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An Efficient Method Based on Framelets for Solving Fractional Volterra Integral Equations.

Mutaz Mohammad1, Alexander Trounev2, Carlo Cattani3

  • 1Department of Mathematics & Statistics, Zayed University, Abu Dhabi 144543, UAE.

Entropy (Basel, Switzerland)
|December 8, 2020
PubMed
Summary
This summary is machine-generated.

This study demonstrates the effectiveness of tight frame systems, specifically framelets generated from B-splines, for solving fractional Volterra integral equations (FVIEs). Numerical results show rapid convergence and accurate solutions for these complex equations.

Keywords:
fractional calculusframeletsgeneralization of Unequal Error Protection (UEP)harmonic numerical analysisnumerical solutionvolterra integral equationswavelets

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Area of Science:

  • Numerical Analysis
  • Applied Mathematics
  • Fractional Calculus

Background:

  • Fractional Volterra Integral Equations (FVIEs) are crucial in modeling complex systems.
  • Caputo fractional order derivatives are frequently involved in these models.
  • Traditional methods may face challenges in solving such equations efficiently.

Purpose of the Study:

  • To explore the advantages of tight frame systems (framelets) for solving FVIEs with Caputo derivatives.
  • To utilize framelet systems generated from non-negative B-spline functions.
  • To provide accurate and efficient numerical solutions for specific FVIEs.

Main Methods:

  • Framelet systems, a generalization of orthonormal bases, were employed.
  • B-splines, a type of refinable non-negative function, were used to generate framelets.
  • A collocation discretization technique was applied to reduce FVIEs to linear systems.
  • Numerical solutions were obtained for various FVIE examples.

Main Results:

  • The proposed framelet method successfully reduced FVIEs to solvable linear systems.
  • Accurate and efficient numerical solutions were achieved for several FVIE examples.
  • The numerical results demonstrated very rapid convergence towards the exact solutions.

Conclusions:

  • Tight frame systems, particularly B-spline generated framelets, offer significant advantages for solving FVIEs.
  • The collocation discretization technique combined with framelets provides an effective numerical approach.
  • The method ensures high accuracy and rapid convergence, outperforming traditional approaches for these equations.