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A discretization for the nonlinear parabolic evolution equation of fractional order in space

Chien-Hong Cho, Hisashi Okamoto

Abstract

We consider a nonlinear parabolic equation of fractional order in space and propose its numerical discretization. The fractional derivative is defined through a functional analytic setting, rather than the traditional definition of fractional derivatives such as the Riemann-Liouville derivative. Numerical experiments are reported and some conjectures are presented.

A discretization for the nonlinear parabolic evolution equation of fractional order in space

Abstract

We consider a nonlinear parabolic equation of fractional order in space and propose its numerical discretization. The fractional derivative is defined through a functional analytic setting, rather than the traditional definition of fractional derivatives such as the Riemann-Liouville derivative. Numerical experiments are reported and some conjectures are presented.

Paper Structure

This paper contains 12 sections, 6 theorems, 55 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Derivative of fractional order
  3. A quick review
  4. Preliminaries from functional analysis
  5. Examples
  6. Theoretical considerations
  7. Numerical results and discussion
  8. Discretization for $t$ variable
  9. Numerical experiments
  10. Explicit scheme
  11. Implicit scheme
  12. Blow-up time as a function of $\alpha$

Key Result

Theorem 1

If $A$ satisfies the hypothesis (H), the evolution equation eq:ev1 has a strong solution, local in time, for all $u_0 \in {\cal D}(A^{\alpha})$.

Figures (2)

  • Figure 1: Profiles of $u(0.6,\cdot)$. $\alpha = 0.5,0.6,0.7$ and $u(0,x) = \cos x + 1$.
  • Figure 2: Profiles of $u(0.6,\cdot)$. $\alpha = 0.7,1.1,1.3$ and $u(0,x) = \cos x + 1$.

Theorems & Definitions (6)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Theorem 6