T. Gnana Bhaskar

Set Differential Equations and Flow Invariance

The study of Set Differential Equations is initiated in the metric space of nonempty, compact, convex subsets of , endowed with the Hausdorff metric. The existence of a generalized solution for the associated initial value problems is proved when the function involved does not satisfy any continuity assumptions. Utilizing the ideas of nonsmooth analysis, a proximal aiming condition is employed to investigate the weak and strong invariance for these solutions.

Stability criteria for set differential equations

Notions of stability for the solutions of set differential equations, using Lyapunov-like functions are considered. Criteria for the equistability, equiasymptotic stability, uniform and uniform asymptotic stability are presented.

Comparison theorem for a nonlinear boundary value problem on time scales

We prove a comparison theorem for the lower and upper solutions of a nonlinear two point boundary value problem on time scales. This theorem plays an important role in the development of the method of generalized quasilinearization on time scales.

Solutions of initial Value Problems Associated with a Pair of Mixed Linear Ordinary Differential Equations

= C;=O (&xl&) = 0,x defined on an interval I, = [a, b], and My = CT=, QJ&y/&) = B,y defined on the adjacent interval Z2 = [b, c], where 01, 6, are constants and the functions x, y are required to satisfy certain mixed conditions at the interface point

Set differential systems and vector lyapunov functions

We prove a comparison result in terms of vector Lyapunov-like functions relative to a set differential system. Using this, we provide sufficient conditions in terms of vector Lyapunov-like functions for the stability properties of the trivial solutions of set differential systems.

Computation of Green's matrices for boundary value problems associated with a pair of mixed linear regular ordinary differential operators

An algorithm for the computation of Greens matrices for boundary value problems associated with a pair of mixed linear regular ordinary differential operators is presented and two examples from the studies of acoustic waveguides in ocean and transverse vibrations in nonho- mogeneous strings are discussed.

New uniqueness results for fractional differential equations

We develop the Krasnoselskii–Krein type of uniqueness theorem for an initial value problem of the Riemann–Liouville type fractional differential equation which involves a function of the form f (t, x(t), D q−1 x(t)), for 1<q<2 and establish the convergence of successive approximations. We prove a few other uniqueness theorems.

Nonuniform stability and boundedness criteria for set differential equations

Notions of nonuniform stability and boundedness criteria for the solutions of Set Differential Equations (SDEs), using Lyapunov-like functions, under less restrictive assumptions are studied in this article.

Existence and uniqueness results for fuzzy linear differential-algebraic equations

Abstract We discuss the existence results for a fuzzy initial value problem of linear differential-algebraic equations and provide an explicit representation for the solution. A few illustrative examples are given.

Heatlet approach to diffusion equation on unbounded domains

We develop Heatlets, the fundamental solutions of heat equation using wavelets, for numerically solving inhomogeneous and homogeneous initial value problems of diffusion equation on unbounded domains. Unlike finite difference and finite element methods, diffusion into an infinite medium is satisfied analytically, avoiding the need for artificial boundary conditions on a finite computational domain. The approach is applied to a number of examples and the numerical results confirm the theoretical findings.