Substitution Method for Fractional Differential Equations

TL;DR

This paper introduces a substitution-based method for solving fractional differential equations that avoids increasing the equation's order and ensures stable, accurate numerical solutions with at least second-order precision.

Contribution

It proposes a novel substitution approach to eliminate singularities in fractional derivatives, improving stability and accuracy over traditional methods.

Findings

The substitution method does not increase the equation's order.

It provides well-conditioned difference approximations.

Solutions achieve at least second-order accuracy.

Abstract

Numerical solving differential equations with fractional derivatives requires elimination of the singularity which is inherent in the standard definition of fractional derivatives. The method of integration by parts to eliminate this singularity is well known. It allows to solve some equations but increases the order of the equation and sometimes leads to wrong numerical results or instability. We suggest another approach: the elimination of singularity by substitution. It does not increase the order of equation and its numerical implementation provides the opportunity to define fractional derivative as the limit of discretization. We present a sufficient condition for the substitution-generated difference approximation to be well-conditioned. We demonstrate how some equations can be solved using this method with full confidence that the solution is accurate with at least second order of approximation.