Sage-Tips

Why we need boundary and initial conditions in differential equations?

WHY DO WE NEED INITIAL AND BOUNDARY CONDITIONS: Boundary value problems are extremely important as they model a vast amount of phenomena and applications, from solid mechanics to heat transfer, from fluid mechanics to acoustic diffusion.

Why are initial conditions needed?

For a system of order k (the number of time lags in discrete time, or the order of the largest derivative in continuous time) and dimension n (that is, with n different evolving variables, which together can be denoted by an n-dimensional coordinate vector), generally nk initial conditions are needed in order to trace …

What is initial condition and boundary conditions?

A boundary condition expresses the behavior of a function on the boundary (border) of its area of definition. An initial condition is like a boundary condition, but then for the time-direction. Not all boundary conditions allow for solutions, but usually the physics suggests what makes sense.

What is initial and boundary value problems?

A boundary value problem has conditions specified at the extremes (“boundaries”) of the independent variable in the equation whereas an initial value problem has all of the conditions specified at the same value of the independent variable (and that value is at the lower boundary of the domain, thus the term “initial” …

What is initial boundary?

A boundary condition expresses the behavior of a function on the boundary (border) of its area of definition. An initial condition is like a boundary condition, but then for the time-direction. As before the maximal order of the derivative in the boundary condition is one order lower than the order of the PDE.

What is initial and boundary conditions?

What are the initial conditions for a boundary value problem?

For instance, for a second order differential equation the initial conditions are, With boundary value problems we will have a differential equation and we will specify the function and/or derivatives at different points, which we’ll call boundary values.

What are the boundary and initial conditions in heat conduction?

Boundary and Initial Conditions in Heat Condution. As for another differential equation, the solution is given by boundary and initial conditions. Four kinds of boundary conditions: As for another differential equation, the solution is given by boundary and initial conditions.

What is initial condition in differential equations?

Initial condition is instead of specifying value of the differential equation at some location, we specify it at some initial time. Ex, ds/dt=2 where s is distance, t is time.

Is the differential equation the only one used in boundary value problems?

The answers to these questions are fairly simple. First, this differential equation is most definitely not the only one used in boundary value problems. It does however exhibit all of the behavior that we wanted to talk about here and has the added bonus of being very easy to solve.