Numerical Method - Old Questions

10. Explain about boundary value problem with example? Differentiate it with initial value problem

Boundary Value Problem

Consider the following linear second order differential equation,

Suppose we are interested in solving this differential equation between the values x = a & z = b. Hence a & b are two values such that a<b. Let us divide the interval [a, b] into n equal subintervals of length h each.

Let x₀, x₁,.....,x_n be the pivotal points and a & b are called the boundary points. Solving the differential equation means finding the values of y₀, y₁, ....., y_n. Suppose y₀ and y_n are given. That is the solution values at the boundary points are given. Then the differential equation is called a Boundary Value Problem. So, the following is the general form of a boundary value problem.

Difference between Initial value and Boundary value problem

An initial value problem has all of the conditions specified at the same value of the independent variable in the equation (and that value is at the lower boundary of the domain, thus the term “initial” value). On the other hand, a boundary value problem has conditions specified at the extremes of the independent variable. For example, if the independent variable is time over the domain [0,1], an initial value problem would specify a value of y(t) and y‘(t) at time t = 0, while a boundary value problem would specify values for y(t) at both t = 0 and t = 1.