Traditional Functional-Discrete Methods for the Problems of Mathematical Physics: New Aspects (Mathematics and Statistics)

✍ Scribed by Volodymyr Makarov (editor), Nataliya Mayko (editor)

• Publisher: Wiley-ISTE
• Year: 2024
• Tongue: English
• Leaves: 346
• Edition: 1
• Category: Library

✦ Synopsis

This book is devoted to the construction and study of approximate methods for solving mathematical physics problems in canonical domains. It focuses on obtaining weighted a priori estimates of the accuracy of these methods while also considering the influence of boundary and initial conditions. This influence is quantified by means of suitable weight functions that characterize the distance of an inner point to the boundary of the domain.

New results are presented on boundary and initial effects for the finite difference method for elliptic and parabolic equations, mesh schemes for equations with fractional derivatives, and the Cayley transform method for abstract differential equations in Hilbert and Banach spaces. Due to their universality and convenient implementation, the algorithms discussed throughout can be used to solve a wide range of actual problems in science and technology. The book is intended for scientists, university teachers, and graduate and postgraduate students who specialize in the field of numerical analysis.

✦ Table of Contents

Cover
Title Page
Copyright Page
Contents
Preface
Introduction
Chapter 1. Elliptic Equations in Canonical Domains with the Dirichlet Condition on the Boundary or its Part
1.1. A standard finite-difference scheme for Poisson’s equation with mixed boundary conditions
1.1.1. Discretization of the BVP
1.1.2. Properties of the finite-difference operators
1.1.3. Discrete Green’s function
1.1.4. Accuracy with the boundary effect
1.1.5. Conclusion
1.2. A nine-point finite-difference scheme for Poisson’s equation with the Dirichlet boundary condition
1.2.1. Discretization of the BVP
1.2.2. Properties of the finite-difference operators
1.2.3. Discrete Green’s function
1.2.4. Accuracy with the boundary effect
1.2.5. Conclusion
1.3. A finite-difference scheme of the higher order of approximation for Poisson’s equation with the Dirichlet boundary condition
1.3.1. Auxiliary results
1.3.2. Accuracy with the boundary effect
1.3.3. Conclusion
1.4. A finite-difference scheme for the equation with mixed derivatives
1.4.1. Discretization of the BVP
1.4.2. Properties of the finite-difference operators
1.4.3. Discrete Green’s function
1.4.4. Accuracy with the boundary effect
1.4.5. Conclusion
Chapter 2. Parabolic Equations in Canonical Domains with the Dirichlet Condition on the Boundary or its Part
2.1. A standard finite-difference scheme for the one-dimensional heat equation with mixed boundary conditions
2.1.1. Discretization of the problem
2.1.2. Accuracy with the boundary effect
2.1.3. Accuracy with the initial effect
2.1.4. Conclusion
2.2. A standard finite-difference scheme for the two-dimensional heat equation with mixed boundary conditions
2.2.1. Discretization of the differential problem and properties of the finite-difference operators
2.2.2. Discrete Greenfs function
2.2.3. Accuracy with the boundary effect
2.2.4. Conclusion
2.3. A standard finite-difference scheme for the two-dimensional heat equation with the Dirichlet boundary condition
2.3.1. Discretization of the differential problem
2.3.2. Accuracy with the boundary effect
2.3.3. Accuracy with the initial effect
2.3.4. Conclusion
Chapter 3. Differential Equations with Fractional Derivatives
3.1. BVP for a differential equation with constant coefficients and a fractional derivative of order
3.1.1. A weighted estimate for the exact solution
3.1.2. Weighted estimates for approximate solutions
3.1.3. Conclusion
3.2. BVP for a differential equation with constant coefficients and a fractional derivative of order ƒ¿ ¸ (0,1
3.2.1. A scale of weighted estimates for the exact solution
3.2.2. The scale of weighted estimates for approximate solutions
3.2.3. A numerical example and conclusion
3.3. BVP for a differential equation with variable coefficients and a fractional derivative of order ƒ¿ ¸ (0,1)
3.3.1. Differential properties of the exact solution
3.3.2. The accuracy of the mesh scheme
3.3.3. Conclusion
3.4. Two-dimensional differential equation with a fractional derivative
3.4.1. A weighted estimate for the exact solution
3.4.2. A mesh scheme of the first order of accuracy
3.4.3. A mesh scheme of the second order of accuracy
3.4.4. Conclusion
3.5. The Goursat problem with fractional derivatives
3.5.1. Properties of the exact solution
3.5.2. The accuracy of the mesh scheme
3.5.3. Conclusion
Chapter 4. The Abstract Cauchy Problem
4.1. The approximation of the operator exponential function in a Hilbert space
4.2. Inverse theorems for the operator sine and cosine functions
4.3. The approximation of the operator exponential function in a Banach space
4.4. Conclusion
Chapter 5. The Cayley Transform Method for Abstract Differential Equations
5.1. Exact and approximate solutions of the BVP in a Hilbert space
5.1.1. Auxiliary results
5.1.2. The exact solution of the BVP
5.1.3. The approximate method without saturation of accuracy
5.1.4. The approximate method with a super-exponential rate of convergence
5.1.5. Conclusion
5.2. Exact and approximate solutions of the BVP in a Banach space
5.2.1. The BVP for the homogeneous equation
5.2.2. The BVP for the inhomogeneous equation
5.2.3. Conclusion
References
Index