Domains of Dependence and Influence

In calculus, we may come across various partial differential equations based on their order and degree. However, we know that the wave equation is the second-order linear partial differential equation representing a wave. Similarly, we can write PDEs for different geometrical shapes, such as a parabola, hyperbola, etc. We can identify the domains of dependence and influence for any point subject to the initial conditions for all these PDEs. In this article, you will learn how to represent a domain of dependence with an interval and the region of influence of a point within this interval.

Check: Partial differential equations

Domain of Dependence and Region of Influence

Consider one-dimensional wave equation,

u_tt − c² u_xx = 0, −∞ < x < ∞

Thus,

u(x, 0) = f(x), u_t(x, 0) = g(x).

From the d’Alembert’s formula for the solution, we have;

We can see that the solution of this expression at any point, say (x₀, t₀) will be depend on the following two statements.

1. The values of the function f(x) at two points, i.e., x₀ + ct₀ and x₀ − ct₀
2. The values of the function g(x) in the closed interval [x₀ − ct₀, x₀ + ct₀]. This can be expressed as x ≥ x₀ − ct₀ and x ≤ x₀ + ct₀

Thus, the characteristic triangle’s base is formed by the closed interval [x₀ − ct₀, x₀ + ct₀]. Also, the triangle obtained is an isosceles triangle with vertices (x₀, t₀), (x₀ − ct₀, 0) and (x₀ + ct₀, 0). Here, the interval or the base [x₀ − ct₀, x₀ + ct₀] is also called the domain of dependence of u at the point (x₀, t₀). Moreover, we can observe that u(x₀, t₀) = 0 when f(x) and g(x) disappear inside the domain of dependence. This can be understood more clearly with the help of the figure below.

As we mentioned above, the functions f(x) and g(x) determine the initial conditions, i.e. u(x, 0) = f(x) and ut(x, 0) = g(x).

From this, it can be said that u(x₀, t₀) is the solution at any point, say (x₀, t₀) is determined entirely by the functions f(x) and g(x) on the triangle’s base and vertices of this triangle’s base of the domain of dependence.

Similarly, the solution u(x, t) at any point (x, t) within the domain of dependence is pinned down by f(x) and g(x) on the base of the triangle and the base vertices of the obtained triangle.

Now, consider the region of influence given by the interval [x₁, x₂]. The region of influence of an interval [x₁, x₂] comprises those points (x₀, t₀) in the xt-plane such that their domains of dependence coincides with [x₁, x₂]. It also obeys that if the point (x₀, t₀) is outside the region of influence of [x₁, x₂], then no initial data will be there in the interval [x₁, x₂] that can define u(x₀, t₀). In order to specify whether a point (x, t) contained in the region of influence of the interval [x₁, x₂], we can use its domain of dependence, i.e. [x − ct, x + ct] and express that the point (x, t) lies inside the region of influence if x − ct ≤ x₂, x + ct ≥ x₁.

Therefore, the region of influence bounded by [x₁, x₂] on the x-axis, and the lines x + ct = x₁ and x − ct = x₂ for t > 0, will be a truncated characteristic cone.

Hence, the range of influence can be shown as:

Importance of Domain of Dependence and Influence

We can illustrate the dissimilarities between the types of partial differential equations (PDEs) by drawing their respective domains of dependence.

For instance, in the hyperbolic case, i.e. for the equation u_tt = c² u_xx, the point P(x₀, t₀) in the figure below can be influenced by only those points lying inside the region and bounded by the two characteristics line equations x + ct = constant, x – ct = constant and t < t₀. This region is known as the domain of dependence. Also, the point P that influences the points in this interval is known as the region of influence here. This can be shown as given below:

Now, consider the parabolic case, where the equation is u_t = k u_xx. Here, the domain of dependence of the point P(x₀, t₀) is the area t < t_0, and the region of influence will be the set of all points for which t > t₀. This is illustrated in the below figure.

When we consider the elliptic equation, u_xx + u_yy = f(x, y), the domain of dependence and the region of influence can be obtained as given below.

Similarly, we can form the domain of dependence and regions of influence for the given partial differential equation subject to the initial conditions.