Piecewise linear approximate solution of fractional order non-stiff and stiff differential-algebraic equations by orthogonal hybrid functions

TL;DR

This paper introduces a new numerical method using orthogonal hybrid functions to accurately solve both stiff and non-stiff fractional order differential-algebraic equations, demonstrating convergence and superior performance especially for stiff problems.

Contribution

The paper develops a novel numerical approach based on orthogonal hybrid functions and generalized operational matrices for fractional differential-algebraic equations, including stiff cases.

Findings

Method converges to the exact solution as step size decreases.

Results agree with semi-analytical methods for non-stiff problems.

Successfully solves stiff fractional differential-algebraic equations where other methods fail.

Abstract

A simple yet effective numerical method using orthogonal hybrid functions consisting of piecewise constant orthogonal sample-and-hold functions and piecewise linear orthogonal triangular functions is proposed to solve numerically fractional order non-stiff and stiff differential-algebraic equations. The complementary generalized one-shot operational matrices, which are the foundation for the developed numerical method, are derived to estimate the Riemann-Liouville fractional order integral in the new orthogonal hybrid function domain. It is theoretically and numerically shown that the numerical method converges the approximate solutions to the exact solution in the limit of step size tends to zero. Numerical examples are solved using the proposed method and the obtained results are compared with the results of some popular semi-analytical techniques used for solving fractional order differential-algebraic equations in the literature. Our results are in good accordance with the results of those semi-analytical methods in case of non-stiff problems and our method provides valid approximate solution to stiff problem (fractional order version of Chemical Akzo Nobel problem) which those semi-analytical methods fails to solve.