An exponentially convergent numerical method for solving a differential equation with a right-hand fractional Riemann-Liouville time-derivative and an unbounded operator coefficient in Banach space is proposed and analysed for a homogeneous/inhomogeneous equation of the Hardy-Tichmarsh type. We employ a solution representation by the Danford-Cauchy integral on hyperbola that envelopes spectrum of the operator coefficient with a subsequent application of an exponentially convergent quadrature. To do that, parameters of the hyperbola are chosen so that the integration function has an analytical extension into a strip around the real axis and then apply the Sinc-quadrature. We show the exponential accuracy and illustrate the results by a numerical example confirming the a priori estimate. Existence conditions for the solution of the inhomogeneous equation are established.

Problem for the first order differential equation with an unbounded operator coefficient in Banach space and nonlinear nonlocal condition is considered. A numerical method is proposed and justified for the solution of this problem under assumptions that the mentioned operator coefficient A is strongly positive and some existence and uniqueness conditions are fulfilled. The method is based on the reduction of the given problem to an abstract Hammerstein equation. The later one is discretized by collocation and then solved via the fixed-point iteration method. Each iteration of the method involves Sinc-based numerical evaluation of the operator exponential represented by a Dunford-Cauchy integral along hyperbola enveloping the spectrum of A.

On October 26, 2023, the distinguished Ukrainian mathematician Roman Chapko, Doctor of Sciences, Professor in the Department of Computational Mathematics of the Faculty of Applied Mathematics and Informatics at Ivan Franko National University of Lviv, Ukraine, has turned 60. He is renowned in the broad mathematical community in Ukraine and beyond for his significant contributions to numerical analysis, computational mathematics, and mathematical modeling. His decades-long scientific activity has earned him a high reputation and has significantly elevated the standing of Ukrainian mathematics and science as a whole.

We propose and analyze an exponentially convergent numerical method for solving differential equations with right-hand fractional Riemann–Liouville derivative and unbounded operator coefficient in Banach spaces. We use a representation of the solution in the form of the Danford–Cauchy integral on a hyperbola that covers the spectrum of the operator coefficient with subsequent application of the exponentially convergent quadrature. To this end, we choose the parameters of the hyperbola in order to guarantee the possibility of analytic extension of the integrand in a strip containing the real axis and then apply the Sinc-quadrature. We prove the exponential accuracy of the method and present a numerical example that confirms the obtained a priori estimate.

ВОЛОДИМИР ЛЕОНІДОВИЧ МАКАРОВ(до 80-річчя від дня народження)Одинадцятого серпня 2021 року виповнилося 80 років відомому українському математику Володимиру Леонідовичу Макарову, провідному фахівцю в галузі обчислювальної та прикладної математики, академіку НАН України, лауреату Державної премії України в галузі науки і техніки, заслуженому діячу науки і техніки України

УДК 519.62, 519.63Запропоновано та проаналiзовано експоненцiально збiжний наближений метод розв’язування диференцiального рiвняння з правосторонньою дробовою похiдною Рiмана – Лiувiлля i необмеженим операторним коефiцiєнтом у банаховому просторi. Застосовано зображення розв’язку за допомогою iнтеграла Данфорда – Кошi по гiперболi, що охоплює спектр операторного коефiцiєнта, з подальшим застосуванням експоненцiально збiжної квадратурної формули. Для цього вибрано параметри гiперболи таким чином, щоб пiдiнтегральна функцiя мала аналiтичне продовження в смугу навколо дiйсної осi, а потiм застосовано Sinc-квадратуру. Показано експоненцiальну точнiсть методу i наведено числовi розрахунки тестового прикладу, що пiдтверджують апрiорну оцiнку.

