Differential operator in the context of Boundary value problem

In mathematical analysis, integral equations are equations in which an unknown function appears under an integral sign. In mathematical notation, integral equations may thus be expressed as being of the form: $${\displaystyle f(x_{1},x_{2},x_{3},\ldots ,x_{n};u(x_{1},x_{2},x_{3},\ldots ,x_{n});I^{1}(u),I^{2}(u),I^{3}(u),\ldots ,I^{m}(u))=0}$$where ${\displaystyle I^{i}(u)}$ is an integral operator acting on u. Hence, integral equations may be viewed as the analog to differential equations where instead of the equation involving derivatives, the equation contains integrals. A direct comparison can be seen with the mathematical form of the general integral equation above with the general form of a differential equation which may be expressed as follows:$${\displaystyle f(x_{1},x_{2},x_{3},\ldots ,x_{n};u(x_{1},x_{2},x_{3},\ldots ,x_{n});D^{1}(u),D^{2}(u),D^{3}(u),\ldots ,D^{m}(u))=0}$$where ${\displaystyle D^{i}(u)}$ may be viewed as a differential operator of order i. Due to this close connection between differential and integral equations, one can often convert between the two. For example, one method of solving a boundary value problem is by converting the differential equation with its boundary conditions into an integral equation and solving the integral equation. In addition, because one can convert between the two, differential equations in physics such as Maxwell's equations often have an analog integral and differential form. See also, for example, Green's function and Fredholm theory.

In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a scalar function on Euclidean space. It is usually denoted by the symbols ⁠${\displaystyle \nabla \cdot \nabla }$⁠, ${\displaystyle \nabla ^{2}}$ (where ${\displaystyle \nabla }$ is the nabla operator), or ⁠${\displaystyle \Delta }$⁠. In a Cartesian coordinate system, the Laplacian is given by the sum of second partial derivatives of the function with respect to each independent variable. In other coordinate systems, such as cylindrical and spherical coordinates, the Laplacian also has a useful form. Informally, the Laplacian Δf (p) of a function f at a point p measures by how much the average value of f over small spheres or balls centered at p deviates from f (p).

The Laplace operator is named after the French mathematician Pierre-Simon de Laplace (1749–1827), who first applied the operator to the study of celestial mechanics: the Laplacian of the gravitational potential due to a given mass density distribution is a constant multiple of that density distribution. Solutions of Laplace's equation Δf = 0 are called harmonic functions and represent the possible gravitational potentials in regions of vacuum.

The telegrapher's equations (or telegraph equations) are a set of two coupled, linear partial differential equations that model voltage and current along a linear electrical transmission line. The equations are important because they allow transmission lines to be analyzed using circuit theory. The equations and their solutions are applicable from 0 Hz (i.e. direct current) to frequencies at which the transmission line structure can support higher order non-TEM modes. The equations can be expressed in both the time domain and the frequency domain. In the time domain the independent variables are distance and time. In the frequency domain the independent variables are distance ${\displaystyle x}$ and either frequency, ${\displaystyle \omega }$, or complex frequency, ${\displaystyle s}$. The frequency domain variables can be taken as the Laplace transform or Fourier transform of the time domain variables or they can be taken to be phasors in which case the frequency domain equations can be reduced to ordinary differential equations of distance. An advantage of the frequency domain approach is that differential operators in the time domain become algebraic operations in frequency domain.

The equations come from Oliver Heaviside who developed the transmission line model starting with an August 1876 paper, On the Extra Current. The model demonstrates that the electromagnetic waves can be reflected on the wire, and that wave patterns can form along the line. Originally developed to describe telegraph wires, the theory can also be applied to radio frequency conductors, audio frequency (such as telephone lines), low frequency (such as power lines), and pulses of direct current.

A vector operator is a differential operator used in vector calculus. Vector operators include:

• Gradient is a vector operator that operates on a scalar field, producing a vector field.
• Divergence is a vector operator that operates on a vector field, producing a scalar field.
• Curl is a vector operator that operates on a vector field, producing a vector field.

