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A shooting-Newton procedure for solving fractional terminal value problems

Luigi Brugnano, Gianmarco Gurioli, Felice Iavernaro

Abstract

In this paper we consider the numerical solution of fractional terminal value problems (FDE-TVPs). In particular, the proposed procedure uses a Newton-type iteration which is particularly efficient when coupled with a recently-introduced step-by-step procedure for solving fractional initial value problems (FDE-IVPs), able to produce spectrally accurate solutions of FDE problems. Some numerical tests are reported to make evidence of its effectiveness.

A shooting-Newton procedure for solving fractional terminal value problems

Abstract

In this paper we consider the numerical solution of fractional terminal value problems (FDE-TVPs). In particular, the proposed procedure uses a Newton-type iteration which is particularly efficient when coupled with a recently-introduced step-by-step procedure for solving fractional initial value problems (FDE-IVPs), able to produce spectrally accurate solutions of FDE problems. Some numerical tests are reported to make evidence of its effectiveness.
Paper Structure (15 sections, 5 theorems, 96 equations, 3 figures, 7 tables)

This paper contains 15 sections, 5 theorems, 96 equations, 3 figures, 7 tables.

Table of Contents

  1. Introduction
  2. The shooting-Newton procedure
  3. A simplified Newton-iteration
  4. Implementing the algorithm
  5. Piecewise quasi-polynomial approximation
  6. Error estimation
  7. The simplified shooting-Newton algorithm for semi-linear problems
  8. Numerical Tests
  9. Example 1
  10. Example 2
  11. Example 3
  12. Example 4
  13. Example 5
  14. Example 6
  15. Conclusions

Key Result

Theorem 1

For $t\in[0,T]$, one has: which is the solution of the fractional variational problem As is usual, $f'(y)$ denotes the Jacobian matrix of $f(y)$. explicitly given by:

Figures (3)

  • Figure 1: Reference solution for problem (\ref{['prob3']}).
  • Figure 2: Reference solution for problem (\ref{['prob5']}) (solid line). The circle denotes the actual initial condition, whereas the pluses denote the final approximate solution.
  • Figure 3: Execution times of Algorithm \ref{['alg1']} for solving problem (\ref{['ex6']}) with $y(5)$ given, for dimensions ranging from 2 to 70. See the text for details.

Theorems & Definitions (16)

  • Theorem 1
  • proof
  • Remark 1
  • Remark 2
  • Theorem 2
  • proof
  • Theorem 3
  • proof
  • Theorem 4
  • proof
  • ...and 6 more