Mathematics Research Developments
Partial Differential EquationsTheory, Numerical Methods and Ill-Posed Problems
Contents
Preface
Acknowledgments
Chapter 1Introduction
Part I: Representation formulas for solutions
Part II:Weak/generalized solutions of PDEs with variable coefficients
Part III. Beginning of the theory of ill-posed problems
Part IV. FDM/FEM for PDEs
1.1 Terminology and Notation
Topological Notation
Functional Notation
Differential Notation
1.2 Examples of Linear PDEs
1.2.1. Scalar PDEs
1. Linear Transport Equation
2. Liouville’s Equation
3. Laplace’s Equation
4. Helmholtz’s Equation
5. Heat (or Diffusion) Equation
6. Kolmogorov’s Equation
7. Fokker-Planck Equation
8. Wave Equation
9. Wave Equation with Variable Coefficients
10. Telegrapher Equation
11. Biharmonic Equation
12. Beam Equation
1.2.2. System of PDEs
1. Schr¨odinger Equation
2. Maxwell’s Equations
3. Equilibrium Equations of Linear Elasticity
4. Evolution Equations of Linear Elasticity
1.3 Strategies for Studying PDEs
1.3.1. Well Posed Problems: Classical Solutions
1.3.2. Weak Solutions and Regularity
1.3.3. Typical Difficulties
1.4 Generalized Functions
Chapter 2Analytic Approaches toLinear PDEs
2.1 Transport Equation with Constant Coefficients
2.1.1. Initial Value Problem
2.1.2. Non-Homogeneous Problem
2.1.3. Initial Boundary Value Problem
2.2 Laplace’s Equation
2.2.1. Fundamental Solution
2.2.2. Mean Value Property
2.2.3. Applications
a. Strong Maximum Principle; Uniqueness
b. Regularity
c. Estimates for Harmonic Functions
d. Liouville’s Theorem
e. Harnack’s Inequality
f. A Probabilistic Representation Formula for Harmonic Functions
2.2.4. Green’s Function for Laplace’s Equation
a. Green’s Function for General Domain
b. Green’s Function for the Unit Ball
c. A Representation Formula for Harmonic Function on Balls:Poisson’s Integral
2.2.5. Energy Methods
a. Uniqueness
b. Dirichlet’s Principle
2.3 Heat Equation
2.3.1. Fundamental Solution: Duhamel Principle
2.3.2. The Mean Value Property
2.3.3. Applications
a. Strong Maximum Principle; Uniqueness
b. Regularity
c. Estimates for Solutions of Heat Equation
2.4 Wave Equation
2.4.1. Solution by Spherical Means
a. Solution for n = 1; d’Alembert Formula
b. Spherical Means
c. Solution for Odd n
d. Solution for Even n; Method of Descent; Huygens Phenomenon
2.4.2. Solution of Non-HomogeneousWave Equation:Duhamel Principle
2.4.3. Energy Methods
a. Uniqueness
b. Domain of Dependence
Chapter 3Transformation Approachesto Certain PDEs
3.1 Separation of Variables
3.2 Similarity Solutions
3.2.1. Travelling and planeWaves, Solutions
a. Exponential Solutions
(i) Heat Equation
(ii)Wave equation
(iii) Dispersion Equation
b. Solitons
3.3 Fourier and Laplace Transforms
3.3.1. Fourier Transform
3.3.2. Laplace Transform
Chapter 4Function Spaces
4.1 Spaces of Continuous and ContinuouslyDifferentiable Functions, CQand CkQ
4.1.1. The Space CQ
4.1.2. The Space CkQ
4.2 The Space L1 (Q)
4.3 The Space L2 (Q)
4.4 The Space CQis Dense in L1 (Q) and L2 (Q) .Spaces L1 (Q) and L2 (Q) Are Separable. Continuityin the Mean of Functions from L1 (Q) andL2 (Q)
4.5 Averaging of Functions of L1 (Q) and L2 (Q)
4.6 Linear Spaces L1,loc and L2,loc
4.7 Generalized Derivatives
4.8 Sobolev Spaces Hk (Q)
4.9 Traces of Functions From Hk (Q)
4.10 Equivalent Norms in Spaces H1 (Q)and H10 (Q)
4.10.1. Integration by Parts for Functions in H1 (Q)
4.10.2. Equivalent Norms
4.10.3. Representation of Functions via Integrals
