Introduction

It is well known that the vast majority of boundary value and initial value problems cannot be solved exactly and require the use of appropriate approximate methods. An important characteristic of any approximate method is its accuracy. To estimate the accuracy, we traditionally use a certain discretization parameter: a mesh step, the number of terms of the partial sum of the series, etc.

However, for both theoretical and practical reasons, it is also important to take into account the influence of other factors, for example, the so-called boundary and initial effects. Precisely, the boundary effect means that due to the Dirichlet boundary condition for a differential equation in the canonical domain, the accuracy of the approximate solution near the boundary of the domain is higher compared to the accuracy further from the boundary. A similar situation is observed for non-stationary equations near those mesh nodes where the initial condition is set.

For the quantitative characteristics of the boundary or initial effect, we can take an a priori error estimate (in a certain mesh norm) with a certain weight function, which characterizes the distance of a point inside the domain to the boundary of the domain. The idea of such estimates was first announced by Volodymyr Makarov in Makarov (1987) for an elliptic equation in case of generalized solutions from Sobolev spaces and developed further in publications for quasilinear stationary and non-stationary equations. Since the concept was quite new, there were (and still are) very few publications on this subject. In some respects, the same issue is studied in the works of Galba (1985) and Molchanov and Galba (1990). However, they assume only the classical smoothness of solutions and do not consider time-dependent problems.

In this book, we develop our previous studies and present some new results on the impact of initial and boundary conditions on the accuracy of the following methods: the finite-difference method for elliptic and parabolic equations, the discrete method for solving equations with fractional derivatives, and the Cayley transform method for abstract differential equations in Hilbert and Banach spaces. Regardless of the type of problem or method, our main focus is always on obtaining weighted estimates with a proper weight function.

For a better understanding of the reasoning and easier navigation through the computation, some information is assumed to be known to the reader from classical mathematics courses, while the rest is provided directly in the text. Some of the formulas may seem a bit long and cumbersome, but this is partly because we are trying to be as detailed as possible and help the reader follow the calculations with ease.

This book consists of five chapters. Chapters 1 and 2 are devoted to the study of the accuracy of finite-difference schemes for stationary and non-stationary equations respectively, taking into account the influence of boundary and initial conditions (in the sense of Makarov as mentioned above).

The finite-difference method is historically one of the first and most recognized numerical methods for solving problems of mathematical physics, mainly due to its universality and convenience in practical implementation. In recent decades, it has gained considerable popularity due to growing interest in the study of nonlinear processes in various fields of physics, chemistry, seismology, ecology, etc. Mathematical models of such phenomena involve nonlinear partial differential equations. For example, in aerodynamics and hydrodynamics, the one-dimensional quasilinear Burgers parabolic equation arises as an adequate mathematical model of turbulence. A special case of the Burgers equation is the quasilinear transport equation (the Hopf equation), which is the simplest equation describing discontinuous flows or flows with shock waves. In biology, ecology, physiology, combustion theory, crystallization theory, plasma physics, etc., the Fisher–Kolmogorov–Petrovsky–Piskunov equation (the Fisher–KPP equation) plays an important role as the simplest semi-linear parabolic equation. The propagation of shallow water waves that weakly and nonlinearly interact, ion acoustic waves in plasma, acoustic waves on crystal lattices, etc. are often modeled by the Korteweg–de Vries equation (the KdV equation). Many publications are devoted to finite-difference schemes for solving problems for elliptic and parabolic equations with dynamic conjugation conditions at the contact boundary (which is associated with the presence of concentrated heat capacities in a heat-conducting medium) and/or dynamic boundary conditions (which model heat conduction in a solid body in contact with fluid, as well as processes in semiconductor devices). In the mathematical modeling of some processes in ecology, physics and technology, when it is impossible to set the exact values of the desired solution at the boundary of a domain, problems with non-local boundary conditions usually arise.

These and many other examples demonstrate that the finite-difference method is actively developing and is widely used to solve current scientific and technical problems. At the same time, there are very few publications dedicated to the study of the initial and boundary effects in the above sense, and our book is a certain step towards filling this gap. One of the first such works is the announcement (Makarov 1989) that deals with the problem

where

and Ω = {(x1,x2) : 0 < xα < 1, α = 1,2} is a unit square. The problem is discretized by the finite-difference scheme

where ω = ω1×ω2, ωα = {xα = iαhα : iα = 1,2,…, Nα −1, hα = 1/Nα}, α = 1,2; γ is a boundary of the mesh ω. Some traditional notations for finite-difference schemes from Samarskii (2001) are used here, for example: ,

The main result was presented in the following statement.

THEOREM.– Let and . Then, there exists h0 > 0 such that for all h ∈ (0, h0] the accuracy of the finite-difference scheme [I.1] is characterized by the weighted estimate

with the weight function ρ(x) = min{x1x2, x1(1 − x2), (1 − x1)x2, (1 − x1)(1 − x2)}.

This idea is further developed in the present book for other types of boundary conditions for elliptic and parabolic equations. It is worth mentioning that the important stages in obtaining such weighted estimates are the evaluation of discrete Green’s functions and the analysis of approximation errors. Each time, when it is necessary to estimate discrete Green’s functions, we apply the following proposition, which is formulated and proved in Samarskii et al. (1987, p. 54).

MAIN LEMMA.– Let the following assumptions be fulfilled: 1) A : H → H is a self-adjoint operator acting in a Hilbert space H; 2) B : H* → H is a linear operator; 3) the inverse operator A−1 exists; 4) ║B*v║* ≤ γ║Av║ for all v ∈ H, where B* : H → H* is the adjoint operator of B, (y, v)* and are an inner product and an associate norm in H* respectively. Then, ║A−1Bv║ ≤ γ║v║* for all v ∈ H*.

Similarly, when it comes to estimating an approximation error for a generalized solution from Sobolev spaces, we refer to the Bramble–Hilbert lemma (e.g. Samarskii et al. (1987, p. 29)). We recall it here for convenience.

LEMMA (BRAMBLE–HILBERT).– Let be an open convex bounded set of the diameter d > 0, let l(u) be a bounded linear functional in the space with , where is a positive non-negative number and 0 < λ ≤ 1, namely:

and let l(u) turn into zero on polynomials of degree of variables x1, x2,…,xn . Then, there exists a positive constant , which is dependent on Ω and independent of u(x), such that the following inequality holds true:

The study of the boundary and initial effects is also of great interest for new classes of problems, for example, related to the application of fractional integro-differentiation. In Chapter 3, we address the accuracy of the mesh methods for solving boundary value problems for differential equations with fractional derivatives.

For almost 300 years (from 1695 until recently) this branch of classical analysis was no more than an abstract mathematical theory. However, over the past several decades, fractional analysis has found wide applications in the construction of adequate mathematical models of many natural and social phenomena, as evidenced by a considerable number of publications (e.g. Kilbas et al. (2006); Sabatier et al. (2007); Nakagawa et al. (2010), to mention a few). Due to the ability to model...