Well-Posedness of Parabolic Difference Equations

verfasst von: A. Ashyralyev, P. E. Sobolevskii

Verlag: Birkhäuser Basel

Buchreihe : Operator Theory: Advances and Applications

Enthalten in: Professional Book Archive

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A well-known and widely applied method of approximating the solutions of problems in mathematical physics is the method of difference schemes. Modern computers allow the implementation of highly accurate ones; hence, their construction and investigation for various boundary value problems in mathematical physics is generating much current interest. The present monograph is devoted to the construction of highly accurate difference schemes for parabolic boundary value problems, based on Padé approximations. The investigation is based on a new notion of positivity of difference operators in Banach spaces, which allows one to deal with difference schemes of arbitrary order of accuracy. Establishing coercivity inequalities allows one to obtain sharp, that is, two-sided estimates of convergence rates. The proofs are based on results in interpolation theory of linear operators. This monograph will be of value to professional mathematicians as well as advanced students interested in the fields of functional analysis and partial differential equations.

Inhaltsverzeichnis

Frontmatter

Chapter 1. The Abstract Cauchy Problem

Abstract

Consider the following Cauchy problem in an arbitrary Banach space E:

$$ v'\left( t \right) + Av\left( t \right) = f\left( t \right),{\text{ }}0 \leqslant t \leqslant 1,{\text{ }}v\left( 0 \right) = v_0 . $$

(1.1)

Here v(t) and f(t) are the unknown and the given function, respectively, defined on [0,1] with values in E. The derivative v′(t) is understood as the limit in the norm of E of the corresponding ratio of differences. A is a linear operator acting in E, with domain D(A). Finally, v₀ is a given element of E.

A. Ashyralyev, P. E. Sobolevskii

Chapter 2. The Rothe Difference Scheme

Abstract

Let us associate to the Cauchy problem (1.1) of Chapter 1 the corresponding difference problem

$$ D_{uk} + A_{uk} = \varphi _k ,{\text{ }}1 \leqslant k \leqslant N,u_0 = u_o (\tau ). $$

(0.1)

Here N is a fixed positive integer, τ = 1/N, Du_k = (u_k-u_k-1)/τ; u^τ = {u_k} ₁ ^N , φ^τ = {φ_k} ₁ ^N are the unknown and the given grid functions with values in the Banach space E. It is assumed that the function (φ^τ and the elements u₀(τ) approximate f(t) and v₀, respectively, in a specified way.

A. Ashyralyev, P. E. Sobolevskii

Chapter 3. Padé Difference Schemes

Abstract

Let us consider the problem of approximating the function e^-z near z = 0 by rational functions

$$ \begin{gathered} R_{j,l} (z) = \frac{{P_{j,l} (z)}} {{Q_{j,l} (z)}} = \frac{{a_0 + a_1 z + ... + a_j z^j }} {{}}, \hfill \\ a_r = a_r (j,l),r = 1...j,b_r = b_r (j,l),r = 1...,l,a_j \ne 0,b_l \ne 0,b_0 \ne 0. \hfill \\ \end{gathered} $$

A. Ashyralyev, P. E. Sobolevskii

Chapter 4. Difference Schemes for Parabolic Equations

Abstract

Let us consider a differential operator with constant coefficients of the form

$$ A = \sum\limits_{r = \left| m \right|} {{a_r}} {{{\partial ^{{r_{1 + \cdots + {r_n}}}}}} \over {\partial x_1^{{r_1}} \cdots \partial x_n^{{r_n}}}}, $$

(1.1)

acting on functions defined on the entire space Rⁿ. Here r ∊ Rⁿ is a vector with nonnegative integer components, |r| = r₁+…+r_n. If φ(y) (y = (y₁, …, y_n) ∊ Rⁿ) is an infinitely differentiable function that decays at infinity together with all its derivatives, then by means of the Fourier transformation one establishes the equality

$$ F(A\varphi )(\xi ) = A(\xi )F(\varphi )(\xi ). $$

(1.2)

A. Ashyralyev, P. E. Sobolevskii

Backmatter

Titel: Well-Posedness of Parabolic Difference Equations
verfasst von: A. Ashyralyev
P. E. Sobolevskii
Verlag: Birkhäuser Basel
Electronic ISBN: 978-3-0348-8518-8
Print ISBN: 978-3-0348-9661-0
DOI: https://doi.org/10.1007/978-3-0348-8518-8

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Inhaltsverzeichnis

Frontmatter

Chapter 1. The Abstract Cauchy Problem

Chapter 2. The Rothe Difference Scheme

Chapter 3. Padé Difference Schemes

Chapter 4. Difference Schemes for Parabolic Equations

Backmatter