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

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