Initial Value Problems vs Boundary Value Problems

Often, in addition to the differential equation, the unknown function is required to satisfy certain conditions at one or more points within the interval of interest.

For example, considering an oscillator connected to a spring (see the following figure), in addition to the governing differential equation

$$m{\displaystyle \frac{d^{2}y}{d{t}^2}}+ky=F(t),$$

where m is the mass of the body, k is the spring constant, and F(t) is the applied force, we are required to know its initial position and initial velocity

$$y(0)\text{and}{{\displaystyle \frac{dy}{dt}}|}_{t=0}=\dot{y}(0),$$

in order to determine its position ($y(t)$) at certain time.

Similarly, we need to know the initial position (x(0), y(0)) and initial velocity components ($v_x(0)=\dot{x}(0)$ and $v_y(0)=\dot{y}(0)$) of a projectile in order to determine its path ($x(t)$ and $y(t)$).

Another example would be the deformation of a beam. In addition to the governing equation $EI{\displaystyle \frac{d^{4}y}{d{x}^4}}=w(x)$ where $EI$ is flexural rigidity and $w(x)$ is the applied load, we need to know the support conditions. For example, if at one end ($x=0$), the beam is clamped, then we know that there is no displacement $y(0)=0$ and there is no rotation $y^{\prime }(0)=0$. If the other end is pinned, then there is no displacement $y(L)=0$ and the beam does not withstand a moment there, meaning $EI{y}^{″}(L)=0$ (see the following figure). Therefore, for this example, we need to solve

$$EI\frac{d^{4}y}{d{x}^4}=0,\text{with }y(0)=y^{\prime }(0)=0,y(L)=y^{″}(L)=0.$$

What is the difference between the first two examples and the last one? In the first two examples, the additional conditions are specified at the same value of the independent variable, say $t=0$. Such additional conditions are called initial conditions. A differential equation together with initial conditions is called an initial-value problem (IVP). Initial-value problems are very common in dynamic problems.

In the last example, the additional conditions are specified at the endpoints (or boundaries) of the interval of interest ($[0,L]$). Such conditions are called boundary conditions. A differential equation together with the boundary conditions is called a boundary value problem (BVP).

When these conditions are specified at the same value of $x$ (or $t$ or any other independent variable)—typically at the beginning of the interval where the solution is sought—they are called initial conditions. A differential equation together with initial conditions is called an initial-value problem (IVP). On the other hand, if the conditions are specified at the endpoints (or boundaries) of $x$ (or any independent variable), they are called boundary conditions. A differential equation together with the boundary conditions is called a boundary value problem (BVP).

BVPs (Boundary Value Problems) and IVPs (Initial Value Problems) are two fundamental types of differential equation problems that specify different kinds of conditions for the solutions of the differential equations.

Initial Value Problem (IVP)

• An Initial Value Problem involves a differential equation along with some additional conditions specified at a single point. For example, given a differential equation like $\frac{dy}{dx}=f(x,y)$, an IVP would also provide the value of $y$ at a particular point $x_0$, such as $y(x_0)=y_0$. The three equations

  $$y^{\prime \prime }+4y=0,y(0)=0.2,y^{\prime }(0)=0.1$$

  for example constitute an initial value problem because both conditions are specified at $x=0$.

• The key characteristic of an IVP is that the initial conditions are given at one specific point, and the solution is typically sought within an interval starting from this point.

• IVPs are common in problems where the state at a starting time is known, and the objective is to predict the future behavior, such as the motion of a projectile at the moment it is launched.

Boundary Value Problem (BVP)

• A Boundary Value Problem involves a differential equation along with conditions specified at two or more points. For instance, for the equation $\frac{d^{2}y}{d{x}^2}=f(x,y,y^{\prime })$, a BVP would specify values at two points, such as $y(a)=\alpha$ and $y(b)=\beta$, where $a$ and $b$ are the endpoints of the interval of interest. An example of such a problem is

  $$y^{\prime \prime }+4y=0,y(0)=-0.1,y(\pi /4)=0.2$$

• BVPs are characterized by the requirement to satisfy the conditions at multiple boundaries, which can make them more complex to solve than IVPs.

• Such problems are typical in steady-state phenomena where conditions are known at spatial boundaries, like the temperature distribution along a rod that is heated at both ends.

The numerical methods for solving IVPs and BVPs can differ significantly, primarily due to the nature of the conditions that need to be satisfied. Numerical methods for IVPs often proceed forward from the known initial condition, while BVPs might require iterative techniques that adjust the solution until it satisfies conditions at all specified boundaries.

Mixed Problems

In some partial differential equations, both initial conditions and boundary conditions are required. Consider the vibration of a string or a heat distribution problem. In such a problem, not only do we need to know the conditions of the string (position of the string points) or the temperature of every point, but we also need the conditions at both ends, say the temperature on the boundary or the fact that the string is fixed at both ends.

Neither IVP nor BVP

Not every differential equation problem comes with initial conditions and/or boundary conditions. But these two types of conditions are the most common ones.