ABSTRACT
Initial value problems are the first class of differential equation problems we are interested in solving. Ordinary differential equations are complicated by the fact they do not provide sufficient information to solve them, normally. The elementary method of solving initial value problems is the Euler method. The Euler method is the elementary form of what are broadly called Runge–Kutta methods. Runge–Kutta methods, like Newton–Cotes methods, use an increasing number of calculation points to estimate the position of a function from its first derivative. The fourth-order Runge–Kutta method is widely considered the standard approach to numerical solutions of initial value problems. Linear multistep methods were developed to accommodate the need for fewer function evaluations. The implementation of adamsbashforth is itself simplistic, like the implementations of Euler and the Runge–Kutta methods. The heat equation is a classic example of a partial differential equation. The wave equation is another classic example of a partial differential equation.
