Partial Differential Equations: Theory, Numerical Methods and Ill-posed Problems

Description

The laws of nature are written in the language of partial differential equations. Therefore, these equations arise as models in virtually all branches of science and technology. Our goal in this book is to help you to understand what this vast subject is about. The book is an introduction to the field suitable for senior undergraduate and junior graduate students. Introductory courses in partial differential equations (PDEs) are given all over the world in various forms. The traditional approach to the subject is to introduce a number of analytical techniques, enabling the student to derive exact solutions of some simplified problems. Students who learn about computational techniques in other courses subsequently realize the scope of partial differential equations beyond paper and pencil. Our book is significantly different from the existing ones. We introduce both analytical theory, including the theory of classical solutions and that of weak solutions, and introductory techniques of ill-posed problems with reference to weak solutions. Besides, since computational techniques are commonly available and are currently used in all practical applications of partial differential equations, we incorporate classical finite difference methods and finite element methods in our book. 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, C􀀀Qand Ck􀀀Q 4.1.1. The Space C􀀀Q 4.1.2. The Space Ck􀀀Q 4.2 The Space L1 (Q) 4.3 The Space L2 (Q) 4.4 The Space C􀀀Qis 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 Blank Page