Constant coefficients
Encyclopedia
In mathematics
, constant coefficients is a term applied to differential operator
s, and also some difference operators, to signify that they contain no functions of the independent variables, other than constant function
s. In other words, it singles out special operators, within the larger class of operators having variable coefficients. Such constant coefficient operators have been found to be the easiest to handle, in several respects. They include for example the Laplacian of potential theory
and other major examples of mathematical physics
.
In the case of ordinary differential equation
s, writing
the general constant-coefficient differential operator is
where p is any polynomial with complex number
coefficients. The solution of equations
with a given function g(x) was given already in the eighteenth century, by Leonhard Euler
.
For partial differential equation
s, the constant-coefficient operators are characterised geometrically by their translation invariance, and algebraically as polynomials in the partial derivative
s. According to the Ehrenpreis–Malgrange theorem, they all have fundamental solution
s.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, constant coefficients is a term applied to differential operator
Differential operator
In mathematics, a differential operator is an operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation, accepting a function and returning another .This article considers only linear operators,...
s, and also some difference operators, to signify that they contain no functions of the independent variables, other than constant function
Constant function
In mathematics, a constant function is a function whose values do not vary and thus are constant. For example the function f = 4 is constant since f maps any value to 4...
s. In other words, it singles out special operators, within the larger class of operators having variable coefficients. Such constant coefficient operators have been found to be the easiest to handle, in several respects. They include for example the Laplacian of potential theory
Potential theory
In mathematics and mathematical physics, potential theory may be defined as the study of harmonic functions.- Definition and comments :The term "potential theory" was coined in 19th-century physics, when it was realized that the fundamental forces of nature could be modeled using potentials which...
and other major examples of mathematical physics
Mathematical physics
Mathematical physics refers to development of mathematical methods for application to problems in physics. The Journal of Mathematical Physics defines this area as: "the application of mathematics to problems in physics and the development of mathematical methods suitable for such applications and...
.
In the case of ordinary differential equation
Ordinary differential equation
In mathematics, an ordinary differential equation is a relation that contains functions of only one independent variable, and one or more of their derivatives with respect to that variable....
s, writing
- D = d/dx
the general constant-coefficient differential operator is
- L = p(D),
where p is any polynomial with complex number
Complex number
A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...
coefficients. The solution of equations
- Lf = g
with a given function g(x) was given already in the eighteenth century, by Leonhard Euler
Leonhard Euler
Leonhard Euler was a pioneering Swiss mathematician and physicist. He made important discoveries in fields as diverse as infinitesimal calculus and graph theory. He also introduced much of the modern mathematical terminology and notation, particularly for mathematical analysis, such as the notion...
.
For partial differential equation
Partial differential equation
In mathematics, partial differential equations are a type of differential equation, i.e., a relation involving an unknown function of several independent variables and their partial derivatives with respect to those variables...
s, the constant-coefficient operators are characterised geometrically by their translation invariance, and algebraically as polynomials in the partial derivative
Partial derivative
In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant...
s. According to the Ehrenpreis–Malgrange theorem, they all have fundamental solution
Fundamental solution
In mathematics, a fundamental solution for a linear partial differential operator L is a formulation in the language of distribution theory of the older idea of a Green's function...
s.