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Operational matrices based on the shifted fifth-kind Chebyshev polynomials for solving nonlinear variable order

H Jafari1,2,3, S Nemati1, R M Ganji1

  • 1Department of Applied Mathematics, Faculty of Mathematical Sciences, University of Mazandaran, P.O. Box: 47416-95447, Babolsar, Iran.

Advances in Difference Equations
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Summary

This study introduces a novel numerical method using shifted fifth-kind Chebyshev polynomials (SFKCPs) to solve variable order integro-differential equations (VO-IDEs). The method transforms complex equations into algebraic systems, demonstrating high accuracy in numerical tests.

Keywords:
Convergence analysisNonlinear integro-differential equationsOperational matrixShifted fifth-kind Chebyshev polynomialsVariable order

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Area of Science:

  • Numerical Analysis
  • Applied Mathematics
  • Computational Science

Background:

  • Variable order integro-differential equations (VO-IDEs) present significant challenges in mathematical modeling.
  • Existing numerical methods may lack efficiency or accuracy for these complex equations.

Purpose of the Study:

  • To develop and analyze a novel numerical scheme for solving general VO-IDEs.
  • To utilize shifted fifth-kind Chebyshev polynomials (SFKCPs) for approximating solutions.
  • To transform VO-IDEs into a system of algebraic equations for efficient computation.

Main Methods:

  • Expansion of the unknown function and its derivatives using SFKCPs.
  • Calculation of operational matrices based on SFKCPs.
  • Application of collocation points to convert the integro-differential problem into algebraic equations.
  • Analysis of the method's convergence and error estimation.

Main Results:

  • The proposed SFKCP-based method effectively transforms VO-IDEs into solvable algebraic systems.
  • Numerical tests confirm the high accuracy and reliability of the developed scheme.
  • The method provides accurate approximate solutions for the studied class of equations.

Conclusions:

  • The SFKCP-based numerical scheme offers an accurate and efficient approach for solving VO-IDEs.
  • The method's convergence and error analysis provide a theoretical foundation for its application.
  • This research contributes a valuable tool for researchers and practitioners working with VO-IDEs.