• Nonlinear Local and Nonlocal Boundary Value Problems: Studying how solutions to ODEs behave and how to find them using both traditional and integral-type boundary conditions.
• Numerical Methods for Singular Boundary Value Problems: Designing and implementing efficient numerical schemes that can handle singularities at boundary or interior points, ensuring stability and convergence.
• Wavelet Methods: Using wavelet techniques to solve differential equations numerically, taking advantage of their ability to work at different levels of detail and focus on specific areas.
• Compact Finite Difference Methods: Developing high-order compact schemes that offer improved accuracy while preserving computational efficiency, especially in stiff or singular problems.
• Population Balance Models: Using and improving numerical methods for population balance equations, which are equations that describe how particles behave in systems like aerosols, crystallizers, and bioreactors.
• Dr. Julee Shahni: Numerical Methods for Singular Boundary Value problems: (2018-2023): December (2023) Awarded
• Nirupam Sahoo: Numerical Solutions of singular differential equations with local and nonlocal boundary conditions: (2021-2025): August (2025) Awarded
• Nikita Saha: Delay-Pantograph Singular Differential equation and nonlocal singular boundary value problems: (Joined 2021): July (2026) Awarded
• Anupam: Wavelet Methods for Singular Fractional Differential Equations (Joined 2024): Ongoing [External]
• Vipul: PINN and Wavelet Methods for Differential Equations and Integro-Differental Equation (Joined 2024): Ongoing

Ordinary differential equations (ODEs) model many boundary value problems encountered in physical sciences and applied mathematics, subject to two-point boundary conditions. While solving linear boundary value problems can often be accomplished using classical methods, the exact solution of nonlinear boundary value problems (BVPs) is, in general, much more challenging. The complexity increases significantly when the BVPs are also singular in nature, i.e., when the differential equation or the boundary conditions exhibit singular behavior at one or more points within the domain. These singularities make the problem mathematically intricate and numerically unstable, requiring specialized techniques for both analysis and computation.

My primary research interests lie in the numerical study of nonlinear singular differential equations, which frequently arise in diverse scientific fields, such as fluid dynamics, chemical kinetics, astrophysics, and population biology. These problems are particularly compelling because their solutions often exhibit non-trivial behavior near the singular points, necessitating the development of robust and accurate numerical schemes.

Developing and analyzing numerical and analytical methods for such nonlinear singular BVPs is my particular focus. My work includes exploring both local and nonlocal boundary conditions, as these arise in various models of physical phenomena, such as heat conduction with memory effects or diffusion with spatial constraints.

Key areas of my research include:

Through this research, I aim to contribute to both the theoretical understanding and the practical computation of complex differential systems, particularly those characterized by nonlinearity and singularity.

PhD supervision record (submitted or awarded):