Abstract This paper is inspired by recently proposed approach for interpreting data of Electrochemical Impedance Spectroscopy (EIS) in terms of Distribution of Diffusion Times (DDT). Such an interpretation requires to solve a Fredholm integral equation of the first kind, which may have a non-square-integrable kernel. We consider a class of equations with above-mentioned peculiarity and propose to regularize them in weighted functional spaces. One more issue associated with DDT-problem is that EIS data are available only for a finite number of frequencies. Therefore, a regularization should unavoidably be combined with a collocation. In this paper we analyze a regularized collocation in weighted spaces and propose a scheme for its numerical implementation. The performance of the proposed scheme is illustrated by numerical experiments with synthetic data mimicking EIS measurements.

We consider a nonlocal problem for the first-order differential equation with unbounded operator coefficient in a Banach space and a nonlinear integral nonlocal condition. We propose an exponentially convergent method for the numerical solution of this problem and justified this method under the assumptions that the indicated operator coefficient A is sectorial and that certain conditions for the existence and uniqueness of the solution are satisfied. This method is based on the reduction of the posed problem to an abstract Hammerstein-type equation, discretization of this equation by the method of collocation, and its subsequent solution by the method of simple iterations. Each iteration of the method is based on the Sinc-quadrature approximation of the exponential operator function represented by the Dunford–Cauchy integral over the hyperbola enveloping the spectrum of A. The integral part of the nonlocal condition is approximated by using the Clenshaw–Curtis quadrature formula.

The two-pointed nonlocal problem for the first order differential equation with an unbounded operator coefficient in a Banach space X is considered. The nonlocal condition involves a bounded operator coefficient. A new exponentially convergent method is proposed and justified in the case when the operator coefficient A in equatuion is strongly positive and some existence and uniqueness conditions are fulfilled. This method is based on representations of operator functions by a Dunford-Cauchy integral along a hyperbola enveloping the spectrum of A and on the proper quadratures involving short sums of resolvents. The efficiency of proposed method is demonstrated by numerical examples.

For the first-order differential equation with unbounded operator coefficient in a Banach space, we study the nonlocal problem with integral condition. An exponentially convergent algorithm for the numerical solution of this problem is proposed and justified under the assumption that the operator coefficient A is strongly positive and certain existence and uniqueness conditions are satisfied. The algorithm is based on the representations of operator functions via the Dunford–Cauchy integral along a hyperbola covering the spectrum of A and the quadrature formula containing a small number of resolvents. The efficiency of the proposed algorithm is illustrated by several examples.

This work is devoted to the study of a nonlocal-in-time evolutional problem for the first order differential equation in Banach space. Our primary approach, although stems from the convenient technique based on the reduction of a nonlocal problem to its classical initial value analogue, uses more advanced analysis. That is a validation of the correctness in definition of the general solution representation via the Dunford-Cauchy formula. Such approach allows us to reduce the given existence problem to the problem of locating zeros of a certain entire function. It results in the necessary and sufficient conditions for the existence of a generalized (mild) solution to the given nonlocal problem. Aside of that we also present new sufficient conditions which in the majority of cases generalize existing results.

Problem for the elliptic differential equation with an unbounded operator coefficient in Banach space and integral nonlocal condition is considered. An exponentially convergent algorithm is proposed and justified for the numerical solution of this problem under an assumption that operator coefficient A is strongly positive and some existence and uniqueness conditions are fulfilled. This algorithm is based on the representation of operator functions by a Dunford-Cauchy integral along a hyperbola, enveloping the spectrum of A, and on the proper quadratures involving small number of resolvents. The efficiency of the proposed algorithm is demonstrated on numerical example.

Two-points nonlocal problem for the first order differential evolution equation with an operator coeffcient in a Banach space X is considered. An exponentially convergent algorithm is proposed and justified under the assumption that the operator coefficient is strongly positive and some existence and uniqueness conditions hold. This algorithm leads to a system of linear equations that can be solved by fixed-point iteration. The algorithm provides exponentially convergence in time that in combination with fast algorithms on spatial variables can be efficient for solving such problems. The efficiency of the proposed algorithms is demonstrated through numerical examples.