Defined in terms of del:

Del, or nabla, is an operator used in mathematics (particularly in vector calculus) as a vector differential operator, usually represented by ∇ (the nabla symbol). When applied to a function defined on a one-dimensional domain, it denotes the standard derivative of the function as defined in calculus. When applied to a field (a function defined on a multi-dimensional domain), it may denote any one of three operations depending on the way it is applied: the gradient or (locally) steepest slope of a scalar field (or sometimes of a vector field, as in the Navier–Stokes equations); the divergence of a vector field; or the curl (rotation) of a vector field.

Del is a very convenient mathematical notation for those three operations (gradient, divergence, and curl) that makes many equations easier to write and remember. The del symbol (or nabla) can be formally defined as a vector operator whose components are the corresponding partial derivative operators. As a vector operator, it can act on scalar and vector fields in three different ways, giving rise to three different differential operations: first, it can act on scalar fields by a formal scalar multiplication—to give a vector field called the gradient; second, it can act on vector fields by a formal dot product—to give a scalar field called the divergence; and lastly, it can act on vector fields by a formal cross product—to give a vector field called the curl. These formal products do not necessarily commute with other operators or products. These three uses are summarized as:

In the study of differential equations, a boundary-value problem is a differential equation subjected to constraints called boundary conditions. A solution to a boundary value problem is a solution to the differential equation which also satisfies the boundary conditions.

Boundary value problems arise in several branches of physics as any physical differential equation will have them. Problems involving the wave equation, such as the determination of normal modes, are often stated as boundary value problems. A large class of important boundary value problems are the Sturm–Liouville problems. The analysis of these problems, in the linear case, involves the eigenfunctions of a differential operator.

In mathematics, an operator is generally a mapping or function that acts on elements of a space to produce elements of another space (possibly and sometimes required to be the same space). There is no general definition of an operator, but the term is often used in place of function when the domain is a set of functions or other structured objects. Also, the domain of an operator is often difficult to characterize explicitly (for example in the case of an integral operator), and may be extended so as to act on related objects (an operator that acts on functions may act also on differential equations whose solutions are functions that satisfy the equation). (see Operator (physics) for other examples)

The most basic operators are linear maps, which act on vector spaces. Linear operators refer to linear maps whose domain and range are the same space, for example from ${\displaystyle \mathbb {R} ^{n}}$ to ${\displaystyle \mathbb {R} ^{n}}$.Such operators often preserve properties, such as continuity. For example, differentiation and indefinite integration are linear operators; operators that are built from them are called differential operators, integral operators or integro-differential operators.

In mathematics and computer science, a higher-order function (HOF) is a function that does at least one of the following:

• takes one or more functions as arguments (i.e. a procedural parameter, which is a parameter of a procedure that is itself a procedure),
• returns a function as its result.

All other functions are first-order functions. In mathematics higher-order functions are also termed operators or functionals. The differential operator in calculus is a common example, since it maps a function to its derivative, also a function. Higher-order functions should not be confused with other uses of the word "functor" throughout mathematics, see Functor (disambiguation).

In mathematics, the spectral theory of ordinary differential equations is the part of spectral theory concerned with the determination of the spectrum and eigenfunction expansion associated with a linear ordinary differential equation. In his dissertation, Hermann Weyl generalized the classical Sturm–Liouville theory on a finite closed interval to second order differential operators with singularities at the endpoints of the interval, possibly semi-infinite or infinite. Unlike the classical case, the spectrum may no longer consist of just a countable set of eigenvalues, but may also contain a continuous part. In this case the eigenfunction expansion involves an integral over the continuous part with respect to a spectral measure, given by the Titchmarsh–Kodaira formula. The theory was put in its final simplified form for singular differential equations of even degree by Kodaira and others, using von Neumann's spectral theorem. It has had important applications in quantum mechanics, operator theory and harmonic analysis on semisimple Lie groups.

In mathematics, a Green's function (or Green function) is the impulse response of an inhomogeneous linear differential operator defined on a domain with specified initial conditions or boundary conditions.

This means that if ${\displaystyle L}$ is a linear differential operator, then