Stochastic Galerkin Method for Fractional Boundary Value Problems: Convergence Analysis and Numerical Treatment
Abstract
We study two-point fractional boundary value problems with uncertain input data, where randomness may enter through the coefficients and boundary conditions. To quantify the resulting uncertainty in the solution, we employ the generalized polynomial chaos (gPC) framework and develop a stochastic Galerkin formulation of the problem. A particular focus of this work is the convergence analysis of the resulting approximation. Rather than imposing assumptions directly on the stochastic coefficients appearing in the gPC representation, we introduce minimal regularity assumptions on the input data and use them to establish the properties required for the convergence analysis. Based on these results, we prove the convergence of the stochastic Galerkin approximation to the corresponding gPC solution. Numerical experiments are presented to illustrate the theoretical findings and to investigate the influence of random coefficients and boundary conditions on the statistical behavior of the solution.
Related
- Numerical approximation of nonlinear fractional dynamical systems for coastal wave modeling: rigorous analysis of convergence and computational efficiency
- Convergence and error analysis for numerical approximations of stochastic Leray-Lions equations
- Consistency and Convergence of the Backward-Euler Scheme for Stochastic Functional Differential Equations Driven by Fractional Brownian Motion
- Inverse problems with fractional operators: an analytical study and its numerical linear algebra treatment via the GLT theory
- A class of high-order discontinuous-Galerkin methods satisfying infinitely many entropy conditions with provable error estimates and strong convergence for general nonlinear conservation laws
BibTeX
@misc{singh2026stochastic,
title = {{Stochastic Galerkin Method for Fractional Boundary Value Problems: Convergence Analysis and Numerical Treatment}},
author = {Abhishek Kumar Singh and Dhyan Divakar Laad and Vaibhav Mehandiratta and Roland Pulch},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.14669},
eprint = {2609.14669},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.14669},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Stochastic Galerkin Method for Fractional Boundary Value Problems: Convergence Analysis and Numerical Treatment
AU - Abhishek Kumar Singh
AU - Dhyan Divakar Laad
AU - Vaibhav Mehandiratta
AU - Roland Pulch
PY - 2026
DA - 2026/09/13
PB - arXiv
DO - 10.48550/arxiv.2609.14669
UR - https://arxiv.org/abs/2609.14669
AB - We study two-point fractional boundary value problems with uncertain input data, where randomness may enter through the coefficients and boundary conditions. To quantify the resulting uncertainty in the solution, we employ the generalized polynomial chaos (gPC) framework and develop a stochastic Galerkin formulation of the problem. A particular focus of this work is the convergence analysis of the resulting approximation. Rather than imposing assumptions directly on the stochastic coefficients appearing in the gPC representation, we introduce minimal regularity assumptions on the input data and use them to establish the properties required for the convergence analysis. Based on these results, we prove the convergence of the stochastic Galerkin approximation to the corresponding gPC solution. Numerical experiments are presented to illustrate the theoretical findings and to investigate the influence of random coefficients and boundary conditions on the statistical behavior of the solution.
ER -