a. Convergence of an Improper Integral
b. The First Representation Formula
c. The Second Representation Formula
4.11 Sobolev Embedding Theorem
Chapter 5Elliptic PDEs
5.1 Weak Solution of a Boundary Value Problemforan Elliptic Equation in the Simplest Case
5.1.1. The Riesz Representation Theorem
5.1.2. Weak Solution
5.2 Dirichlet Boundary Value Problem for aGeneral Elliptic Equation
5.2.1. Equation with a Compact Operator
5.2.2. Fredholm’s Theorems
a. Preliminaries
b. Fredholm’s Theorems
c. Consequences for Elliptic Equations
5.3 Volterra Integral Equation and Gronwall’sInequalities
5.3.1. Volterra Integral Equation of the Second Kind
a. Equation of the First Kind
b. Equation of the Second Kind
5.3.2. Gronwall’s Inequalities
Chapter 6Hyperbolic PDEs
6.1 Wave Equation: Energy Estimate and Domainof Influence
6.2 Energy Estimate for the Initial Boundary ValueProblem for a General Hyperbolic Equation
Chapter 7Parabolic PDEs
7.1 Parabolic Equations
7.1.1. Heat Equation
7.1.2. The Maximum Principle in the General Case
7.1.3. A General Uniqueness Theorem
7.2 Construction of the Weak Solution of Problem(7.1.1)-(7.1.3)
7.2.1. Weak Solution: Definition
7.2.2. Existence of theWeak Solution
7.2.3. Continuation of the Proof of Theorem 7.2.1
Chapter 8Introduction to Ill-PosedProblems
8.1 Some Definitions
8.2 Some Examples of Ill-Posed Problems
8.3 The Foundational Theorem of A. N. Tikhonov[8]
8.4 Classical Correctness and ConditionalCorrectness
8.5 Quasi-Solution
8.6 Regularization
8.7 The Tikhonov Regularization Functional
8.7.1. The Tikhonov Functional
8.7.2. Approximating the Exact Solution x
8.7.3. Regularized Solution and Accuracy Estimate in a FiniteDimensional Space
Chapter 9Finite Difference Method
9.1 Finite Difference (FD) Methods forOne-Dimensional Problems
9.1.1. A Simple Example of a FD Method
9.1.2. Fundamentals of FD Methods
a. Forward, Backward and Central FD Formulas for u0(x)
9.1.3. Deriving FD Formulas Using the Methodof Undetermined Coefficients
a. FD Formulas for Second Order Derivatives
b. FD Formulas for Higher Order Derivatives
9.1.4. Consistency, Stability, Convergence and ErrorEstimates of FD Methods
a. Global Error
b. Local Truncation Error
c. Round-Off Errors
9.1.5. FD Methods for Elliptic Equations
a. 1-D Self-Adjoint Elliptic Equations
b. General 1D Elliptic Equations
9.1.6. The Ghost Point Method for Boundary ConditionsInvolving Derivatives
9.1.7. The Grid Refinement Analysis Technique
9.2 Finite DifferenceMethods for 2D Elliptic PDEs
9.2.1. Boundary and Compatibility Conditions
9.2.2. The Central FD Method for Poisson Equations
9.2.3. The Maximum Principle and Error Analysis
a. The Discrete Maximum Principle
b. Error Estimates of the FD Method for Poisson Equations
9.2.4. Finite Difference Methods for General 2nd Order EllipticPDEs
9.2.5. Solving the Resulting Linear System of AlgebraicEquations
a. The Jacobi IterativeMethod
b. The Gauss-Seidel IterativeMethod
c. The Successive Over-Relaxation Method
c. Convergence of Stationary IterativeMethods
9.2.6. A Fourth-Order Compact FD Scheme forPoisson Equations
9.3 Finite Difference Methods for Linear ParabolicPDEs
9.3.1. The Euler Methods
a. Forward Euler Method (FT-CT)
b. Backward Euler Method (BW-CT)
9.3.2. The Method of Lines (MOL)
9.3.3. The Crank-Nicolson Scheme
9.3.4. Stability Analysis for Time-Dependent Problems
a. Review of the Fourier Transform (FT)
b. The Discrete Fourier Transform
