Fredholm theoryIn mathematics, Fredholm theory is a theory of integral equations. In the narrowest sense, Fredholm theory concerns itself with the solution of the Fredholm integral equation. In a broader sense, the abstract structure of Fredholm's theory is given in terms of the spectral theory of Fredholm operators and Fredholm kernels on Hilbert space. It therefore forms a branch of operator theory and functional analysis. The theory is named in honour of Erik Ivar Fredholm.
Linear equationIn mathematics, a linear equation is an equation that may be put in the form a1x1+…+anxn+b=0,{\displaystyle a_{1}x_{1}+\ldots +a_{n}x_{n}+b=0,} where x1,…,xn{\displaystyle x_{1},\ldots ,x_{n}} are the variables (or unknowns), and b,a1,…,an{\displaystyle b,a_{1},\ldots ,a_{n}} are the coefficients, which are often real numbers.
Integral transformMapping involving integration between function spaces In mathematics, an integral transform is a type of transformation that maps a function from its original function space into another function space via integration, where some of the properties of the original function might be more easily characterized and manipulated than in the original function space. The transformed function can generally be mapped back to the original function space using the inverse transform.
IntegralOperation in calculusA definite integral of a function can be represented as the signed area of the region bounded by its graph and the horizontal axis; in the above graph as an example, the integral of f(x){\displaystyle f(x)} between a{\displaystyle a} and b{\displaystyle b} is the yellow (−) area subtracted from the blue (+) area
Boundary value problemType of problem involving ODEs or PDEs Shows a region where a differential equation is valid and the associated boundary values Differential equations Scope Fields Natural sciencesEngineering Astronomy Physics Chemistry Biology Geology Applied mathematics Continuum mechanics Chaos theory Dynamical systems Social sciences Economics Population dynamics Classification Types Ordinary Partial Differential-algebraic Integro-differential Fractional Linear Non-linear By variable type Dependent and independent variables Autonomous Coupled / Decoupled Exact Homogeneous / Nonhomogeneous Features Order Operator Notation Relation to processes Difference (discrete analogue) Stochastic Stochastic partial Delay Solution Existence and uniqueness Well-posed problem Picard–Lindelöf theorem Peano existence theorem Carathéodory's existence theorem Cauchy–Kovalevskaya theorem General topics Initial values Boundary values Dirichlet Neumann Robin Cauchy Periodic Wronskian Abel's identity Sturm–Liouville theory Floquet theory Phase portrait Stability Lyapunov Asymptotic Exponential Series solutions Rate of convergence Asymptotic series Special functions Numerical integration Dirac delta function Solution methods Inspection Method of characteristics Ansatz Euler Exponential response formula Finite difference Crank–Nicolson Finite element Infinite element Finite volume Galerkin Petrov–Galerkin Green's function Integrating factor Integral transforms Perturbation theory Reduction of order Runge–Kutta Separation of variables Undetermined coefficients Variation of parameters WKB approximation People List Newton Leibniz Jacob Bernoulli d'Alembert Clairaut Euler Lagrange Laplace Wroński Fourier Cauchy Green Dirichlet Sturm Liouville Neumann Robin Boole Kovalevskaya Runge Kutta Lipschitz Lindelöf Picard Jeffreys Nicolson Crank Hrennikoff Courant Named equationsvte In the study of differential equations, a boundary-value problem is a differential equation subjected to constraints called boundary conditions.Differential equationIn mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, and the differential equation defines a relationship between the two.Integral operatorAn integral operator is an operator that involves integration.
Differential operatorIn mathematics, a differential operator is an operator defined as a function of the differentiation operator (which is an operation that accepts a function and returns another function, i.e. a higher-order function). This article considers mainly linear differential operators, which are the most common type. However, non-linear differential operators also exist, such as the Schwarzian derivative.
Mathematical analysisMathematical analysis is the branch of mathematics that studies functions, spaces, and operators through methods of approximation and convergence. It grew out of calculus, especially the use of derivatives and integrals to study variable quantities, and in the 19th century its foundations were reformulated with greater rigor. Basic objects of study in mathematical analysis include the real numbers, functions, sequences, series, and limits.Function (mathematics)In mathematics, a function from a set X to a set Y assigns to each element of X exactly one element of Y. The set X is called the domain of the function and the set Y is called the codomain of the function. Functions were originally the idealization of how a varying quantity depends on another quantity. For example, the position of a planet is a function of time.
Maxwell's equationsMaxwell's equations are a set of coupled partial differential equations that describe how electric and magnetic fields are generated by electric charges and currents. Together with the Lorentz force law, they form the foundation of classical electromagnetism, classical optics, electric and magnetic circuits.
Green's functionMethod of solution to differential equations*As an Amazon Associate I earn from qualifying purchases.