Partial Differential Equations. First Order Equations and Characteristics. Weak Solutions to Hyperbolic Equations. Diffusion Processes. Hyperbolic Systems. Reaction--Diffusion Equations. Models for Chemically Reacting Fluids. Elliptic Equations. Index.
1. Introduction 2. First-order equations 3. Second-order linear equations 4. The 1D wave equation 5. Separation of variables 6. Sturm-Liouville problem 7. Elliptic equations 8. Green's function and integral representation 9. Equations in high dimensions 10. Variational methods 11. Numerical methods 12. Solutions of odd-numbered problems.
Preface 1. A brief discussion of integral equations 2. Degenerate kernel methods 3. Projection methods 4. The Nystrom method 5. Solving multivariable integral equations 6. Iteration methods 7. Boundary integral equations on a smooth planar boundary 8. Boundary integral equations on a piecewise smooth planar boundary 9. Boundary integral equations in three dimensions Discussion of the literature Appendix Bibliography Index.
This is an introduction to stochastic integration and stochastic differential equations written in an understandable way for a wide audience, from students of mathematics to practitioners in biology, chemistry, physics, and finances. The presentation is based on the naïve stochastic integration, rather than on abstract theories of measure and stochastic processes. The proofs are rather simple for practitioners and, at the same time, rather rigorous for mathematicians. Detailed application examples in natural sciences and finance are presented. Much attention is paid to simulation diffusion processes. The topics covered include Brownian motion; motivation of stochastic models with Brownian motion; Itô and Stratonovich stochastic integrals, Itô’s formula; stochastic differential equations (SDEs); solutions of SDEs as Markov processes; application examples in physical sciences and finance; simulation of solutions of SDEs (strong and weak approximations). Exercises with hints and/or solutions are also provided.
PART I: ORDINARY DIFFERENTIAL EQUATIONS Advances in the Asymptotic and Numerical Solution of Linear Ordinary Differential Equations, F.W.J. Olver Some Unsolved Problems in Asymptotics, R. Wong Periodic Solutions and Heteroclinic Cycles in the Convection Model of a Rotating Fluid Layer, J. Li and X.H. Zhao The Equivalence of Exponential Stability for Impulsive Time-Delay Differential Systems, Z.-H. Guan, Y.-C. Zhou, and X.-P. He Conditions for Identity of Bifurcations in Cubic Hamiltonian Systems with Symmetry or Nonsymmetry Perturbations, Z. Liu, H. Cao, and J. Li PART II: PARTIAL DIFFERENTIAL EQUATIONS Long Time Behavior for the Generalized Ginzburg-Landau Equations, B. Guo The Inverse Scattering Transform for a Variable-Coefficient KdV Equations (with Applications to Shallow Water Waves), H.-H. Dai The Semigroup Theory and Abstract Linear Equations, G. Yang A Unified Approach Towards Nonlinear Parabolic Equations with Strong Reaction in Rn, Y.-W. Qi Global Existence of Smooth Solution to Boltzmann-Poisson System in Semiconductor Physics, G. Cui and Y. Wang Analytical Methods for a Selection of Elliptic Singular Perturbation Problems, N.M. Temme Exponential Attractors of the Strongly Damped Nonlinear Wave Equations, Z. Dai and B. Guo Generalized Isovorticity Principle for Ideal Magnetohydrodynamics, V.A. Vladimirov and K.I. Ilin Scroll Waves in Excitable Media and the Motion of Organization Center, Q. Lu and S. Liu Transport Equations for a General Class of Evolution Equations, M.Z. Guo and X.P. Wang a-Times Integrated Cosine Function, G. Yang Identifying Parameters in Elliptic Systems by Finite Element Methods with Multi-Level Initializing, Y.F. Seid and J. Zou Techniques Monotone Difference Schemes for Two Dimensional Nonhomogeneous Conservation Laws, T. Tang and Z.-H. Teng
A. The title is designed to indicate those particular aspects of stochastic differential equations which will be considered here: these are almost equally valid when the manifold in question is â n (although compactness is often a useful simplifying assumption). In fact one of the main themes here will be that stochastic differential equations, even on â n , induce non-trivial differential-geometric structures and these structures are an important tool in analyzing the behaviour of the solutions of the s.d.e.
1. Mathematical toolbox 2. Basic difference equations models and their analysis 3. Basic differential equations models 4. Qualitative theory for a single equation 5. From discrete to continuous models and back Bibliography Index.
