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Numerical Solution of Nonclassical Boundary Value Problems

Paola Boito, Yuli Eidelman, Luca Gemignani

TL;DR

The paper develops a mixed polynomial-rational expansion framework to solve first-order evolution problems with nonlocal integral boundary conditions in Banach spaces. It proves a convergent, operator-based representation of the solution and derives a practical algorithm (Algorithm 1) for computing accurate approximations by combining a polynomial Bernoulli-like component with a rational tail that requires solving shifted operator equations. The approach is validated through finite-dimensional matrix experiments and a model one-dimensional parabolic PDE, with two discretization paths: a purely numerical spatial discretization and a functional approach that applies the expansion directly to the infinite-dimensional operator. Results indicate the method is robust and flexible, offering adaptive control over accuracy and clear pathways for parallelization, while highlighting conditioning challenges and the potential benefits of symbolic-numeric hybrids. The work lays groundwork for extending to higher dimensions and more complex operators, with avenues for improved residual estimates and acceleration schemes.

Abstract

We provide a new approach to obtain solutions of certain evolution equations set in a Banach space and equipped with nonlocal boundary conditions. From this approach we derive a family of numerical schemes for the approximation of the solutions. We show by numerical tests that these schemes are numerically robust and computationally efficient.

Numerical Solution of Nonclassical Boundary Value Problems

TL;DR

The paper develops a mixed polynomial-rational expansion framework to solve first-order evolution problems with nonlocal integral boundary conditions in Banach spaces. It proves a convergent, operator-based representation of the solution and derives a practical algorithm (Algorithm 1) for computing accurate approximations by combining a polynomial Bernoulli-like component with a rational tail that requires solving shifted operator equations. The approach is validated through finite-dimensional matrix experiments and a model one-dimensional parabolic PDE, with two discretization paths: a purely numerical spatial discretization and a functional approach that applies the expansion directly to the infinite-dimensional operator. Results indicate the method is robust and flexible, offering adaptive control over accuracy and clear pathways for parallelization, while highlighting conditioning challenges and the potential benefits of symbolic-numeric hybrids. The work lays groundwork for extending to higher dimensions and more complex operators, with avenues for improved residual estimates and acceleration schemes.

Abstract

We provide a new approach to obtain solutions of certain evolution equations set in a Banach space and equipped with nonlocal boundary conditions. From this approach we derive a family of numerical schemes for the approximation of the solutions. We show by numerical tests that these schemes are numerically robust and computationally efficient.
Paper Structure (5 sections, 1 theorem, 67 equations, 9 figures, 2 tables, 1 algorithm)

This paper contains 5 sections, 1 theorem, 67 equations, 9 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. The Abstract Nonlocal Differential Problem
  3. The matrix case
  4. The model one-dimensional problem for parabolic equation
  5. Conclusion and Future Work

Key Result

Theorem 1

Assume that all the complex numbers are regular points of the operator $A$ and there is a constant $C>0$ such that Also, suppose that $f\in D(A^2)$. Then the problem (l1), (l2) has a unique solution which is given by the formula with Moreover for any $n\ge0$ under the additional assumption $f\in D(A^{2n+2})$ the solution $v(t)$ satisfies the formula with and where $B_m(t)$ are the well-know

Figures (9)

  • Figure 1: Illustration of the measured error \ref{['errmeas']} for Test 1 with $tol=1.0e\!-12$ and $m=10$ for different values of $n\in \{1,2,3,4\}$.
  • Figure 2: Illustration of the measured error \ref{['errmeas']} for Test 2 with $tol=1.0e\!-12$, $m=10$, $n=1$, $a_0=a_1=1, a_2=1+1.0e\!-4$ in (a) and $a_0=a_1=1, a_2=1+1.0e\!-6$ in (b).
  • Figure 3: Illustration of the measured error \ref{['errmeas']} for Test 1 with $a_0=a_1=a_2=1$, $tol=1.0e\!-12$ and $m=10$ for different values of $n\in \{1,2,3,4\}$.
  • Figure 4: Error plots generated by the power-based method and our algorithm applied or solving Test 1 with $a_0=a_1=a_2=1$, $tol=1.0e\!-12$, $n=2$ and $m=30$.
  • Figure 5: Illustration of the measured error \ref{['errmeas']} for Test 3 with $\sigma=1$, $\gamma=100$, $tol=1.0e\!-12$, $m=10$, $n=1$ in (a) and $n=2$ in (b).
  • ...and 4 more figures

Theorems & Definitions (2)

  • Theorem 1
  • proof