Cannon John Rozier. The one-dimensional heat equation

Addison-Wesley Publishing Company, Inc., 1984. — (Encyclopedia of mathematics and its applications; v. 23)Editor’s Statement
Foreword by Felix E. BrowderPreliminaries
Some Inequalities
Sequences and Series of Continuous
Functions, the Ascoli-Arzela Theorem, and
the Weierstrass Approximation Theorem
The Lebesgue Dominated-Convergence Theorem,
Leibniz’s Rule, and Fubini’s Theorem
Complex Analysis
NotesDerivation of the Heat Equation
Equations Reducible to the Heat Equation
Elementary Solutions
Methods of Generating Solutions from Solutions
Basic Definitions
The Weak Maximum Principle and Some Applications
Exercises
NotesThe Cauchy ProblemThe Holmgren Function Classes
An Existence Theorem
An Example
Continuous Dependence upon the Data
Variable Data-Bearing Curves: Existence: An Exercise
NotesThe Initial-Value ProblemThe Fundamental Solution K(x,t)
Convergence of the Convolution
M(X,Z) = /-OO/C(X-<,0/«)^
Limf|Ou(x, t)
An Existence Theorem
Uniqueness
Nonuniqueness
Exercises
The Initial-Boundary-Value Problem for the Quarter Plane with Temperature-Boundary SpecificationProperties of the Convolution
w(x, t) = -2Ц(дК/дх)(х, t - T)g(r)^T
An Existence Theorem
Uniqueness and Nonuniqueness
Exercises
The Initial-Boundary-Value Problem for the Quarter Plane with Heat-Flux-Boundary Specification
Existence
Uniqueness
Nonuniqueness
Exercises
The Initial-Boundary-Value Problem for the Semi-Infinite Strip with Temperature-Boundary Specification and Heat-Flux-Boundary SpecificationSome Properties of в(х, t) = K(x + 2m, t)
An Existence and Uniqueness Theorem
Specification of Heat Flux on the Boundary
Reduction to a System of Integral Equations
Notes on Chapters
References for Chapters
The Reduction of Some Initial-Boundary-Value Problems for the Semi-Infinite Strip, to Integral Equations: Some Exercises
The Boundary Condition u(0, r) with MX(1, t)
The Boundary Condition мл(0, z)+ a(z)w(0, t) with w(l, t)
The Boundary Condition wx(0, t) = F(t, M(0, /)) with MX(1, t)
The Boundary Condition M,(0, ?)+ a(t)ux(0, t) + J8(Z)M(0, t) with w(l, t)
The Boundary Condition /05(г)м(х, t) dx with M(1, t)
NotesIntegral Equations
Volterra Integral Equations of the First Kind and the Abel Integral Equation
Volterra Integral Equations of the Second Kind
Continuous Dependence of the Solutions of Integral Equations upon the Data
A priori Bounds on Solutions of Integral Equations
Exercises
NotesSolutions of Boundary-Value Problems for All Times and Periodic Solutions
The Infinite Strip with Temperature-Boundary Conditions
The Infinite Strip with Flux-Boundary Conditions
The Infinite Strip with Temperature- and
Flux-Boundary Conditions: Some Exercises
Notes ‘Analyticity of Solutions Ill
Introduction Ill
The Analyticity of the Solution of the Initial-Value Problem in the Space Variable
The Analyticity of the Solution of the Initial-Value Problem in the Time Variable
The Analyticity of the Solution of the Initial-Boundary-Value Problem for the Quarter Plane in the Space Variable
The Analyticity of the Solution of the Initial-Boundary-Value Problem for the Semi-Infinite Strip in the Space Variable
Some Problems in Estimating the Modulus of
Analytic Functions
Exercises
NotesContinuous Dependence upon the Data for Some State-Estimation ProblemsAn Interior Temperature Measurement in a Finite Conductor with One Unknown Boundary Condition
An Interior Temperature Measurement on a Characteristic in a Finite Conductor with No Initial Data
The Cauchy Problem: The Specification of Temperature and Heat Flux on a Boundary
Specification of an Interior Temperature and Flux along with Boundary Temperatures: An Exercise
The Specification of an Interior Temperature in a Finite Conductor along with the Temperature on the Boundaries: An Exercise
Interior Temperature Specification on a Characteristic for an Infinite Conductor with Unknown
Initial Temperature
NotesSome Numerical Methods for Some State-Estimation
Problems
Some Numerical Results for the Solution of the Heat Equation Backwards in Time
