|
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 \left()+q(x)y=-\lambda w(x)y, |}} 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 (), 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 () 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 () 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 () 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 (), and has ''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 ()-()-() are real and can be ordered such that :: * Corresponding to each eigenvalue λ''n'' 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 ()-()-(). * The normalized eigenfunctions form an orthonormal basis :: :in the Hilbert space ''L''2((), ''w''(''x'')''dx''). Here δ''mn'' 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 form == The differential equation () is said to be in Sturm–Liouville form or self-adjoint form. All second-order linear ordinary differential equations can be recast in the form on the left-hand side of () by multiplying both sides of the equation by an appropriate integrating factor (although the same is not true of second-order partial differential equations, or if ''y'' is a vector.) 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Sturm–Liouville theory」の詳細全文を読む スポンサード リンク
|