Numerical Solution of Ordinary Differential Equations
Picard s Method, Euler s Method, Taylor s Method, Runge-Kutta Methods, Predictor-Corrector Methods, Milne s Method, AdamsMoulton Formula, Stability in the Solution of Ordinary Differential Equations.
A physical situation concerned with the rate of change of one quantity with respect to another gives rise to a differential equation.
Consider the first order ordinary differential equation
with the initial condition
Many analytical techniques exist for solving such equations, but these methods can be applied to solve only a selected class of differential equations.
However, a majority of differential equations appearing in physical problems cannot be solved analytically. Thus it becomes imperative to discuss their solution by numerical methods.
In numerical methods, we do not proceed in the hope of finding a relation between variables but we find the numerical values of the dependent variable for certain values of independent variable.
It must be noted that even the differential equations which are solvable by analytical methods can be solved numerically as well.
Problems in which all the conditions are specified at the initial point only are called initial-value problems. For example, the problem given by eqns. (1) and (2) is an initial value problem.
Problems involving second and higher order differential equations, in which the conditions...
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