My research focuses on nonlinear differential equations with applications to complex natural phenomena ranging from population dynamics to chemical reactions. I work in three interconnected areas that bridge pure mathematical theory with computational applications and real-world modeling.

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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 and numerical exploration.

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My work in nonlinear dynamic equations on time scales bridges continuous and discrete mathematics, with applications to population modeling, epidemic dynamics, and control systems. I investigate singular second-order dynamic equations with mixed boundary conditions and am currently extending this to higher-order cases and problems with nonlinear boundary conditions, work that advances theory in this relatively new field established through Hilger's 1988 framework.

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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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