Mathematical Modeling of N-Body Quantum Scattering Processes Microhydrodynamics of Sharp Corners and Edges Acoustic Scattering by Irregular Obstacles On Generalized Inverse Steklov Problems A Hybrid Root Finder Volterra Type Integral Geometry Problems Inversionof the X-Ray Transform and the Radon Transform with Incomplete Data Reconstructing a Function by means of Integrals over a Family of Conical Surfaces On the use of Wavelet Expansions and the Conjugate Gradient Method for Solving First Kind Integral Equations Optimal Algorithms for the Calculation of Singular Integrals An Order One Approximate Method of Solution of Differential and Integral Equations A Multigrid Method for Solving Reaction-Diffusion Systems Time-Dependent Bending of Plates with Transverse Shear Deformation Polarization Gradient in Piezoelectric Micropolar Elasticity Thermoelastic Stress Separation via Poisson Equation Solution by means of the Boundary Element Method On Neutral Functional Differential Equations with Causal Operations A Semi-Analytic Method for the Study of Acoustic Pulse Propagation in Inhomogeneous Elastic 1-D Media Efficient Finite Elements for the Numerical Approximation of Cylindrical Shells Error Estimates for Computing Fixed Densities of Markov Integral Operators A Modified Monte Carlo Approach to the Approximation of Invariant Measures Acceleration Waves in von Karman Plate Theory Dynamics and Resonance of a Nonlinear Mechanical Oscillator Subjected to Parametric and External Excitation On the Vibration of Helical Springs A Two-Dimensional Numerical Model of Chemotaxis Asymptotic Analysis of Fracture Theory for Layered Composites in Compression (The Plane Problem) Existence and Regularity of Weak Solutions to the Displacement Boundary Value Problem of Nonlinear Elastostatics On Convergence and Uniqueness of Microscale Heat Transfer Equation On Numerical Approximations of a Frictionless Contact Problem for Elastic-Viscoplastic Materials A Panel Clustering Method for 3-D Elastostatics using Spherical Harmonics Extensions of Constrained Least-Squares for Obtaining Regularized Solutions to First-Kind Integral Equations Existence and Nonexistence Results for some Boundary Value Problems at Resonance Analytic Investigation of Thick Anisotropic Plates with Undulating Surfaces On Stoke's Nonlinear Integral Wave Equation Dynamics of the Spurt Phenomenon for Single History Integral Constitutive Equations A Boundary Integral Equation Method for the Heat Equation Computational Simulation and Interfacial Shear Models for Downward Annular Wavy-Interface Condensing Flow in a Vertical Pipe Compact Fourth-Order Approximation for a Nonlinear Reaction-Diffusion Equation arising in Population Genetics Dynamic Deformation of a Layered Continuum Surrounding a Cylindrical or Spherical Cavity Boundary Value Problems for Harmonic Vector Fields on Non-Smooth Domains The Oblique Derivative Problem for General Elliptic Systems in Lipschitz Domains Viscous Aerodynamic Optimal Design of Flying Configurations via an Enlarged Variational Method Interface Crack Problem for a Piecewise Homogeneous Anisotropic Plane Micromechanics of Heterogeneous Materials Admissibility and Optimal Control for Stochastic Difference Equations Linear and Sublinear Tricomi via DFI Multidimensional Fractional Integrals on Spaces of Smooth Functions Piecewise Polynomial Projection Methods for Nonlinear, Multidimensional, Weakly Singular Integral Equations Asymptotics and Inequalities for a Class of Infinite Sums A New Theorem Concerning the Buckingham-Rayleigh Methods and General Dimensional Analysis Energy Decay for a Weak Solution of the Non-Newtonian Fluid Equations with Slowly Varying External Forces An Integral Representation for the Solution of 2-D Non-Stationary Flow of Micropolar Fluids A Monotonically Convergent Adaptive Method for Nonlinear Combustion Problems Stationary Oscillations of Elastic Plates with Robin Boundary Conditions Adapter and Driver Design for Rotary Encoders Expert System Design for Fault Diagnosing in CNC Machine Tools Non-Ideal Liquid Solutions Modeling by means of Integral Methods Symmetry Groups, Conservation Laws, and Group-Invariant Solutions of the Marguerre-von Karman Equations Modeling the Motion of an Underwater Explosion Bubble On some New Systems of N-ary Integral Equations Computational Simulations and Flow Domain Classification for Laminar/Laminar annular/Stratified Condensing Flows
Picard's researches — which we shall quote in their place — are also essential in several parts of the present work. Such is also the case for Le Roux.
A comprehensive approach to numerical partial differential equations Spline Collocation Methods for Partial Differential Equations combines the collocation analysis of partial differential equations (PDEs) with the method of lines (MOL) in order to simplify the solution process. Using a series of example applications, the author delineates the main features of the approach in detail, including an established mathematical framework. The book also clearly demonstrates that spline collocation can offer a comprehensive method for numerical integration of PDEs when it is used with the MOL in which spatial (boundary value) derivatives are approximated with splines, including the boundary conditions. R, an open-source scientific programming system, is used throughout for programming the PDEs and numerical algorithms, and each section of code is clearly explained. As a result, readers gain a complete picture of the model and its computer implementation without having to fill in the details of the numerical analysis, algorithms, or programming. The presentation is not heavily mathematical, and in place of theorems and proofs, detailed example applications are provided. Appropriate for scientists, engineers, and applied mathematicians, Spline Collocation Methods for Partial Differential Equations: Introduces numerical methods by first presenting basic examples followed by more complicated applications Employs R to illustrate accurate and efficient solutions of the PDE models Presents spline collocation as a comprehensive approach to the numerical integration of PDEs and an effective alternative to other, well established methods Discusses how to reproduce and extend the presented numerical solutions Identifies the use of selected algorithms, such as the solution of nonlinear equations and banded or sparse matrix processing Features a companion website that provides the related R routines Spline Collocation Methods for Partial Differential Equations is a valuable reference and/or self-study guide for academics, researchers, and practitioners in applied mathematics and engineering, as well as for advanced undergraduates and graduate-level students.
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