BOUNDARY & INITIAL CONDITIONS WITH RESPECT TO PARTIAL DIFFERENTIAL EQUATIONS

An initial value problem is a differential equations problem in which you are given the the value of the function and sufficient of its derivatives at ONE VALUE OF X.

Now, if you have a second order equation, you are given the value of the function and its first derivative at some value of x. (That value of x being the "initial" time- often it is x= 0 but doesn't have to be.)

A boundary value problem is a differential equations problem in which you are given the value of the function at several different values of x.

Typically, if you have a second order equation, you are given the value of the function at the endpoints of an interval. (Those endpoints being the "boundary".)

There is a critical theoretical difference between the two: if f(x,y) is well behaved (continuous in both x and y) then the initial value problem dy/dx= f(x,y) with y(x0)= y0 has a unique solution.

There is no such theorem for boundary value problems. For example, the differential equation d2y/dx2= -y With initial conditions y(0)= A, y'(0)= B has the unique solution y(x)= A cos(x)+ B sin(x) no matter what A and B are.

The differential equation d2y/dx2= -y with the boundary value conditions y(0)= 0, y( π π)= 0 has an infinite number of solutions (y= A sin(x) for any value of A). The differential equation d2y/dx2= -y with the boundary value conditions y(0)= 0, y( π π)= 1 has NO solution.