  In the present work, an attempted was made to develop a numerical algorithm
by the use of new orthogonal hybrid functions formed from hybrid of piecewise
constant orthogonal sample-and-hold functions and piecewise linear orthogonal
triangular functions to obtain the numerical solution of multi-order fractional
differential equations. The construction of numerical algorithm involves a
formula, consisting of generalized one-shot operational matrices, expressing
explicitly the Riemann-Liouville fractional order integral of the new
orthogonal hybrid functions in terms of new orthogonal hybrid functions
themselves. A set of test problems comprising of linear and nonlinear
multi-order fractional differential equations with constant and variable
coefficients was considered to demonstrate the accuracy and computational
efficiency of our algorithm and to compare our results with those acquired by
some other well-known methods used for solving multi-order fractional
differential equations in the literature. The comparison highlighted that our
algorithm exhibits superior performance to those methods.
