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- Sturm–Liouville_theory abstract "In mathematics and its applications, a classical Sturm–Liouville equation, named after Jacques Charles François Sturm (1803–1855) and Joseph Liouville (1809–1882), is a real second-order linear differential equation of the form</dl>where y is a function of the free variable x. Here the functions p(x), q(x), and w(x) > 0 are specified at the outset. In the simplest of cases all coefficients are continuous on the finite closed interval [a,b], and p has continuous derivative. In this simplest of all cases, this function "y" is called a solution if it is continuously differentiable on (a,b) and satisfies the equation (1) at every point in (a,b). In addition, the unknown function y is typically required to satisfy some boundary conditions at a and b. The function w(x), which is sometimes called r(x), is called the "weight" or "density" function. The value of λ is not specified in the equation; finding the values of λ for which there exists a non-trivial solution of (1) satisfying the boundary conditions is part of the problem called the Sturm–Liouville (S–L) problem. Such values of λ when they exist are called the eigenvalues of the boundary value problem defined by (1) and the prescribed set of boundary conditions. The corresponding solutions (for such a λ) are the eigenfunctions of this problem. Under normal assumptions on the coefficient functions p(x), q(x), and w(x) above, they induce a Hermitian differential operator in some function space defined by boundary conditions. The resulting theory of the existence and asymptotic behavior of the eigenvalues, the corresponding qualitative theory of the eigenfunctions and their completeness in a suitable function space became known as Sturm–Liouville theory. This theory is important in applied mathematics, where S–L problems occur very commonly, particularly when dealing with linear partial differential equations that are separable.A Sturm–Liouville (S–L) problem is said to be regular if p(x), w(x) > 0, and p(x), p'(x), q(x), and w(x) are continuous functions over the finite interval [a, b], and have separated boundary conditions of the form </dl></dl>Under the assumption that the S–L problem is regular, the main tenet of Sturm–Liouville theory states that: The eigenvalues λ1, λ2, λ3, ... of the regular Sturm–Liouville problem (1)-(2)-(3) are real and can be ordered such that Corresponding to each eigenvalue λn is a unique (up to a normalization constant) eigenfunction yn(x) which has exactly n − 1 zeros in (a, b). The eigenfunction yn(x) is called the n-th fundamental solution satisfying the regular Sturm–Liouville problem (1)-(2)-(3). The normalized eigenfunctions form an orthonormal basisin the Hilbert space L2([a, b], w(x)dx). Here δmn is a Kronecker delta.Note that, unless p(x) is continuously differentiable and q(x), w(x) are continuous, the equation has to be understood in a weak sense.".
- Sturm–Liouville_theory abstract "In mathematics and its applications, a classical Sturm–Liouville equation, named after Jacques Charles François Sturm (1803–1855) and Joseph Liouville (1809–1882), is a real second-order linear differential equation of the formwhere y is a function of the free variable x. Here the functions p(x), q(x), and w(x) > 0 are specified at the outset. In the simplest of cases all coefficients are continuous on the finite closed interval [a,b], and p has continuous derivative. In this simplest of all cases, this function "y" is called a solution if it is continuously differentiable on (a,b) and satisfies the equation (1) at every point in (a,b). In addition, the unknown function y is typically required to satisfy some boundary conditions at a and b. The function w(x), which is sometimes called r(x), is called the "weight" or "density" function. The value of λ is not specified in the equation; finding the values of λ for which there exists a non-trivial solution of (1) satisfying the boundary conditions is part of the problem called the Sturm–Liouville (S–L) problem. Such values of λ when they exist are called the eigenvalues of the boundary value problem defined by (1) and the prescribed set of boundary conditions. The corresponding solutions (for such a λ) are the eigenfunctions of this problem. Under normal assumptions on the coefficient functions p(x), q(x), and w(x) above, they induce a Hermitian differential operator in some function space defined by boundary conditions. The resulting theory of the existence and asymptotic behavior of the eigenvalues, the corresponding qualitative theory of the eigenfunctions and their completeness in a suitable function space became known as Sturm–Liouville theory. This theory is important in applied mathematics, where S–L problems occur very commonly, particularly when dealing with linear partial differential equations that are separable.A Sturm–Liouville (S–L) problem is said to be regular if p(x), w(x) > 0, and p(x), p'(x), q(x), and w(x) are continuous functions over the finite interval [a, b], and have separated boundary conditions of the form Under the assumption that the S–L problem is regular, the main tenet of Sturm–Liouville theory states that: The eigenvalues λ1, λ2, λ3, ... of the regular Sturm–Liouville problem (1)-(2)-(3) are real and can be ordered such that Corresponding to each eigenvalue λn is a unique (up to a normalization constant) eigenfunction yn(x) which has exactly n − 1 zeros in (a, b). The eigenfunction yn(x) is called the n-th fundamental solution satisfying the regular Sturm–Liouville problem (1)-(2)-(3). The normalized eigenfunctions form an orthonormal basisin the Hilbert space L2([a, b], w(x)dx). Here δmn is a Kronecker delta.Note that, unless p(x) is continuously differentiable and q(x), w(x) are continuous, the equation has to be understood in a weak sense.".