c. Definition of the Stability of a FD Scheme
d. The von Neumann Stability Analysis for FD Methods
e. Simplification of the von Neumann Stability Analysis for One-StepTime Marching Methods
9.3.5. FD Methods and Analysis for 2DParabolic Equations
9.3.6. The Alternating Directional Implicit (ADI) Method
a. Implementation of the ADI Algorithm
b. Consistency of the ADIMethod
c. Stability Analysis for the ADI Method
9.3.7. An Implicit-Explicit Method for Diffusionand Advection Equations
9.4 Finite DifferenceMethods for LinearHyperbolic PDEs
9.4.1. FD Methods
a. Lax-Friedrichs Methods
b. The Upwind Scheme
c. The Leap-Frog Scheme
9.4.2. Modified PDEs and Numerical Diffusion/Dispersion
9.4.3. The Lax-Wendroff Scheme
9.4.4. Numerical Boundary Conditions (NBC)
9.4.5. FD Methods for Second Order Linear Hyperbolic PDEs
a. A Finite Difference Method (CT-CT) for the Second Order WaveEquation
b. Transforming Second Order Wave Equation to a First OrderSystem
c. Initial and Boundary Conditions for the System
9.4.6. Some Commonly Used FD Methods for a Linear Systemof Hyperbolic PDE
Chapter 10Finite Element Method
10.1 Finite ElementMethods for 1D EllipticProblems
10.1.1. An Example of the Finite Element for a Model Problem
10.1.2. Finite Element Methods for One DimensionalElliptic Equations
a. Physical Reasoning
b. Mathematical Equivalence
10.1.3. Finite Element Method for the 1D Model Problem:Method and Programming
a. Galerkin Method
b. The Ritz Method
10.1.4. FEM Programming for 1D Problem
10.2 Theoretical Foundations of the Finite ElementMethod
10.2.1. FE Analysis for the 1D Model Problem
a. Conforming FEMethods
b. FE Analysis for a 1D Sturm-Liouville Problem
c. The Bilinear Form
d. The FEMethod for the 1D Sturm-Liouville Problem UsingPiecewise Linear Basis Functions in H10 (xl , xr)
e. Local Stiffness Matrix and Load Vector Using the Hat BasisFunctions
10.2.2. Error Analysis for the FE Method
a. Interpolation Functions and Error Estimates
b. Error Estimates of the FEMethods Using the InterpolationFunction
c. Error Estimates in Pointwise Norm
10.3 Issues of the Finite ElementMethod in 1D
10.3.1. Boundary Conditions
a. Mixed Boundary Conditions
b. Non-Homogeneous Dirichlet Boundary Conditions
10.3.2. The FE Method for Sturm-Liouville Problems
a. Numerical Treatments of Dirichlet BC
b. Contributions from Neumann or Mixed BC
10.3.3. High Order Elements
a. Piecewise Quadratic Basis Functions
b. Assembling the Stiffness Matrix and the Load Vector
c. The Cubic Functions in H1(xl, xr) Space
10.3.4. The Lax-Milgram Lemma and the Existenceof FE Solutions
a. General Settings: Assumptions and Conditions
b. The Lax-Milgram Lemma
c. An Example of the Lax-Milgram Theorem
d. Abstract FE Methods
10.4 Finite ElementMethods for 2D Problems
10.4.1. The Second Green’s Theorem and Integration by Partsin 2D
a. Boundary Conditions
b. Weak Form of the Second Order Self-Adjoint Elliptic PDE
c. Verification of the Conditions of the Lax-Milgram Lemma
10.4.2. Triangulation and Basis Functions
a. Triangulation and Mesh Parameters
b. FE Space of Piecewise Linear Functions over a Mesh
c. Global Basis Functions
d. The Interpolation Function and Error Analysis
e. Error Estimates of the FE Solution
10.4.3. Transforms, Shape Functions and QuadratureFormulas
10.4.4. Quadrature Formulas
10.4.5. Simplification of the FE Method for PoissonEquations
Chapter 11Conclusion
References
Author Contact Information
Index