A Numerical Method for the Cauchy Problem
A Least-Squares Method
An Exercise
NotesDetermination of an Unknown Time-Dependent Diffusivity a (t) from Overspecified DataSome Problems with Explicit Solutions
Some Problems That Reduce to the Solution of
F(Ha(y)dy)-h(t)
A Nonlinear Integral Equation for a(t)
Exercises
NotesInitial- and/or Boundary-Value Problems for General Regions with Holder Continuous Boundaries
Introduction—Function Spaces
The Single-Layer Potential н^(х, ty s) = &K(x - S(T), t - т)<р(т) dr
The Initial-Boundary-Value Problem with
Temperature-Boundary Specification
The Existence of ux at the Boundary for the Solution of the Initial-Boundary-Value Problem with Temperature-Boundary Specification
Exercises
NotesSome Properties of Solutions in General Domains
Uniform Convergence of Solutions
Analyticity in the Spatial Variable
The Strong Maximum Principle
Behavior of ux at a Boundary Point Where It Assumes Its Maximum or Minimum Value
Exercises
NotesThe Solution in a General Region with Temperature-Boundary Specification: The Method of Perron-PoincareSuperparabolic and Subparabolic Functions
Existence of и Satisfying the Heat Equation in DT
Boundary Behavior of и
Exercises
NotesThe One-Phase Stefan Problem with Temperature-Boundary SpecificationMonotone Dependence upon the Data and Uniqueness of the Solution
Existence
Some Special Solutions
Asymptotic Behavior of the Free Boundary at t = + oo
Asymptotic Behavior of the Free Boundary at t =
Continuous Dependence upon the Data
Regularity of the Free Boundary
Exercises
The One-Phase Stefan Problem with Flux-Boundary Specification: Some ExercisesMonotone Dependence upon the Data and Uniqueness of the Solution
Existence
Asymptotic Behavior of the Free Boundary at t = + oo
Continuous Dependence of the Free Boundary upon the Data
Notes on Chapters 17 and 18*
References for Chapters 17 and 18*
The Inhomogeneous Heat Equation ut = uxx + /(x, t)The Volume Potential z(x, 0 = fit-^x -It- r)f(& r)d^dr
Solutions for Some Initial-Boundary-Value Problems for ut = uxx + f(x,t)
Initial-Boundary-Value Problems for General Domains
Exercises
NotesAn Application of the Inhomogeneous Heat Equation: The Equation ut = uxx + F(x,t,u,ux)The Initial-Value Problem
Some Initial-Boundary-Value Problems for
Elementary Domains
Exercises
NotesSome References to the Literature on <&(u) = uxx - ut
Classical: 1800-1950
Semi-Classical: 1950-1973
Properties of Solutions
The Maximum Principle, a priori estimates, parabolic inequalities, and nonexistence of solutions
Uniqueness and representation theorems
Transforms, polynomials, and analyticity of solutions
Potentials and Green’s functions
Behavior of solutions at the boundary and regularity of boundary points
Some roots of reaction diffusion theory
Periodic solutions
Initial-Boundary-Value Problems
Stefan and Free-Boundary-Value Problems
Not-Well-Posed and Inverse Problems
Recent
Properties of Solutions
A priori estimates and maximum principles
Initial-value problem and uniqueness
Representations, transformations, special functions, and Runge approximations
Potentials, Green’s functions, capacity, boundary behavior, and some applications
Asymptotic behavior of solutions
Periodic solutions
Applications to statistics
Initial-Boundary-Value Problems
Stefan and Free-Boundary-Value Problems
Control of Various Heat Problems
Not-Well-Posed and Inverse Problems
More Recent: 1981-1982
Properties of Solutions
A priori estimates, maximum principle, comparison results, means
Green’s functions, potential theory, transforms, elementary solutions
Continuity and differentiability of solutions
Boundary behavior of solutions
Periodic solutions
Reaction-diffusion problems
Bifurcation, nonuniqueness, and chaos in reaction-diffusion problems
Stability of solutions
Asymptotic behavior
Applications to probability and statistics
Initial-Boundary-Value Problems
Stefan-Free Boundary
Control of Various Heat Problems
Not-Well-Posed and Inverse Problems
Symbol Index
Subject Index