Piecewise linear approximate solution of fractional order non-stiff and stiff differential-algebraic equations by orthogonal hybrid functions (1801.06970v2)

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.