{'fraction_non_alphanumeric': 0.04485887096774194, 'fraction_numerical': 0.012600806451612902, 'mean_word_length': 3.7716346153846154, 'pattern_counts': {'":': 0, '<': 1, '<?xml version=': 0, '>': 1, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
CHAPTER I - NUMERICAL METHODS FOR THE SOLUTION OF DIFFERENTIAL EQUATIONS

where f(y, t) = [f1(y, t), .., fn(y, t)] ∈ R is a vector valued function commonly referred to as a vector field. Thus for every y ∈ R it yields a time dependent vector f(y, t) in R. The fundamental theorem of ODE tells us that the equation (1.1) has a unique solution both forward and backward in time when the value of y(t) is specified at a certain time say t0, so that y(t0) = y 0 is given. In most applications in applied mathematics we wish to solve (1.1) forward in time i.e. for t ≥ t0. In that case we say the condition y(t0) = y is the initial condition for the differential equation (1.1). An example of this occurs in fluid flow. When we place a light particle such as a leaf in a stream it will be carried along by the fluid. In that case f(y, t) is the fluid velocity at the position y at time t, and y is the position we put the leaf at the initial time t0. We shall also be interested in solving differential equations backward in time i.e. for t ≤ t0. In that case the condition y(t0) = y 0 is called the terminal condition for the equation (1.1). Such problems come up in control theory, a subject of which mathematical finance is a part. A familiar example of such a problem for an infinite dimensional ODE is the problem of finding the value of an option such as a call option. The value of the option as a function of the stock price S is known at the expiration date T > 0 of the option, and the bank wishes to compute the value of the option today which is time t = 0 < T say. In the Black-Scholes (BS) theory one finds the value of the option by solving an infinite dimensional linear ODE like (1.1) backward in time until time t = 0 with the terminal condition given by the option price at the expiration date T . It is sometimes possible to solve the differential equation (1.1) explicitly. The most common situation where that occurs is when the vector field f(y, t) is linear so that