Abstract  

In the first project (with Dr. Young You at IU East), Dr. Karki has studied a solution to a semi-linear pseudo-differential equation involving fractional power of Laplacian using a method analogous to the direct method of calculus of variations. They have studied the existence of a weak solution to this equation as a minimizer of a suitable energy-type functional and have also discussed the possibility of regularity of such a weak solution so that it will be a solution to the semi-linear equation.  

In the second project (with Dr. Ashok Aryal, MSU Moorhead), Dr. Karki has constructed a linearly growing finitely many future times in a bounded interval and improved the result of Devore and Zuazua [Recovery of an initial temperature from discrete sampling, 2014] of recovering initial temperature profile from known temperature measurements at a fixed location of a body and at finitely many later times.   

Biographical Statement 

Ramesh Karki, PhD, is Assistant Professor of Mathematics in the school of Natural Science and Mathematics at Indiana University East.  His research interests include: Partial Differential Equations, Dynamical Systems, and the Scholarship of Teaching and Learning.  His publications include: R. Karki, Y. You,  A solution to fraction order semi-linear equations using variational method, Mathematics in Applied Sciences and Engineering, Volume 1, Number 4, 275-285 (Supported by: Summer Faculty Fellowship 2020). Also, R. Aceska, A. Arsie, R. Karki,  On near-optimal time sampling for initial data best approximation,  Le  Matematiche  (Catania) 72 (2019), Number 1, 173-190.  And, R. Karki,  Sobolev  Gradients  & Applications to Nonlinear Pseudo-differential Equations, Neural, Parallel &  Scientific  Computations, Vol 22 (2014), Number 3, 359-373, September 2014.  His submitted manuscripts include: A. Aryal, R. Karki,  An approach for recovering initial temperature via bounded linear sampling, Journal of Mathematical Analysis and Applications (Supported by: Summer Faculty Fellowship 2020).  And, R. Karki,  On Approximating Initial Data in Some Linear Evolution Equations Involving Fractional Laplacian, Mathematics in Applied Science and Engineering [under revision]  (Supported by: Summer Faculty Fellowship 2020)