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 Theory and Technique, 1998, pp 219-255 from Springer

Abstract: Abstract In Section 2.6 we defined the differential operator L, 1 $$Lt=\left( {{a}_{n}}\left( x \right)\frac{{{d}^{n}}}{d{{x}^{n}}}+{{a}_{n-1}}\left( x \right)\frac{{{d}^{n-1}}}{d{{x}^{n-1}}}+\cdot \cdot \cdot +{{a}_{1}}\left( x \right)\frac{d}{dx}+{{a}_{0}} \right)t=\sum\limits_{m=0}^{n}{{{a}_{m}}\left( x \right)}\frac{{{d}^{m}}t}{d{{x}^{m}}}$$ and its formal adjoint L*, 2 $$L*\phi =\sum\limits_{m=0}^{n}{{{{\left( -1 \right)}^{m}}{{d}^{m}}\left( {{a}_{m}}\left( x \right)\phi \right)}/{d{{x}^{m}}}\;}$$ where the coefficients am (x) are infinitely differentiable functions, t is a distribution, and ø is a test function. These operators are related by the equation 3 $$\left\langle Lt,\phi \right\rangle =\left\langle t,{{L}^{*}}\phi \right\rangle$$ This means that the action of Lt on φ is equivalent to the action of t on the test function ψ =L*ø.

Keywords: Ordinary Differential Equation; Classical Solution; Fundamental Solution; Jump Discontinuity; Bessel Equation (search for similar items in EconPapers)
Date: 1998
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DOI: 10.1007/978-1-4684-0035-9_9

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