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: where 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:where 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.
Integral equations in the context of David Hilbert
David Hilbert (/ˈhɪlbərt/; German: [ˈdaːvɪt ˈhɪlbɐt]; 23 January 1862 – 14 February 1943) was a German mathematician and philosopher of mathematics and one of the most influential mathematicians of his time.
Hilbert discovered and developed a broad range of fundamental ideas including invariant theory, the calculus of variations, commutative algebra, algebraic number theory, the foundations of geometry, spectral theory of operators and its application to integral equations, mathematical physics, and the foundations of mathematics (particularly proof theory). He adopted and defended Georg Cantor's set theory and transfinite numbers. In 1900, he presented a collection of problems that set a course for mathematical research of the 20th century.
View the full Wikipedia page for David HilbertIntegral equations in the context of Functional analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (for example, inner product, norm, or topology) and the linear functions defined on these spaces and suitably respecting these structures. The historical roots of functional analysis lie in the study of spaces of functions and the formulation of properties of transformations of functions such as the Fourier transform as transformations defining, for example, continuous or unitary operators between function spaces. This point of view turned out to be particularly useful for the study of differential and integral equations.
The usage of the word functional as a noun goes back to the calculus of variations, implying a function whose argument is a function. The term was first used in Hadamard's 1910 book on that subject. However, the general concept of a functional had previously been introduced in 1887 by the Italian mathematician and physicist Vito Volterra. The theory of nonlinear functionals was continued by students of Hadamard, in particular Fréchet and Lévy. Hadamard also founded the modern school of linear functional analysis further developed by Riesz and the group of Polish mathematicians around Stefan Banach.
View the full Wikipedia page for Functional analysisIntegral equations in the context of Sine and cosine transforms
In mathematics, the Fourier sine and cosine transforms are integral equations that decompose arbitrary functions into a sum of sine waves representing the odd component of the function plus cosine waves representing the even component of the function. The modern, complex-valued Fourier transform concisely contains both the sine and cosine transforms. Since the sine and cosine transforms use sine and cosine waves instead of complex exponentials and don't require complex numbers or negative frequency, they more closely correspond to Joseph Fourier's original transform equations and are still preferred in some signal processing and statistics applications and may be better suited as an introduction to Fourier analysis.
View the full Wikipedia page for Sine and cosine transforms