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Generalized capillary-rise models: existence and fast solvers in integral Hölder spaces

Josefa Caballero, Łukasz Płociniczak, Kishin Sadarangani

TL;DR

The paper addresses nonlinear Volterra integral equations that generalize the capillary-rise model by accommodating nonsmooth kernels and nonlinearities. It develops existence theory in integral Hölder spaces $J_{\alpha,\beta}[0,1]$ and introduces two collocation solvers: a robust piecewise linear method for low-regularity and a Legendre-node spectral method for smooth solutions, with rigorous error analysis. A key contribution is sharp interpolation bounds in $J_{\alpha,\beta}$ that enable convergence proofs for the collocation schemes, together with numerical experiments confirming the theory and demonstrating fast performance. This work provides a rigorous framework and practical fast solvers for weakly regular nonlinear Volterra equations, with potential impact on capillary-rise modeling and other memory-dependent processes.

Abstract

We study a class of nonlinear Volterra integral equations that generalize the classical capillary rise models, allowing for nonsmooth kernels and nonlinearities. To accommodate such generalities, we work in two families of function spaces: spaces with prescribed modulus of continuity and integral Hölder spaces. We establish existence results for solutions within the integral Hölder space framework. Furthermore, we analyze the behavior of linear interpolation in these spaces and provide, for the first time, sharp error estimates, demonstrating their optimality. Building on this foundation, we propose a piecewise linear collocation method tailored to solutions in integral Hölder spaces and prove its convergence. For problems admitting smoother solutions, we develop an efficient spectral collocation scheme based on Legendre nodes. Finally, several numerical experiments illustrate the theoretical results and highlight the performance of the proposed methods.

Generalized capillary-rise models: existence and fast solvers in integral Hölder spaces

TL;DR

The paper addresses nonlinear Volterra integral equations that generalize the capillary-rise model by accommodating nonsmooth kernels and nonlinearities. It develops existence theory in integral Hölder spaces and introduces two collocation solvers: a robust piecewise linear method for low-regularity and a Legendre-node spectral method for smooth solutions, with rigorous error analysis. A key contribution is sharp interpolation bounds in that enable convergence proofs for the collocation schemes, together with numerical experiments confirming the theory and demonstrating fast performance. This work provides a rigorous framework and practical fast solvers for weakly regular nonlinear Volterra equations, with potential impact on capillary-rise modeling and other memory-dependent processes.

Abstract

We study a class of nonlinear Volterra integral equations that generalize the classical capillary rise models, allowing for nonsmooth kernels and nonlinearities. To accommodate such generalities, we work in two families of function spaces: spaces with prescribed modulus of continuity and integral Hölder spaces. We establish existence results for solutions within the integral Hölder space framework. Furthermore, we analyze the behavior of linear interpolation in these spaces and provide, for the first time, sharp error estimates, demonstrating their optimality. Building on this foundation, we propose a piecewise linear collocation method tailored to solutions in integral Hölder spaces and prove its convergence. For problems admitting smoother solutions, we develop an efficient spectral collocation scheme based on Legendre nodes. Finally, several numerical experiments illustrate the theoretical results and highlight the performance of the proposed methods.

Paper Structure

This paper contains 14 sections, 10 theorems, 105 equations, 3 figures.

Table of Contents

  1. Introduction
  2. The integral equation
  3. Function space preliminaries
  4. Existence of solutions
  5. Example.
  6. Collocation numerical methods
  7. Linear interpolation in the $J_{\alpha,\beta}$ space
  8. Example.
  9. Linear collocation scheme
  10. Spectral collocation scheme
  11. Numerical illustration
  12. Non-smooth solutions
  13. Smooth solutions
  14. Conclusion

Key Result

Lemma 1

Suppose that $x\in C[0,1]$, then for $0<\sigma \leq 1$ we have

Figures (3)

  • Figure 1: Exact non-smooth solutions chosen for testing the piecewise linear collocation scheme: \ref{['eqn:NumericalHolder']} (solid line), \ref{['eqn:NumericalLogarithm']} (dashed line).
  • Figure 2: $L^\infty$ error of the linear collocation scheme applied to \ref{['eqn:NumericalLogarithm']}. On the left: Hölder solution \ref{['eqn:NumericalHolder']}, on the right: logarithmic solution \ref{['eqn:NumericalLogarithm']}.
  • Figure 3: $L^\infty$ error of the spectral collocation scheme applied to \ref{['eqn:NumericalSmooth']}. On the left: smooth solution \ref{['eqn:NumericalSmooth']}, on the right: logarithmic solution \ref{['eqn:NumericalLogarithm']}.

Theorems & Definitions (28)

  • Definition 1
  • Definition 2
  • Remark 1
  • Definition 3
  • Remark 2
  • Remark 3
  • Lemma 1
  • proof
  • Proposition 1
  • proof
  • ...and 18 more