# 数学代写|差分方程作业代写difference equation代考|An Introduction to the Finite Difference Method

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## 数学代写|差分方程作业代写difference equation代考|INTRODUCTION AND OBJECTIVES

Part II introduces the finite difference method (FDM). The chapters in this part focus on producing accurate and robust schemes for second-order parabolic and first-order hyperbolic partial differential equations in two independent variables, usually called $x$ and $t$. The first variable $x$ plays the role of a space coordinate and the second variable $t$ plays the role of time. We model the partial differential equations by approximating the derivatives using divided differences. These latter quantities are defined at so-called discrete mesh points. Having motivated FDM in a generic setting we then apply the resulting finite difference schemes to the one-factor Black-Scholes model in Part III.

In this chapter we investigate the application of FDM to ordinary differential equations (ODEs). An ODE has one independent variable and hence it is conceptually easier to understand and to approximate than equations in two or more variables. In particular, we examine a special kind of problem in this chapter. This is called first-order initial value problems (IVP). They are useful objects of study in their own right and our objective is to approximate them using FDM in order to pave the way for more complex applications later in the book. In particular, the added value is:

• Initial value problems provide the motivation for finite difference schemes that will be used to approximate the time dimension in the Black-Scholes partial differential equation.
• In this chapter we introduce notation that will be used throughout the book. We aim to be as consistent as possible in our use of notation.

We shall also introduce the concept of divided differences and how we use them to approximate the first- and second-order derivatives of real-valued functions of one variable. The chapter should be read and understood before embarking on the other chapters. It is fundamental.

## 数学代写|差分方程作业代写difference equation代考|FUNDAMENTALS OF NUMERICAL DIFFERENTIATION

In this section let us look at a real-valued function of a real variable, as follows:
$$y=f(x)$$
In general we are interested in finding approximations to the first and second derivatives of the function $f$. This is needed because, in general, the form of the function $f$ is unknown and it is thus impossible to calculate its derivatives analytically. To this end, we must resort to numerical approximations. Suppose that we wish to approximate the first derivative of $y$ at some point $a$ (see Figure 6.1) and assume that $h$ is a (small) positive number. The first approximation (called the centred difference formula) is given by
$$f^{\prime}(a) \approx \frac{f(a+h)-f(a-h)}{2 h}$$
Another approximation is called the forward difference formula given by
$$f^{\prime}(a) \approx \frac{f(a+h)-f(a)}{h}$$
Finally, the backward difference formula is given by
$$f^{\prime}(a) \approx \frac{f(a)-f(a-h)}{h}$$
For future work, we use the following notation:
\begin{aligned} &D_{0} f(a) \equiv \frac{f(a+h)-f(a-h)}{2 h} \ &D_{+} f(a) \equiv \frac{f(a+h)-f(a)}{h} \ &D_{-} f(a) \equiv \frac{f(a)-f(a-h)}{h} \end{aligned}

## 数学代写|差分方程作业代写DIFFERENCE EQUATION代考|INTRODUCTION AND OBJECTIVES

• 初值问题为有限差分方案提供了动力，该方案将用于逼近 Black-Scholes 偏微分方程中的时间维度。
• 在本章中，我们将介绍将在整本书中使用的符号。我们的目标是在使用符号时尽可能保持一致。

## 数学代写|差分方程作业代写DIFFERENCE EQUATION代考|FUNDAMENTALS OF NUMERICAL DIFFERENTIATION

$$f^{\prime}(a) \approx \frac{f(a+h)-f(a-h)}{2 h}$$
Another approximation is called the forward difference formula given by
$$f^{\prime}(a) \approx \frac{f(a+h)-f(a)}{h}$$
Finally, the backward difference formula is given by
$$f^{\prime}(a) \approx \frac{f(a)-f(a-h)}{h}$$
For future work, we use the following notation:
\begin{aligned} &D_{0} f(a) \equiv \frac{f(a+h)-f(a-h)}{2 h} \ &D_{+} f(a) \equiv \frac{f(a+h)-f(a)}{h} \ &D_{-} f(a) \equiv \frac{f(a)-f(a-h)}{h} \end{aligned}

## Matlab代写

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