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.
Eigenvalues and eigenvectorsConcepts from linear algebraIn linear algebra, an eigenvector ( ) or characteristic vector is a (nonzero) vector that has its direction unchanged (or reversed) by a given linear transformation. More precisely, an eigenvector v{\displaystyle \mathbf {v} } of a linear transformation T{\displaystyle T} is scaled by a constant factorλ{\displaystyle \lambda } when the linear transformation is applied to it: Tv=λv{\displaystyle T\mathbf {v} =\lambda \mathbf {v} }.
Spectrum (functional analysis)Set of eigenvalues of a matrixIn mathematics, particularly in functional analysis, the spectrum of a bounded linear operator (or, more generally, an unbounded linear operator) is a generalisation of the set of eigenvalues of a matrix.
Vector spaceAlgebraic structure in linear algebraVector addition and scalar multiplication: a vector v (blue) is added to another vector w (red, upper illustration). Below, w is stretched by a factor of 2, yielding the sum v + 2w.In mathematics, a vector space (also called a linear space) is a set whose elements, often called vectors, can be added together and multiplied ("scaled") by numbers called scalars.
Exponential functionMathematical function, denoted exp(x) or e^xIn mathematics, the exponential function is the unique real function which maps zero to one and has a derivative everywhere equal to its value. It is denoted ex{\displaystyle e^{x}} or expx{\displaystyle \exp x}; the latter is preferred when the argument x{\displaystyle x} is a complicated expression.
Degenerate energy levelsIn quantum mechanics, an energy level is degenerate if it corresponds to two or more different measurable states of a quantum system. Conversely, two or more different states of a quantum mechanical system are said to be degenerate if they give the same value of energy upon measurement. The maximum number of linearly independent states corresponding to a particular energy level is known as the degree of degeneracy (or simply the degeneracy) of the level.
Linear mapMathematical function, in linear algebraIn mathematics, and more specifically in linear algebra, a linear map, linear mapping, or linear operator is a particular kind of function between vector spaces, which respects the basic operations of vector addition and scalar multiplication.Function spaceIn mathematics, a function space is a set of functions between two fixed sets. Often, the domain and/or codomain will have additional structure which is inherited by the function space. For example, the set of functions from any set X into a vector space has a natural vector space structure given by pointwise addition and scalar multiplication. In other scenarios, the function space might inherit a topological or metric structure, hence the name function space.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.
Scalar (mathematics)In mathematics, more specifically in linear algebra, a scalar is an element of a field which is used to define a vector space through the operation of scalar multiplication: a vector (denoted v) multiplied by a scalar (denoted a) produces another vector (av). Real numbers and complex numbers may be used as scalars in real and complex vector spaces, respectively.
MathematicsMathematics is a field of knowledge concerned with abstract concepts such as numbers, geometric shapes, sets, functions, and probabilities. It uses logical reasoning and proof to study and establish their properties, often expressed as theorems, formulas, and equations. Mathematics is used to model and solve problems in science, engineering, technology, economics, and everyday life.
Inner product spaceIn mathematics, an inner product space is a real or complex vector space endowed with an operation called an inner product. The inner product of two vectors in the space is a scalar, often denoted with angle brackets such as in ⟨a,b⟩{\displaystyle \langle a,b angle }. Inner products allow formal definitions of intuitive geometric notions, such as lengths, angles, and orthogonality (zero inner product) of vectors.*As an Amazon Associate I earn from qualifying purchases.