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A new framework for polynomial approximation to differential equations.

Luigi Brugnano1, Gianluca Frasca-Caccia2, Felice Iavernaro3

  • 1Università di Firenze, Florence, Italy.

Advances in Computational Mathematics
|November 21, 2022
PubMed
Summary
This summary is machine-generated.

This study introduces a polynomial approximation framework for solving differential equations, extending it to handle delay differential equations and enabling the derivation of Runge-Kutta methods.

Keywords:
Delay differential equationsLocal Fourier expansionOrdinary differential equationsOrthogonal polynomialsPolynomial approximationsRunge-Kutta methods

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Area of Science:

  • Numerical Analysis
  • Differential Equations
  • Computational Mathematics

Background:

  • Initial value problems for differential equations require robust numerical solutions.
  • Existing approximation methods may have limitations in handling complex equation types.
  • Polynomial approximation offers a flexible approach for differential equation solutions.

Purpose of the Study:

  • To present a novel framework for polynomial approximation of differential equation solutions.
  • To extend the framework to address constant delay differential equations.
  • To demonstrate the derivation of Runge-Kutta methods within this framework.

Main Methods:

  • Expansion of the vector field along an orthonormal basis.
  • Utilization of perturbation results for approximation accuracy.
  • Generalization of the framework for ordinary and delay differential equations.

Main Results:

  • A generalized framework for polynomial approximation of differential equations.
  • Successful extension to constant delay differential equations.
  • Derivation of relevant Runge-Kutta methods from the framework.

Conclusions:

  • The proposed framework provides an effective method for approximating solutions to differential equations.
  • The generalization enhances applicability to problems with constant delays.
  • This approach facilitates the development of new numerical integration methods.