Graduation Year

2013

Document Type

Dissertation

Degree

Ph.D.

Degree Granting Department

Mathematics and Statistics

Major Professor

Gangaram S. Ladde

Abstract

Two of the most well-known nonlinear methods for investigating nonlinear dynamic processes in sciences and engineering are nonlinear variation of constants parameters and comparison method. Knowing the existence of solution process, these methods provide a very powerful tools for investigating variety of problems, for example, qualitative and quantitative properties of solutions, finding error estimates between solution processes of stochastic system and the corresponding nominal system, and inputs for the designing engineering and industrial problems. The aim of this work is to systematically develop mathematical tools to undertake the mathematical frame-work to investigate a complex nonlinear nonstationary stochastic systems of differential equations.

A complex nonlinear nonstationary stochastic system of differential equations are decomposed into nonlinear systems of stochastic perturbed and

unperturbed differential equations. Using this type of decomposition, the fundamental properties of solutions of nonlinear stochastic unperturbed

systems of differential equations are investigated(1). The fundamental properties are used to find the representation of solution process of nonlinear stochastic complex perturbed system in terms of solution process of nonlinear stochastic unperturbed system(2).

Employing energy function method and the fundamental properties of It\^{o}-Doob type stochastic auxiliary system of differential equations, we

establish generalized variation of constants formula for solution process of perturbed stochastic system of differential equations(3). Results

regarding deviation of solution of perturbed system with respect to solution of nominal system of stochastic differential equations are developed(4). The obtained results are used to study the qualitative properties of perturbed stochastic system of differential equations(5). Examples are given to illustrate the usefulness of the results.

Employing energy function method and the fundamental properties of It\^{o}-Doob type stochastic auxiliary system of differential equations, we establish

generalized variational comparison theorems in the context of stochastic and deterministic differential for solution processes of perturbed stochastic system of differential equations(6). Results regarding deviation of solutions with respect to nominal stochastic system are also developed(7). The obtained results are used to study the qualitative properties of perturbed stochastic system(8). Examples are given to illustrate the usefulness of the results.

A simple dynamical model of the effect of radiant flux density and CO_2

concentration on the rate of photosynthesis in light, dark

and enzyme reactions are analyzed(9). The coupled system of dynamic equations are solved numerically for some values

of rate constant and radiant flux density. We used Matlab to solve the

system numerically. Moreover, with the assumption that dynamic model of CO_2 concentration

is studied.

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