Differential Operators

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Examples of Differential Operators

Differential operators are a generalization of the operation of differentiation.

The simplest differential operator \(D\) acting on a function \(y,\) "returns" the first derivative of this function:

\[Dy\left( x \right) = y'\left( x \right).\]

Double \(D\) allows to obtain the second derivative of the function \(y\left( x \right):\)

\[{D^2}y\left( x \right) = D\left( {Dy\left( x \right)} \right) = Dy'\left( x \right) = y^{\prime\prime}\left( x \right).\]

Similarly, the \(n\)th power of \(D\) leads to the \(n\)th derivative:

\[{D^n}y\left( x \right) = {y^{\left( n \right)}}\left( x \right).\]

Here we assume that the function \(y\left( x \right)\) is \(n\) times differentiable and defined on the set of real numbers. The function \(y\left( x \right)\) itself can take complex values.

Differential operators may be more complicated depending on the form of differential expression.

For example, the nabla differential operator often appears in vector analysis. It is defined as

\[\nabla = \frac{\partial }{{\partial x}}\mathbf{i} + \frac{\partial }{{\partial y}}\mathbf{j} + \frac{\partial }{{\partial z}}\mathbf{k},\]

where \(\mathbf{i}, \mathbf{j}, \mathbf{k}\) are the unit vectors along the coordinate axes \(x,\) \(y,\) \(z.\)

As a result of acting of the operator \(\nabla\) on a scalar field \(F,\) we obtain the gradient of the field \(F:\)

\[\nabla F = \frac{{\partial F}}{{\partial x}}\mathbf{i} + \frac{{\partial F}}{{\partial y}}\mathbf{j} + \frac{{\partial F}}{{\partial z}}\mathbf{k}.\]

The gradient vector always points in the direction of greatest increase of the function \(F,\) and its length indicates the rate of increase of the function in this direction.

The scalar product of vector \(\nabla\) and the vector field \(\mathbf{V}\) is known as the divergence of the vector \(\mathbf{V}:\)

\[\nabla \cdot \mathbf{V} = \text{div}\,\mathbf{V} = \frac{{\partial {V_x}}}{{\partial x}} + \frac{{\partial {V_y}}}{{\partial y}} + \frac{{\partial {V_z}}}{{\partial z}}.\]

The vector product of vectors \(\nabla\) and \(\mathbf{V}\) gives the curl of the vector \(\mathbf{V}:\)

\[\nabla \times \mathbf{V} = \text{rot}\,\mathbf{V} = \left| {\begin{array}{*{20}{c}} \mathbf{i} & \mathbf{j} & \mathbf{k}\\ {\frac{\partial }{{\partial x}}}&{\frac{\partial }{{\partial y}}}&{\frac{\partial }{{\partial z}}}\\ {{V_x}}&{{V_y}}&{{V_z}} \end{array}} \right|.\]

The scalar product of \(\nabla \cdot \nabla = {\nabla ^2}\) corresponds to a scalar differential operator, called the Laplace operator or Laplacian. It is also denoted by the symbol \(\Delta:\)

\[\Delta = {\nabla ^2} = \frac{{{\partial ^2}}}{{\partial {x^2}}} + \frac{{{\partial ^2}}}{{\partial {y^2}}} + \frac{{{\partial ^2}}}{{\partial {z^2}}}.\]

The introduction of differential operators allows to investigate differential equations in terms of operator theory and functional analysis. This generalized approach turns out powerful and effective. In particular, considering application to higher order linear differential equations, we obtain a compact way of writing equations, and in some cases, the possibility of a quick solution.

Differential Operator \(L\left( D \right)\)

Consider the linear differential equation of the \(n\)th order:

\[{y^{\left( n \right)}}\left( x \right) + {a_1}\left( x \right){y^{\left( {n - 1} \right)}}\left( x \right) + \cdots + {a_{n - 1}}\left( x \right)y'\left( x \right) + {a_n}\left( x \right)y\left( x \right) = f\left( x \right).\]

Using the differential operator \(D,\) this equation can be written as

\[L\left( D \right)y\left( x \right) = f\left( x \right),\]

where \(L\left( D \right)\) is the differential polynomial equal to

\[L\left( D \right) = {D^n} + {a_1}\left( x \right){D^{n - 1}} + \cdots + {a_{n - 1}}\left( x \right)D + {a_n}\left( x \right).\]

In other words, the operator \(L\left( D \right)\) is an algebraic polynomial, in which the differential operator \(D\) plays the role of a variable.

Let us consider some properties of the operator \(L\left( D \right).\)

The operator \(L\left( D \right)\) is linear:
\[L\left( D \right)\left[ {{C_1}{y_1}\left( x \right) + {C_2}{y_2}\left( x \right)} \right] = {C_1}L\left( D \right){y_1}\left( x \right) + {C_2}L\left( D \right){y_2}\left( x \right).\]
In the case of several operators \(L\left( D \right),\) \(M\left( D \right)\) and \(N\left( D \right)\) (the degree of the differential polynomials can be different), the following properties also hold:
Commutative law of addition:
\[L\left( D \right) + M\left( D \right) = M\left( D \right) + L\left( D \right).\]
Associative law of addition:
\[\left[ {L\left( D \right) + M\left( D \right)} \right] + N\left( D \right) = L\left( D \right) + \left[ {M\left( D \right) + N\left( D \right)} \right].\]
For two operators \(L\left( D \right)\) and \(M\left( D \right)\), one can also define the multiplication operation:
\[\left[ {L\left( D \right) \cdot M\left( D \right)} \right]y\left( x \right) = L\left( D \right) \cdot \left[ {M\left( D \right)y\left( x \right)} \right].\]
It is important to note that the multiplication operation is commutative for differential operators with constant coefficients, that is for the operators of the form
\[L\left( D \right) = {D^n} + {a_1}{D^{n - 1}} + \cdots + {a_{n - 1}}D + {a_n},\]
where \({a_1}, \ldots ,{a_n}\) are constant numbers.
For such operators, conditions \(4-6\) are satisfied:
Commutative law of multiplication:
\[L\left( D \right) \cdot M\left( D \right) = M\left( D \right) \cdot L\left( D \right)\]
Associative law of multiplication:
\[\left[ {L\left( D \right) \cdot M\left( D \right)} \right] \cdot N \left( D \right) = L\left( D \right) \cdot \left[ {M\left( D \right) \cdot N\left( D \right)} \right]\]
Distributive law of multiplication over addition:
\[L\left( D \right) \cdot \left[ {M\left( D \right) + N\left( D \right)} \right] = L\left( D \right) \cdot M\left( D \right) + L\left( D \right) \cdot N\left( D \right)\]
We also mention another useful property of the operator \(D:\)
\({D^m}{D^n} = {D^{m + n}}.\)

As it can be seen, the differential operators \(L\left( D \right)\) with constant coefficients have the same properties as ordinary algebraic polynomials. Consequently, as well as algebraic polynomials, we can multiply, factor or divide differential operators \(L\left( D \right)\) with constant coefficients. These properties are used in the operator method of solution of differential equations.