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My research focuses on analytical and numerical study nonlinear differential equations on continuous, discrete and hybrid domain and their application in Ecology and Engineering. In particular, for higher dimension I focus on elliptic BVP and for one dimension I focus on second order BVP. I explore Dirichlet, Neumann, Mixed, Nonlinear boundary conditions. My research is partially funded by AMS Simons Travel Grant 2025- 2027.

Publications [Google Scholar]

Peer Reviewed

  1. A. Acharya, S. Bandyopadhyay, J.T. Cronin, J. Goddard II, A. Muthunayake. "The diffusive Lotka–Volterra competition model in fragmented patches I: Coexistence." Nonlinear Analysis: Real World Applications 70, 103775 (2023). [7 citations]

  2. S. Bandyopadhyay, M. Chhetri, B.B. Delgado, N. Mavinga, R. Pardo. "Maximal and minimal weak solutions for elliptic problems with nonlinearity on the boundary." Electronic Research Archive 30(6), 2121 (2022). [2 citations]

  3. S. Bandyopadhyay, M. Chhetri, B.B. Delgado, N. Mavinga, R. Pardo. "Bifurcation and multiplicity results for elliptic problems with subcritical nonlinearity on the boundary." Journal of Differential Equations 411, 28-50 (2024).

  4. S. Bandyopadhyay, T. Lewis, 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 (01-??), 43-21.

  5. S. Bandyopadhyay. "Solvability of Nonlinear Elliptic Boundary Value Problems." The University of North Carolina at Greensboro (2023).

Submitted

  1. S. Bandyopadhyay, T. Lewis, D. Nichols. "Numerical Approximation and Bifurcation Results for an Elliptic Problem with Superlinear Subcritical Nonlinearity on the Boundary." arXiv preprint arXiv:2506.08808 (2025).

  2. S. Bandyopadhyay, M. Chhetri, B. B. Delgado, N. Mavinga, R. Pardo. "Positive Solutions of Elliptic Systems with Superlinear Nonlinearities on the Boundary." arXiv preprint arXiv:submit/6962407 (2025).

  3. S. Bandyopadhyay, F. A. Çetinkaya, T. Cuchta. "Prüfer Transformation and Spectral Analysis for a Sturm–Liouville-Type Equation." arXiv preprint arXiv:2408.xxxxx (2025). [Submitted]

  4. S. Bandyopadhyay, S.G. Georgiev. "Nonlinear Higher-Order Dynamic Equation with Polynomial Growth and Mixed Boundary Conditions." arXiv preprint arXiv:2506.08808 (2025).

  5. S. Bandyopadhyay, C.J. Kunkel. “Existence Result for Singular Second Order Dynamic Equations with Mixed Boundary Conditions.” arXiv preprint arXiv:2506.16505 (2025).

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​​For manuscripts written with undergraduate students please check the page UG Research.

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.

Research Collaborators

Current Collaborators

Past Collaborators

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With Maria AmaraKristi Onydio

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With Xiang Wan
 

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With Maria Amarakristi and Ayca Cetinkaya
 

Nonlinear Elliptic PDE with Nonlinear Boundary Condition

Collaborators:

  1. Maya Chhetri

  2. Nsoki Mavinga

  3. Rosa Pardo 

  4. Briceyda Delgado

  5. Maria Amarakristi Onydio

 

In nonlinear elliptic PDEs with nonlinear boundary conditions, I have established foundational results for existence of maximal and minimal weak solutions, extending classical theory to systems where nonlinearity appears on boundaries—crucial for modeling chemical reactions, ecology, and combustion theory. My recent work has progressed from scalar equations to coupled systems and finite difference approximations, with current focus on problems where nonlinearity appears both inside the domain and on the boundary, an area with limited analytical exploration.

Related Work

Numerical Analysis of Nonlinear Elliptic PDE with Nonlinear Boundary Condition

Collaborators:

  1. Thomas L Lewis

  2. Dustin Nichols

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My numerical analysis research focuses on elliptic problems using finite difference methods. I have extensively studied sublinear elliptic problems and am currently extending this work to superlinear cases, where the increased nonlinearity complexity presents new computational challenges. My approach combines rigorous mathematical proofs of convergence and stability with MATLAB implementations, ensuring that numerical methods are both practically effective and theoretically sound. This computational work complements my theoretical research, providing a complete analytical and numerical framework for understanding elliptic problems.

Related Work

Application in Math-Ecology

Collaborators:

  1. Mohan Mallick

  2. Ram Baran Verma

  3. Ratnasingham Shivaji (past)

  4. Ananta Acharya (past)

  5. Amila Muthunayake (past)

  6. Jim Cronin (past)

  7. Jerome Goddard II (past)

My research in mathematical ecology focuses on population dynamics in spatially structured environments. I first developed computational methods using quadrature and shooting techniques in Mathematica to study one-dimensional ecological models, revealing complex bifurcation structures and coexistence patterns. Based on these numerical insights, I proved rigorous existence, uniqueness, and stability results for the diffusive Lotka-Volterra competition model in fragmented patches, showing how dispersal-competition tradeoffs enable species coexistence at intermediate patch sizes. Currently, I am investigating optimal control problems for grazing models with nonlinear grazing terms and density-dependent boundary conditions, combining control theory with PDE analysis to develop sustainable harvesting strategies.

Related Work

Nonlinear Differential Equation 

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 Collaborators:

  1. Tom Cuchta

  2. Ayca Cetinkaya

  3. Serena Mauticci

  4. Pasquale Candito

My research in ordinary differential equations centers on the theoretical analysis and spectral properties of nonlinear ordinary differential equations with various boundary conditions. Working with collaborators, I investigate Sturm-Liouville-type problems involving quasi-derivative operators and non-self-adjoint structures, developing novel analytical frameworks such as generalized Prüfer transformations to study oscillatory behavior and eigenvalue properties. My current work extends classical boundary value theory from Dirichlet conditions to more general Sturm-Liouville separated boundary conditions, establishing variational formulations and constructing Green's functions for both discrete and continuous settings. Future directions include exploring nonlinear boundary conditions, which represent a significant theoretical challenge in extending classical Sturm-Liouville theory to more complex physical and mathematical models.

Nonlinear Dynamic Equation

Collaborators:

  1. Tom Cuchta

  2. Ayca Centinkaya

  3. Svetlin Georgiev

  4. Curtis Kunkel (past)

My research in nonlinear dynamic equations on time scales bridges continuous and discrete mathematics, with applications to population modeling and epidemic dynamics. I have investigated singular second-order dynamic equations with mixed boundary conditions, where nonlinearities exhibit singularities requiring specialized analytical techniques. Currently, I am extending this work to higher-order singular problems and exploring nonlinear boundary conditions, advancing theory in this relatively new field that unifies differential and difference equations through Hilger's framework.

Nonlinear Elliptic Partial Dynamic Equation

Collaborators:

  1. Tom Cuchta

  2. Ayca Centinkaya

Looking toward the future, I am developing the nascent field of partial dynamic equations—creating a unified framework for elliptic PDEs on generalized time scales that captures both continuous and discrete behavior. This innovative direction has significant potential for mathematical biology, physical systems with discrete interference, and mixed continuous-discrete dynamics.

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