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Related Concept Videos

Partial Fractions01:28

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A partial fraction is a component of a rational expression represented as the sum of simpler fractions. When a rational function is expressed as a ratio of two polynomials, it can often be decomposed into a sum of fractions whose denominators are simpler polynomials, typically linear or irreducible quadratic factors. This process is called partial fraction decomposition, and it is used to simplify complex expressions for integration, solving equations, or analysis.Partial fraction decomposition...
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Related Experiment Videos

Initialization, conceptualization, and application in the generalized (fractional) calculus.

Carl F Lorenzo1, Tom T Hartley

  • 1NASA Glenn Research Center, National Aeronautics and Space Administration, 21000 Brookpark Road, Cleveland, Ohio, USA. Carl.F.Lorenzo@nasa.gov

Critical Reviews in Biomedical Engineering
|July 9, 2009
PubMed
Summary
This summary is machine-generated.

This study formalizes initialization in fractional calculus, making it accessible for engineering and science. New definitions and applications demonstrate its broad utility and mathematical foundation.

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Area of Science:

  • Mathematics
  • Engineering
  • Physics

Background:

  • Fractional calculus lacks formalized initialization, hindering its application in science and engineering.
  • Existing definitions do not fully incorporate initial conditions, limiting practical use.

Purpose of the Study:

  • To provide a formalized basis for initialization in fractional calculus.
  • To enhance the accessibility and applicability of fractional calculus in scientific and engineering disciplines.
  • To present new definitions and mathematical tools for generalized fractional calculus.

Main Methods:

  • Modified definitions for fractional calculus incorporating initialization effects.
  • Development of conceptualizations for fractional derivatives and integrals.
  • Derivation of a generalized Laplace transform for the generalized differintegral.
  • Introduction of the concept of a variable order differintegral.

Main Results:

  • Formalized definitions for fractional calculus with initialization.
  • Demonstrated broad applicability through examples in electronics, dynamics, material science, and electrochemistry.
  • Validation of generalized calculus criteria for the generalized fractional calculus.
  • Derivation of a new generalized Laplace transform for differintegrals.
  • Presentation of a variable order differintegral concept.

Conclusions:

  • The formalized basis makes fractional calculus more accessible and applicable.
  • The presented methods and examples highlight the wide-ranging utility of the theory.
  • The study contributes new mathematical tools and concepts to the field of fractional calculus.