Using Schauder's Theorem To Approximate Solutions To Differential Equations (Essay Sample)

USING SCHAUDER'S THEOREM TO APPROXIMATE SOLUTIONS TO DIFFERENTIAL EQUATIONS
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Research summary
The aim of this research proposal is thus to develop a method to be used in approximating differential equation solutions after the solution has been proven to exist using the Schauder’s fixed point theorem. This proposal outlines the introduction to the research, which gives the background information of the intended project. This is followed by the literature review, and then the mechanism outlining how the proposed research is to be carried out. The theorem assets that if K is a nonempty convex subset arising from a Hausdoff topological vector space V, and T maps continuously on K such that T(K) is within the compact subset of K, then there is a fixed point at T.
Introduction
One of the methods of satisfying that there exists a solution to a given differential equation is through the construction of a function that will give solutions at fixed points, and then apply the theorem of fixed points in guaranteeing the existence of such fixed points. The Schauder’s fixed point theorem is one such particular application with reference to differential equations and its results. Although the theorems used are able to show that the solutions do exist, there is little classical proof from their insights on how such solutions can actually be obtained. The aim of this research proposal is thus to develop a method to be used in approximating differential equation solutions after the solution has been proven to exist using the Schauder’s fixed point theorem.
Literature review
The project in question has not been researched upon for publication. However, related areas such as fixed point methods for differential equations together with the numerical methods have been well researched. Xu and Guo (2012) carried out a study on the fixed point analytical method for nonlinear differential equations, in which they constructed a contractive map replacing the nonlinear differential equation, thus forming a series of linear differential equations. Yan (2014) on the other hand researched on Numerical methods of differential equations, in which the initial value theorem (IVP) of ordinary differential equation (ODE) was studied.
Pursuant to this method, there is much background research required in relation to fixed point methods for the differential equations. In addition, the knowledge in topology, fixed point theory and convexity theory will be useful in forming the theoretical basis for the proposed approach. The research will also require findings into the current approximation methods for obtaining solutions to differential equations so as to place this approach in the current context of approximation theory (Gilbarg & Trudinger, 1998).
Proposed research
This research will apply the approximation method which uses a proof of Schaud...