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Efficient computation of the Wright function and its applications to fractional diffusion-wave equations

Lidia Aceto, Fabio Durastante

Abstract

In this article, we deal with the efficient computation of the Wright function in the cases of interest for the expression of solutions of some fractional differential equations. The proposed algorithm is based on the inversion of the Laplace transform of a particular expression of the Wright function for which we discuss in detail the error analysis. We also present a code package that implements the algorithm proposed here in different programming languages. The analysis and implementation are accompanied by an extensive set of numerical experiments that validate both the theoretical estimates of the error and the applicability of the proposed method for representing the solutions of fractional differential equations.

Efficient computation of the Wright function and its applications to fractional diffusion-wave equations

Abstract

In this article, we deal with the efficient computation of the Wright function in the cases of interest for the expression of solutions of some fractional differential equations. The proposed algorithm is based on the inversion of the Laplace transform of a particular expression of the Wright function for which we discuss in detail the error analysis. We also present a code package that implements the algorithm proposed here in different programming languages. The analysis and implementation are accompanied by an extensive set of numerical experiments that validate both the theoretical estimates of the error and the applicability of the proposed method for representing the solutions of fractional differential equations.
Paper Structure (9 sections, 3 theorems, 70 equations, 9 figures)

This paper contains 9 sections, 3 theorems, 70 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Inversion of the Laplace transform
  3. Error estimates for the trapezoidal rule
  4. Selecting the quadrature nodes
  5. Time-fractional evolution equations on the real line
  6. Codes and numerical examples
  7. Computing the Wright function
  8. Solving the time-fractional evolution equation
  9. Conclusions

Key Result

Theorem 1

WeidemanTrefethenParHyp Let $w=u+iv,$ with $u$ and $v$ real. Suppose g(w) is analytic in the strip $-d<v<c,$ for some $c>0,d>0,$ with $g(w)\rightarrow 0$ uniformly as $|w| \rightarrow +\infty$ in that strip. Suppose further that for some $M_+>0, M_->0$ the function $g(w)$ satisfies for all $0<r<c, 0<s<d.$ Then, where

Figures (9)

  • Figure 1: Plots of the function $f_{-\nu,1-\nu}(1;x)$ for $\nu=1/2$, $3/8$, $1/4$, and $1/8$.
  • Figure 2: Parabolic contour , singularity , quadrature nodes $\times$, and branch cut of the $F_{\lambda,\mu}(s;x)$ function given in \ref{['eq:Flm']}.
  • Figure 3: MATLAB® listing of the optimization for $c$ (left panel), values obtained through the optimization procedure together with the corresponding value of $N$ (right panel) when double precision is used, and we pose $\varepsilon = 10^{-15}$ and $\epsilon = 2.2204\times 10^{-16}$. The dashed line represents the threshold value $\Re(\mu) = 2$.
  • Figure 4: Relative error with respect to the number of quadrature points $N$ for cases in which we have an exact alternative representations of the Mainardi function $M_\nu(|x|) = W_{-\nu,1-\nu}(-|x|)$ in \ref{['eq:special-cases']}. Results in double precision (DP) are computed with the Matlab routine.
  • Figure 5: Relative error with respect to the number of quadrature points $N$ for cases in which we have an exact alternative representations of the Mainardi function $M_\nu(|x|) = W_{-\nu,1-\nu}(-|x|)$ in \ref{['eq:special-cases']}. Results in quadruple precision (QP) are computed with the Fortran routine.
  • ...and 4 more figures

Theorems & Definitions (5)

  • Theorem 1
  • Remark 1
  • Theorem 2
  • Remark 2
  • Theorem 3