Mathematical Modeling and Exact Solutions for Non-Newtonian Nano uids and Financial Mathematics

Abstract

The main focus of the thesis is to provide exact solutions to the mathematical models that arise in the realm of non-Newtonian nano uid ow over a at surface and option pricing in nancial mathematics. Exact solutions are always preferred over numerical and experimen tal solutions because these solutions e ciently and accurately demonstrate the mechanism of many complex non-linear formulations. The exact solution shows what variables are im portant in the model and how important they are relative to the others in the solution. This allows researchers to see the e ect of the inputs on the outputs (their inuence on the output and the extent of that inuence). The Lie symmetry method for establishing the transformations leaving a system of ordinary di erential equations (ODEs) or partial di erential equations (PDEs) invariant is one such method that can be used to nd the exact solutions. Lies method o ers a bene t in its applicability in dealing with non-linear di erential equations. It identi es the group transformations that leave a given di erential equation unchanged, thereby determining its symmetries. Speci cally, these symmetries map one solution to another. The investigations presented in the thesis are focused on obtaining the exact solutions through the Lie symmetry method for problems of nano uid ow and the associated processes of heat transfer and the mathematical models arising in nancial mathematics. Many researchers have handled the problems of nano uid ow in the past. The focus of these studies will be on both the aspects of uid ow and heat transfer. They encompass diverse ow geometries, liquid types, boundary conditions, external in uences, surface motion, and more. Moreover, in mathematical nance, the option-pricing theory (like bond pricing) depends on the condition that a standard Brownian motion models the stock prices. The option-pricing value can be uniquely determined by mathematically formulating the problem using equations that incorporate randomness. Under certain limiting assum-ptions these models reduce to parabolic PDEs with variable coe cients. The valuation of an option is the most common derivative contract in modern nancial markets. Within the domain of nano uid ow models incorporating heat transfer processes, the current investigation focuses on the theoretical examination of nano uid ow over surfaces that are either porous or rigid. The models include the ow of non-Newtonian nano uids, including permeability, applied uniform magnetic eld, internal heat source/sink, and lin ear thermal radiation. Two non-Newtonian nano uid models, namely, the third-grade and power-law models, are utilized here. The motion of the uid is induced by the sudden ap plication of a force at the surface of the plate. The basic equations governing the system are derived by applying the principles of mass, momentum, and energy conservation. After that, the classical Lie symmetry method is used to identify all conceivable symmetries for foundational equations. Then, the closed-form solutions invariant under them are obtained and graphically presented. The graphs allow for understanding the physical behavior of emerging parameters in the ow and heat transfer mechanisms. Moreover, the thesis gives the exact solutions for the model of the American put option under the CEV model. The governing (1 + 1) parabolic partial di erential equation is solved using the Lie symmetry method. The symmetry generators are then utilized to nd group-invariant solutions. Pa rameters a ecting the premium of the options are observed and analyzed for the behavior of the expected payo .