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set16 - $ Set 16 Laplace Transform and IVPs Part 1 Kyle A...

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a39 a38 a36 a37 Set 16: Laplace Transform and IVPs Part 1 Kyle A. Gallivan Department of Mathematics Florida State University Ordinary Differential Equations Fall 2009 1
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a39 a38 a36 a37 Integral Transforms Consider a mapping of a function f ( t ) to another function F ( s ) : F ( s ) = integraldisplay β α K ( s, t ) f ( t ) dt K ( s, t ) is the kernel of the transform. α , β may be finite or infinite improper integrals may be needed has the flavor of an infinite dimensional matrix times a vector 2
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a39 a38 a36 a37 Integral Transforms and IVPs We want to transform the IVP from a differential equation into an algebraic equation. Form of ODE determines the choice of integral transform. Effect of integral transform on derivatives must be considered. Kernel relates to fundamental solutions. We will consider second order constant coefficient linear ODEs. 3
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a39 a38 a36 a37 Laplace Transform Consider a mapping of a function f ( t ) to another function F ( s ) : F ( s ) = integraldisplay 0 e st f ( t ) dt K ( s, t ) = e st α = 0 , β = improper integral a special case of a definition over C 4
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a39 a38 a36 a37 Improper Integral integraldisplay a g ( t ) dt = lim A →∞ integraldisplay A a g ( t ) dt = lim A →∞ bracketleftbig G ( A ) G ( a ) bracketrightbig G ( t ) is an antiderivative of g ( t ) integral must exist for each A > a limit as A → ∞ must exist if both exist then the improper integral converges if not then the improper integral diverges 5
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a39 a38 a36 a37 Improper Integral integraldisplay a 1 t dt = lim A →∞ integraldisplay A a 1 t dt = lim A →∞ bracketleftbig ln A ln a bracketrightbig = G ( t ) exists lim A →∞ [ G ( A ) G ( a )] does not improper integral diverges 6
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a39 a38 a36 a37
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