Traditional Functional-Discrete Methods for the Problems of Mathematical Physics

New Aspects

Inbunden, Engelska, 2024

AvVolodymyr Makarov,Volodymyr Makarov,Nataliya Mayko

2 109 kr

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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.

Produktinformation

Utgivningsdatum2024-03-11
Mått156 x 234 x 21 mm
Vikt771 g
FormatInbunden
SpråkEngelska
Antal sidor352
FörlagISTE Ltd and John Wiley & Sons Inc
ISBN9781786309334

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Volodymyr Makarov is Doctor of Physical and Mathematical Sciences, Professor, and Academician at the National Academy of Sciences of Ukraine, Kyiv, where he is also the founder and head of their Computational Mathematics Department.Nataliya Mayko is Doctor of Physical and Mathematical Sciences and Professor in the Department of Computational Mathematics of the Faculty of Computer Science and Cybernetics at Taras Shevchenko National University of Kyiv, Ukraine.

Preface ixIntroduction xiChapter 1 Elliptic Equations in Canonical Domains with the Dirichlet Condition on the Boundary or its Part 11.1 A standard finite-difference scheme for Poisson's equation with mixed boundary conditions 11.2 A nine-point finite-difference scheme for Poisson's equation with the Dirichlet boundary condition 181.3 A finite-difference scheme of the higher order of approximation for Poisson's equation with the Dirichlet boundary condition 311.4 A finite-difference scheme for the equation with mixed derivatives 46Chapter 2 Parabolic Equations in Canonical Domains with the Dirichlet Condition on the Boundary or its Part 692.1 A standard finite-difference scheme for the one-dimensional heat equation with mixed boundary conditions 692.2 A standard finite-difference scheme for the two-dimensional heat equation with mixed boundary conditions 822.3 A standard finite-difference scheme for the two-dimensional heat equation with the Dirichlet boundary condition 102Chapter 3 Differential Equations with Fractional Derivatives 1153.1 BVP for a differential equation with constant coefficients and a fractional derivative of order ½ 1153.2 BVP for a differential equation with constant coefficients and a fractional derivative of order α ∈ (0,1) 1243.3 BVP for a differential equation with variable coefficients and a fractional derivative of order α ∈ (0,1) 1453.4 Two-dimensional differential equation with a fractional derivative 1663.5 The Goursat problem with fractional derivatives 181Chapter 4 The Abstract Cauchy Problem 2134.1 The approximation of the operator exponential function in a Hilbert space 2134.2 Inverse theorems for the operator sine and cosine functions 2304.3 The approximation of the operator exponential function in a Banach space 2364.4 Conclusion 247Chapter 5 The Cayley Transform Method for Abstract Differential Equations 2495.1 Exact and approximate solutions of the BVP in a Hilbert space 2495.2 Exact and approximate solutions of the BVP in a Banach space 282References 307Index 315