On the convergence of a numerical scheme for a boundary controlled 1D linear parabolic PIDE
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arXiv
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866914056339718144 |
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| author | Chatterjee, Soham Natarajan, Vivek |
| author_facet | Chatterjee, Soham Natarajan, Vivek |
| contents | We consider an 1D partial integro-differential equation (PIDE) comprising of an 1D parabolic partial differential equation (PDE) and a nonlocal integral term. The control input is applied on one of the boundaries of the PIDE. Partitioning the spatial interval into $n+1$ subintervals and approximating the spatial derivatives and the integral term with their finite-difference approximations and Riemann sum, respectively, we derive an $n^{\rm th}$-order semi-discrete approximation of the PIDE. The $n^{\rm th}$-order semi-discrete approximation of the PIDE is an $n^{\rm th}$-order ordinary differential equation (ODE) in time. We establish some of its salient properties and using them prove that the solution of the semi-discrete approximation converges to the solution of the PIDE as $n\to\infty$. We illustrate our convergence results using numerical examples. The results in this work are useful for establishing the null controllability of the PIDE considered. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_20960 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the convergence of a numerical scheme for a boundary controlled 1D linear parabolic PIDE Chatterjee, Soham Natarajan, Vivek Systems and Control We consider an 1D partial integro-differential equation (PIDE) comprising of an 1D parabolic partial differential equation (PDE) and a nonlocal integral term. The control input is applied on one of the boundaries of the PIDE. Partitioning the spatial interval into $n+1$ subintervals and approximating the spatial derivatives and the integral term with their finite-difference approximations and Riemann sum, respectively, we derive an $n^{\rm th}$-order semi-discrete approximation of the PIDE. The $n^{\rm th}$-order semi-discrete approximation of the PIDE is an $n^{\rm th}$-order ordinary differential equation (ODE) in time. We establish some of its salient properties and using them prove that the solution of the semi-discrete approximation converges to the solution of the PIDE as $n\to\infty$. We illustrate our convergence results using numerical examples. The results in this work are useful for establishing the null controllability of the PIDE considered. |
| title | On the convergence of a numerical scheme for a boundary controlled 1D linear parabolic PIDE |
| topic | Systems and Control |
| url | https://arxiv.org/abs/2509.20960 |