Αποτελέσματα Αναζήτησης
n-th ORDER INITIAL-VALUE PROBLEM An n-th order initial-value problem consists of an n-th order differential equation F(x,y,y0,y00,...,y(n))=0 together with n (initial) conditions of the form y(c)=k0,y0(c)=k1,y00(c)=k2,...,y(n−1)(c)=kn−1. where c and k0,k1,...,kn−1 are given numbers.
Initial Value Problems. 1 Euler’s Explicit Method (section 10.2.1) Definition . By a first order initial value problem, we mean a problem such as. dy. = f (x;y) dx. y(a) is given. in which we are looking a function y(x) that satisfies these condition. Most IVP’s cannot be solved.
An initial value problem (IVP) in one dimension takes the form. y0 = f(t; y); y(t0) = y0: Typically, we consider solving the ODE forward in `time' (the independent variable), in which case the value y(t) depends on the solution at previous times.
Antiderivatives and Initial Value Problems Definition: An antiderivative of a function f on an interval [a,b] is another function F such that F0(x)=f(x) for all x in [a,b]. Examples: 1. An antiderivative of f(x)=2x is F(x)=x2. 2. Another antiderivative of f(x)=2x is F(x)=x2 +1. 3. There are lots of antiderivatives of f(x)=2x which look like F ...
The equation u0 = f(u;t) starts from an initial value u(0). The key point is that the rate of change u0 is determined by the current state u at any moment t. This model of reality, where all the history is contained in the current state u(t), is a tremendous success throughout science and engineering.
Boundary-value problems, like the one in the example, where the boundary condition consists of specifying the value of the solution at some point are also called initial-value problems (IVP).
i 1. All initial-value problems are solved by integrating forward in x, but there are two main types of numerical procedure: one-step methods: yi+1 depends only on yi. multi-step methods: yi+1 may depend on yi, yi–1, yi–2, .... y new point. new point.
