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My research centers on nonlinear elliptic boundary value problems, especially those in which the nonlinearity appears on the boundary — a setting that arises in chemical reactions, combustion, and population dynamics, and one where the analytical theory is still limited. I study these problems through both analysis and numerical methods, establishing existence, multiplicity, and bifurcation of solutions and developing finite-difference approximations to capture their structure. More recently, I have been extending these boundary-value questions beyond the continuous setting to discrete and hybrid domains through dynamic equations on time scales, which unify the differential and difference cases. A full list of my publications is available on Google Scholar.

If you have trouble accessing any of my papers through the DOI or arXiv links, feel free to email me at sbandyo5@utm.edu for a copy.

Existence, Multiplicity, Bifurcation Theory

Active Collaborators:

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  1. Nsoki Mavinga

  2. Marius N Nkashama

  3. Briceyda Delgado

  4. Maria Amarakristi Onyido

 

My core program concerns nonlinear elliptic problems with nonlinearity on the boundary, which I study through two complementary approaches: the upper–lower solution method, which applies to general nonlinearities and yields existence of maximal and minimal weak solutions, and bifurcation analysis, which I use for superlinear problems. I began with problems that are linear in the domain and nonlinear on the boundary, in the scalar case, and have extended this work from scalar equations to systems and from boundary-only nonlinearity toward problems with nonlinearity both inside the domain and on the boundary. Throughout, the nonlinear boundary condition remains the central thread; I am now extending the program to other types of nonlinearity in this setting.

Related Work

  1. S. Bandyopadhyay, B. B. Delgado, N. Mavinga, and M. A. Onyido, "Existence results for quasimonotone semilinear coupled elliptic systems via sub-supersolution method," Boletín de la Sociedad Matemática Mexicana (accepted, 2025). arXiv:2511.21482. https://arxiv.org/abs/2511.21482
     

  2. S. Bandyopadhyay, M. Chhetri, B. Delgado, N. Mavinga, and R. Pardo, "Positive solutions of elliptic systems with superlinear nonlinearities on the boundary," (under review, 2025). arXiv:2511.04943. https://arxiv.org/abs/2511.04943
     

  3. S. Bandyopadhyay, M. Chhetri, B. B. Delgado, N. Mavinga, and R. Pardo, "Bifurcation and multiplicity results for elliptic problems with subcritical nonlinearity on the boundary," Journal of Differential Equations, 411 (2024), 28–50. https://doi.org/10.1016/j.jde.2024.07.034
     

  4. S. Bandyopadhyay, T. Lewis, and N. Mavinga, "Existence of maximal and minimal weak solutions and finite difference approximations for elliptic systems with nonlinear boundary conditions," Electronic Journal of Differential Equations, 2025 (2025), No. 43, 1–21. https://doi.org/10.58997/ejde.2025.43
     

  5. S. Bandyopadhyay, M. Chhetri, B. B. Delgado, N. Mavinga, and R. Pardo, "Maximal and minimal weak solutions for elliptic problems with nonlinearity on the boundary," Electronic Research Archive, 30 (2022), No. 6, 2121–2137. https://doi.org/10.3934/era.2022107

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Numerical Approximation

Active Collaborators:

  1. Thomas L Lewis

  2. Dustin Nichols

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My numerical analysis research focuses on approximating solutions of elliptic problems using finite difference methods. I have studied both sublinear and superlinear problems. This computational work complements my theoretical research, providing a complete analytical and numerical framework for understanding elliptic problems.

Related Work

  1. S. Bandyopadhyay, T. Lewis, and D. Nichols, "Numerical Approximation and Bifurcation Results for an Elliptic Problem with Superlinear Subcritical Nonlinearity on the Boundary," (under review, 2025). arXiv:2509.08990.https://arxiv.org/abs/2509.08990
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  2. S. Bandyopadhyay, T. Lewis, and N. Mavinga, "Existence of maximal and minimal weak solutions and finite difference approximations for elliptic systems with nonlinear boundary conditions," Electronic Journal of Differential Equations, 2025 (2025), No. 43, 1–21. https://doi.org/10.58997/ejde.2025.43

Nonlinear Elliptic Partial Dynamic Equation

Active Collaborators:
 
Tom Cuchta

A frontier of my program is the study of elliptic PDEs on time scales, which unify continuous and discrete settings. I have begun with the Dirichlet problem and am progressing toward Sturm–Liouville and mixed boundary conditions. A central obstruction is that the notion of an outer normal does not yet exist on time scales, so Neumann and nonlinear boundary conditions — the boundary settings at the heart of the rest of my work — are not yet available in this context. My current effort is aimed at this gap: developing a notion of boundary flux, beginning on lattice domains.

Related Work

  1. S. Bandyopadhyay, F. A. Çetinkaya, and T. Cuchta, "Nonlinear elliptic Dirichlet boundary value problems on time scales," (under review, 2026). arXiv:2602.10335. https://arxiv.org/abs/2602.10335

Other Work

  1. S. Bandyopadhyay, S. G. Georgiev, "Nonlinear Higher-Order Dynamic Equation with Polynomial Growth and Mixed Boundary Conditions," (under review, 2025). arXiv:2506.08808. https://arxiv.org/abs/2506.08808
     

  2. S. Bandyopadhyay, C. J. Kunkel, "Existence Result for Singular Second Order Dynamic Equations with Mixed Boundary Conditions," (under review, 2025). arXiv:2506.16505. https://arxiv.org/abs/2506.16505
     

  3. S. Bandyopadhyay, F. A. Çetinkaya, T. Cuchta, "Prüfer Transformation and Spectral Analysis for a Sturm–Liouville-Type Equation," (under review, 2025). arXiv:2508.12369. https://arxiv.org/abs/2508.12369
     

  4. A. Acharya, S. Bandyopadhyay, J. T. Cronin, J. Goddard II, A. Muthunayake, and R. Shivaji, "The diffusive Lotka–Volterra competition model in fragmented patches I: Coexistence," Nonlinear Analysis: Real World Applications, 70 (2023), 103775. https://doi.org/10.1016/j.nonrwa.2022.103775

Research Collaborators

Current Collaborators​

Past Collaborators

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With Maria AmaraKristi Onyido

With Nsoki Mavinga

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With Xiang Wan
 

With Tom Cuchta

With Dr. Cuchta @MCA2025

I am fortunate to work with Dr. Tom Cuchta from Marshall University as my research mentor through the AMS Simons travel grant. Dr. Cuchta is an Assistant Professor of Mathematics whose research focuses on special functions, difference equations, and time scale calculus. Despite my being new to this field—having not worked on anything similar during my PhD—Dr. Cuchta graciously welcomed me into this research area and has been incredibly resourceful in guiding my development as a researcher.

Since we first met at the SEARCDE conference in Morgantown, West Virginia in November 2024, where he invited me to dinner and we connected over our shared interests, Dr. Cuchta has been instrumental in my research journey. When we began our collaborative project in February 2025, he not only provided expert guidance on the technical aspects of our work but also helped me navigate grant writing and connected me with other researchers and collaborators in the field. His mentorship has been invaluable in helping me establish myself in an entirely new area of mathematics, and his willingness to work with someone inexperienced in the field speaks to his dedication to fostering mathematical research and collaboration.

Member: AMS, MAA, AWM, ISDE

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