined by a symmetric positive-definite matrix; example: L²(Ω) [pages 311-314 of Sec. 6.3]
• Lecture 24 (Thu Apr 10): Hilbert spaces (cont.): example: space l² of sequences; elementary facts about Fourier series; example: a Sobolev space of periodic functions [pages 314, 315 of Sec. 6.3; pages 531-533 of Appendix A]
  Projections and bases: basis of a linear space; dimension of a linear space; orthogonality of vectors in an inner product linear space; orthogonal and orthonormal bases in an inner product linear space; finding the components of a vector in an orthogonal or an orthonormal basis; projections in Rⁿ; orthogonal direct sum Rⁿ=V⊕V^⊥ [pages 316-318 of Sec. 6.4]
  Sobolev spaces: definitions of C^r₀(Ω) (i.e., C^r functions whose support is a compact set contained completely in the interior of Ω), test functions D(Ω), distributions D'(Ω), L¹_loc(Ω) for Ω⊂Rⁿ; weak derivatives - derivatives in the sense of distribution that are in L¹_loc(Ω).
• Lecture 25 (Tue, Apr 15): A digression on the wave equation: waves in a guitar string, flageolets - supressing some harmonics by touching the string at certain positions, see the Wikipedia article Harmonic.
  Sobolev spaces (cont.): examples of weak derivatives; Sobolev spaces W^k,p(Ω), norm in W^k,p(Ω), W^k,p(Ω) is a separable Banach space; Sobolev spaces H^k(Ω)=W^k,2(Ω), H^k(Ω) is a Hilbert space, inner product and norm in H^k(Ω); the space H¹(Ω); functions from H¹(Ω) have finite energy; digression: separability, sketch of the proof of the separability of the space C(Ω) for compact Ω⊂Rⁿ based on Weierstrass Approximation Theorem, every Hilbert space admints an orthonormal basis, separability of the spaces L^p(Ω) for p∈[1,∞); Theorem: H¹(Ω) is a separable Hilbert space, continuously embedded in L²(Ω), and the gradient operator is continuous from H¹(Ω) to L²(Ω) [pages 320, 321 of Sec. 6.4; pages 396, 397 of Sec. 7.7]
• Lecture 26 (Thu, Apr 17): Sobolev spaces (cont.): the function u(x)=(−log(|x|))^a belongs to H¹(Ω) for Ω={x∈|R²:0<|x|<1} and a<1/2; characterization of H¹([a,b]); definition of H¹₀(Ω) as the closure of D(Ω) in H¹(Ω); Poincare inequality in H¹₀(Ω) for bounded domain Ω; inner product (u,v)₁:=(∇u,∇v)₀ and norm ||u||₁:=(u,v)₁^1/2 in H¹₀(Ω) (equivalent to the norm ||u||_1,2:=(u,v)_1,2^1/2 in H¹(Ω)) [pages 397-400 of Sec. 7.7]
• Lecture 27 (Tue, Apr 22): Sobolev spaces (cont.): the dual H⁻¹(Ω) of H¹₀(Ω); the elements of H⁻¹(Ω) are distributions from D'(Ω); characterization of H⁻¹(Ω) as the distributoins of the form F=g+div(h) for g∈L²(Ω) and h∈L²(Ω,Rⁿ); δ₀∈H⁻¹((−a,a)) but δ₀∉H⁻¹(Ω) where Ω is a subset of Rⁿ for n≥2; example: ∇χ_Ω; definition of the spaces H^m(Ω) for m>1 [pages 401-403 of Sec. 7.7]
  Classical, strong, and weak solutions of Poisson equations with homogeneous Dirichlet BCs: Classical solution: given ƒ∈C(cl(Ω)), find u∈C²(Ω)∩C(cl(Ω)) such that −Δu(x)=ƒ(x) classically in Ω and u(x)=0 for x∈∂Ω; strong solution: given ƒ∈L²(Ω), find u∈H²(Ω)∩H¹₀(Ω) such that −Δu=ƒ with derivatives understood in weak sense.
• Lecture 28 (Thu, Apr 24): Classical, strong, and weak solutions of Poisson equations with homogeneous Dirichlet BCs: using Green's formula to derive the weak formulation of the Poisson problem: given ƒ∈H⁻¹(Ω), find u∈H¹₀(Ω) such that (u,v)₁=⟨ƒ,v⟩ for any v∈H⁻¹(Ω), where ⟨ƒ,v⟩ is the dual pairing and the derivatives are understood in weak sense; using Riesz representation theorem to prove existence and uniqueness of the weak solution of Poisson's equation; a generalization: weak formulation of the Poisson's problem −∇⋅[a(x)∇u(x)]=ƒ(x) classically in Ω and u(x)=0 for x∈∂Ω, where 0<μ≤a(x)≤ν; proof of existence and uniqueness by defining an inner product equivalent to the standard inner product in H¹₀(Ω).
• Lecture 29 (Tue, Apr 29): Classical, strong, and weak solutions of Poisson equations with homogeneous Dirichlet BCs (cont.): another generalization: anisotropic heat conductivity - the heat flux density j has components j_i(x,t)=a_ik(x) ∂u(x,t)/∂x_k; abstract variational problems.
  Abstract variational problems: bilinear forms (sesquilinear in the complex case), examples; abstract variational problem: for a bilinear form a on a Hilbert V and a linear functional F∈V^*, find u∈V such that a(u,v)=⟨F,v⟩_* for all v∈V; Lax-Milgram Theorem (Theorem 6.5), stability estimate [pages 334-336 of Sec. 6.6]
• Lecture 30 (Thu, May 1): Abstract variational problems (cont.): for a symmetric bilinear form, the abstract variational problem is equivalent to the problem of minimizing the quadratic functional E(v)=(1/2)a(v,v)-⟨F,v⟩_* (Theorem 6.7); first variation of a linear functional, Euler equation; filtration of nested supspace of a linear space: …⊂V_k⊂V_k+1⊂… projections; projected prolem, stiffness matrix, Galerkin method [pages 339-342 of Sec. 6.6]
• Final exam: Friday, May 9, 1:30-3:30 p.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. 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).

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 February 21, graduate students can withdraw from the course with an automatic "W" (for undergraduate students this period is January 28 to March 28). After this 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.