Differential equation

Question

What are boundary conditions and initial conditions with respect to differential equation? Do they mean the same for partial differential equation? How to BC and IC vary for different types of PDE?

Answer 1

Suppose you have a parameter $Y$.

Suppose that parameter depends only on one variable $x$. You then may derive a differential equation $dY/dx$.

Suppose that parameter depends on more than one variable, say $x$ and $t$. You then must derive a partial differential equation for each variable, in this case $\partial Y/\partial x$ and $\partial Y/\partial t$.

Suppose $x$ represents spatial extent. To solve a first order differential, you will need one boundary condition. This is either the value of $Y$ at a defined point $x$ or the value of $dY/dx$ at a defined point $x$.

Each additional differential order requires an additional boundary condition. For example, a second order differential requires two boundary conditions.

Suppose $t$ represents time. Boundary conditions on time at $t = 0$ are called initial conditions. Initial conditions are a subset (if you will) of boundary conditions; they apply to time. Boundary conditions on time other than $t = 0$ could be called anything. Another special class are those for $t \rightarrow \infty$. These boundary conditions are called "final state" conditions.

Answer 2

In general an infinite number of functions satisfy differential equations (DV). To obtain a specific solution a first order ODE must have an initial condition or constraint that specifies the value of dependent variable at a particular value of independent variable. Because the problems are typically time dependent the constraint is called an intial condition and the problem is called intial value problem (IVP).

I summarise an IVP as: When all the constraint are specified at one value of independent variable.

In many cases we need to solve a DV of higher order, that have constraints specified at different values of independent variable, these problems are called Boundary value problems (BVP) and the constraints are called boundary conditions because the constraints are often specified at the endpoints of boundaries of domain of the solution.

And yes those means the same for partial differential equations, if the variables behave independently or when the equation is called separable. The answer to the third question requires a deep understanding of DV's and PDE's, i suggest you to talk to your professor or aks it somewhere else.