Traditional functional-discrete methods for the problems of mathematical physics : new aspects / Volodymyr Makarov, Nataliya Mayko
- Author
- Makarov, Volodymyr (Volodymyr L.)
- Published
- London, UK : ISTE, Ltd. ; Hoboken, NJ : Wiley, 2023.
- Physical Description
- 1 online resource (352 pages).
- Additional Creators
- Mayko, Nataliya
Access Online
- Series
- 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 -- 1.2 A nine-point finite-difference scheme for Poisson's equation with the Dirichlet boundary condition -- 1.3 A finite-difference scheme of the higher order of approximation for Poisson's equation with the Dirichlet boundary condition -- 1.4 A finite-difference scheme for the equation with mixed derivatives -- 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 -- 2.2 A standard finite-difference scheme for the two-dimensional heat equation with mixed boundary conditions -- 2.3 A standard finite-difference scheme for the two-dimensional heat equation with the Dirichlet boundary condition -- Chapter 3 Differential Equations with Fractional Derivatives -- 3.1 BVP for a differential equation with constant coefficients and a fractional derivative of order ½ -- 3.2 BVP for a differential equation with constant coefficients and a fractional derivative of order α ∈ (0,1) -- 3.3 BVP for a differential equation with variable coefficients and a fractional derivative of order α ∈ (0,1) -- 3.4 Two-dimensional differential equation with a fractional derivative -- 3.5 The Goursat problem with fractional derivatives -- 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 -- 5.2 Exact and approximate solutions of the BVP in a Banach space -- References -- Index.
- Summary
- 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.
- Subject(s)
- ISBN
- 9781394276660 (electronic bk. : oBook)
1394276664 (electronic bk. : oBook)
9781786309334 - Bibliography Note
- Includes bibliographical references and index.
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