A Fractal Approximation Method For Inverse Initial-value Problems Of Differential Equations

The initial-value problems (IVP) for differential equations are widespread in a number of diverse fields in science, and are particularly common in engineering. These problems are characterized by a governing differential equation and its associated initial conditions. Numerous methods for solving direct IVP have been known for many decades in the area of differential equation. However, there is a growing interest in a much more difficult problem related to IVP, i.e., inverse IVP. Inverse IVP occurs in the systems governed by differential equations, in which a response has been measured and a characteristic of the system must be computed. In direct problems, complete (or sufficient) information about the characteristics of a particular system is known, and the resulting response of the system is desired. However, in inverse problems, a response is observed and information is desired about the characteristics of the system.Fractal theory is a rising subject whose applications are very broad. It has provided some new ways to many fields in science and engineering. But its applications to the IVP, especially the inverse IVP of differential equation are almost blank. Based on the ideas of fractal image encoding, a fractal approximation method is developed to obtain approximate solutions to an inverse IVP for differential equation in this dissertation. Numerical computational results demonstrate the effectiveness of this method to solve inverse IVP for a class of specific differential equations. Although more research in this area is obviously needed, the proposed scheme appears to provide a viable and effective method for solving inverse IVP of nonlinear differential equations.The main contents described in this dissertation are as follows:Chapter 1 is the introduction, which consists of an overview of the dissertation and summary of the inverse problem.Chapter 2 introduces the basic elements of Fractal Geometry, including fractal image encoding and basic mathematic theory.Chapter 3 introduces the inverse problem. Chapter 4 proposes the new method of solving the inverse IVP of differential equation m and extends the proposed method.The conclusions are presented in Chapter 5, drawing together the results of the previous research chapter, and presenting the direction of future research.