- Sturm–Liouville_theory wikiPageID "490990".
- Sturm–Liouville_theory wikiPageRevisionID "593656866".
- Sturm–Liouville_theory id "p/s130620".
- Sturm–Liouville_theory title "Sturm-Liouville theory".
- Sturm–Liouville_theory subject Category:Operator_theory.
- Sturm–Liouville_theory subject Category:Ordinary_differential_equations.
- Sturm–Liouville_theory subject Category:Spectral_theory.
- Sturm–Liouville_theory comment "In mathematics and its applications, a classical Sturm–Liouville equation, named after Jacques Charles François Sturm (1803–1855) and Joseph Liouville (1809–1882), is a real second-order linear differential equation of the form</dl>where y is a function of the free variable x. Here the functions p(x), q(x), and w(x) > 0 are specified at the outset. In the simplest of cases all coefficients are continuous on the finite closed interval [a,b], and p has continuous derivative.".
- Sturm–Liouville_theory comment "In mathematics and its applications, a classical Sturm–Liouville equation, named after Jacques Charles François Sturm (1803–1855) and Joseph Liouville (1809–1882), is a real second-order linear differential equation of the formwhere y is a function of the free variable x. Here the functions p(x), q(x), and w(x) > 0 are specified at the outset. In the simplest of cases all coefficients are continuous on the finite closed interval [a,b], and p has continuous derivative.".
- Sturm–Liouville_theory label "Sturm-Liouville-Problem".
- Sturm–Liouville_theory label "Sturm–Liouville theory".
- Sturm–Liouville_theory label "Teoria de Sturm-Liouville".
- Sturm–Liouville_theory label "Teoria di Sturm-Liouville".
- Sturm–Liouville_theory label "Teoría de Sturm-Liouville".
- Sturm–Liouville_theory label "Théorie de Sturm-Liouville".
- Sturm–Liouville_theory label "Задача Штурма — Лиувилля".
- Sturm–Liouville_theory label "スツルムリウビル型微分方程式".
- Sturm–Liouville_theory label "施图姆-刘维尔理论".
- Sturm–Liouville_theory sameAs Sturm%E2%80%93Liouville_theory.
- Sturm–Liouville_theory sameAs Sturm-Liouville-Problem.
- Sturm–Liouville_theory sameAs Teoría_de_Sturm-Liouville.
- Sturm–Liouville_theory sameAs Théorie_de_Sturm-Liouville.
- Sturm–Liouville_theory sameAs Teoria_di_Sturm-Liouville.
- Sturm–Liouville_theory sameAs スツルムリウビル型微分方程式.
- Sturm–Liouville_theory sameAs 스튀름-리우빌_이론.
- Sturm–Liouville_theory sameAs Sturm-liouvillevraagstuk.
- Sturm–Liouville_theory sameAs Teoria_de_Sturm-Liouville.
- Sturm–Liouville_theory sameAs Q1154082.
- Sturm–Liouville_theory sameAs Q1154082.
- Sturm–Liouville_theory wasDerivedFrom Sturm–Liouville_theory?oldid=593656866.