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Numerical method for abstract Cauchy problem with nonlinear nonlocal condition

Volodymyr Makarov, Dmytro Sytnyk, Vitalii Vasylyk

TL;DR

The paper addresses solving a nonlocal first-order problem $u'(t)+A u(t)=0$ in a Banach space, with nonlocal condition $u(-1)-g(u(\cdot))=u_0$, by reducing it to a mild Hammerstein equation $u(t)=T(A,t)[u_0+g(u(\cdot))]$ where $T(A,t)=e^{-A(t+1)}$. It develops a fully discrete, parallelizable scheme using a Dunford-Cauchy integral-based sinc-quadrature to approximate the operator exponential and Chebyshev-Gauss-Lobatto collocation with a modified Hermite-Fejér basis to solve the nonlinear discrete system via a Banach fixed-point iteration; a resolvent correction ensures exponential convergence of the quadrature. The error analysis combines quadrature, interpolation, and fixed-point errors, yielding explicit bounds with exponential convergence in the quadrature size $N$ and collocation size $n$, corroborated by numerical experiments on a second-order elliptic operator. The framework is suitable for multi-core architectures and can be extended to more general nonlinear nonlocal problems on Banach spaces.

Abstract

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 $A$.

Numerical method for abstract Cauchy problem with nonlinear nonlocal condition

TL;DR

The paper addresses solving a nonlocal first-order problem in a Banach space, with nonlocal condition , by reducing it to a mild Hammerstein equation where . It develops a fully discrete, parallelizable scheme using a Dunford-Cauchy integral-based sinc-quadrature to approximate the operator exponential and Chebyshev-Gauss-Lobatto collocation with a modified Hermite-Fejér basis to solve the nonlinear discrete system via a Banach fixed-point iteration; a resolvent correction ensures exponential convergence of the quadrature. The error analysis combines quadrature, interpolation, and fixed-point errors, yielding explicit bounds with exponential convergence in the quadrature size and collocation size , corroborated by numerical experiments on a second-order elliptic operator. The framework is suitable for multi-core architectures and can be extended to more general nonlinear nonlocal problems on Banach spaces.

Abstract

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 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 .
Paper Structure (6 sections, 4 theorems, 45 equations, 2 figures, 1 table)

This paper contains 6 sections, 4 theorems, 45 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Alternative representation of given problem
  3. Discretization
  4. Solution of discretized problem
  5. Error analysis
  6. Conclusions and future work

Key Result

Theorem 1

Assume that $A$ is a linear sectorial operator with the densely defined domain and $v\in D(A^\alpha)$. Let then the error $\eta_N(t)v \equiv \|T(A,t)v - T_N(A,t)v\|$ satisfies the estimate with some positive constant $c$ independent on $A$, $v$, $\alpha$ and $t$.

Figures (2)

  • Figure 1: Graph of $u_0(x)$ for the different values of $\mu$.
  • Figure 2: Graph of experimental error Err as a function of the number of collocation points $n$, drawn in the logarithmic scale for $N=4,8,16,32,64,128,256,512$.

Theorems & Definitions (9)

  • Remark 2.1
  • Theorem 1
  • Theorem 2
  • proof
  • Lemma 3
  • proof
  • Theorem 4
  • Remark 5.1
  • Example 5