Applications to Ordinary Differential Equations

Ram P. Kanwal
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Ram P. Kanwal: The Pennsylvania State University, Department of Mathematics

Chapter Chapter 9 in Generalized Functions, 2004, pp 228-264 from Springer

Abstract: Abstract In Section 2.6 we defined the differential operator L, 1 $$ \begin{array}{*{20}c} {Lt = \left( {a_n (x)\frac{{d^n }} {{dx^n }} + a_{n - 1} \frac{{d^{n - 1} }} {{dx^{n - 1} }} + \cdot \cdot \cdot + a_1 \frac{d} {{dx}} + a_0 } \right)t} \\ { = \sum\limits_{m = 0}^n {a_m (x)\frac{{d^m }} {{dx^m }},} } \\ \end{array} $$ and its formal adjoint L*, 2 $$ L*\varphi = \sum\limits_{m = 0}^n {( - 1)^m d^m (a_m (x)\varphi )/dx^m ,} $$ where the coefficients a m (x) are infinitely differentiable functions, t is a distribution, and ø is a test function. These operators are related by the equation 3 $$ \left\langle {Lt,\varphi } \right\rangle = \left\langle {t,L*\varphi } \right\rangle . $$

Keywords: Ordinary Differential Equation; Classical Solution; Fundamental Solution; Jump Discontinuity; Bessel Equation (search for similar items in EconPapers)
Date: 2004
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DOI: 10.1007/978-0-8176-8174-6_9

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