Complex variable and regularization methods of inversion of the Laplace transform

Abstract

In this paper three methods are derived for approximating <italic>f</italic>, given its Laplace transform <italic>g</italic> on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis 0 comma normal infinity right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mi mathvariant="normal">∞<!-- ∞ --></mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(0,\infty )</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, i.e., <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="integral Subscript 0 Superscript normal infinity Baseline f left-parenthesis t right-parenthesis exp left-parenthesis minus s t right-parenthesis d t equals g left-parenthesis s right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msubsup> <mml:mo largeop="false">∫<!-- ∫ --></mml:mo> <mml:mn>0</mml:mn> <mml:mi mathvariant="normal">∞<!-- ∞ --></mml:mi> </mml:msubsup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>f</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mi>exp</mml:mi> <mml:mo>⁡<!-- ⁡ --></mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mo>−<!-- − --></mml:mo> <mml:mi>s</mml:mi> <mml:mi>t</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mspace width="thinmathspace" /> <mml:mi>d</mml:mi> <mml:mi>t</mml:mi> <mml:mo>=</mml:mo> <mml:mi>g</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>s</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">\smallint _0^\infty {f(t)\exp ( - st)\,dt = g(s)}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Assuming that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="g element-of upper L squared left-parenthesis 0 comma normal infinity right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>g</mml:mi> <mml:mo>∈<!-- ∈ --></mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mi mathvariant="normal">∞<!-- ∞ --></mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">g \in {L^2}(0,\infty )</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, the first method is based on a Sinc-like rational approximation of <italic>g</italic>, the second on a Sinc solution of the integral equation <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="integral Subscript 0 Superscript normal infinity Baseline f left-parenthesis t right-parenthesis exp left-parenthesis minus s t right-parenthesis d t equals g left-parenthesis s right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msubsup> <mml:mo largeop="false">∫<!-- ∫ --></mml:mo> <mml:mn>0</mml:mn> <mml:mi mathvariant="normal">∞<!-- ∞ --></mml:mi> </mml:msubsup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>f</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mi>exp</mml:mi> <mml:mo>⁡<!-- ⁡ --></mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mo>−<!-- − --></mml:mo> <mml:mi>s</mml:mi> <mml:mi>t</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mspace width="thinmathspace" /> <mml:mi>d</mml:mi> <mml:mi>t</mml:mi> <mml:mo>=</mml:mo> <mml:mi>g</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>s</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">\smallint _0^\infty {f(t)\exp ( - st)\,dt = g(s)}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> via standard regularization, and the third method is based on first converting <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="integral Subscript 0 Superscript normal infinity Baseline f left-parenthesis t right-parenthesis exp left-parenthesis minus s t right-parenthesis d t equals g left-parenthesis s right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msubsup> <mml:mo largeop="false">∫<!-- ∫ --></mml:mo> <mml:mn>0</mml:mn> <mml:mi mathvariant="normal">∞<!-- ∞ --></mml:mi> </mml:msubsup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>f</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mi>exp</mml:mi> <mml:mo>⁡<!-- ⁡ --></mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mo>−<!-- − --></mml:mo> <mml:mi>s</mml:mi> <mml:mi>t</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mspace width="0.056em" /> </mml:mrow> <mml:mi>d</mml:mi> <mml:mi>t</mml:mi> <mml:mo>=</mml:mo> <mml:mi>g</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>s</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">\smallint _0^\infty {f(t)\exp ( - st){\mkern 1mu} dt = g(s)}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> to a convolution integral over <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper R"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb {R}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, and then finding a Sinc approximation to <italic>f</italic> via the application of a special regularization procedure to solve the Fourier transform problem. We also obtain bounds on the error of approximation, which depend on both the method of approximation and the regularization parameter.