How can I solve a partial differential equation by only having intial condition?

I have a partial differential equation: dy/dt=A dy/dr ( A is constant). I have only information about the initial condition: y(t=0) = constant. I want to solve this equation at r=a. I would appreciate if someone can help me to solve it in MATLAB.

Answer 1

[Walter Roberson]

You have dy/dt=A dy/dr so y must be a function of two or more variables that include t and r . However, you only have the boundary condition stated in one variable, t=0.

The information you have says that

y = A * r + constant1 + f(t)

where f(t) is independent of r, and that y(t=0) = constant2 . Substituting in t = 0 we get

y = A * r + constant1 = constant2

The only way this can be constant is if A = 0 after which constant1 = constant2 . Therefore your system is

y = the constant for y(t=0)