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Novel hybrid function operational matrices of fractional integration: An application for solving multi-order fractional differential equations

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.


Introduction
Since the fractional calculus was born during the conversation held on 30 September 1695 between Leibniz and L'Hospital (Oldham and Spanier 1974), a number of methods has been devised until the present for solving ordinary differential equations involving derivatives and/or integrals of non-integer order. At first analytical methods such as Fourier transforms (Gaul et al. 1989(Gaul et al. , 1991, modal synthesis and eigenvector expansion (Suarez and Shokooh 1997), Laplace transform method, fractional Green's function, Mellin transform method, power series method (Podlubny 1999), etc. were developed to find the closed form solutions of linear fractional order differential equations. After realizing the fact that the aforesaid analytical methods cannot be applied to get analytical solutions of nonlinear fractional differential equations and even complicated linear fractional order differential equations, numerical techniques like Podlubny's numerical method (Podlubny 1999), Predictor Evaluate-Corrector Evaluate type method (Diethelm et al. 2002), generalized Euler's method (Odibat and Momani 2008) and semi numerical techniques, for instance, Adomian decomposition method (Adomian 1994), variational iteration method (He 1999a), Homotopy perturbation method (He 1999b), fractional differential transform method (Arigoklu and Ozkol 2007), Homotopy analysis method (Zurigat et al. 2010), new iterative method (Varsha and Hossein 2006) were proposed to obtain accurate and stable numerical solutions. Recently there has been an increasing interest in the development of numerical methods using orthogonal functions like block pulse functions, hat functions, Chebyshev polynomials, Muntz polynomials, Laguerre polynomials, Jacobi polynomials, Legendre polynomials, Bernstein polynomials, etc. ( Those numerical algorithms possess an interesting and significant property of transforming the fractional order differential equation under consideration into a system of algebraic equations. Once the fractional order differential equation has been converted into the system of algebraic equations, the problem of approximating the solution of fractional order differential equation becomes the task of solving the system of algebraic equations which is much easier and computationally faster. Therefore, basing numerical methods on orthogonal functions is beneficial. The new orthogonal hybrid functions (HFs), which are linear combinations of the orthogonal piecewise constant sample-andhold functions and the orthogonal piecewise linear triangular functions, belong to the class of orthogonal polynomials of degree 1 and have been proposed by Deb et al. in (Deb et al. 2012) mainly for the analysis of delay and delay-free integer order systems and for the computation of numerical solutions of linear integer order differential equations (Deb et al. 2016). Even though very few applications of this new set of orthogonal hybrid functions has been reported in the literature, as the subsequent sections reveal, it possesses immense strength to be employed in the fractional calculus. The present work brings forward one of the important applications of the new orthogonal hybrid functions for solving multi-order fractional differential equation (FDE) of the form: Here is an arbitrary number,   t y is the unknown function and is the Caputo fractional order derivative defined as (Caputo 1967) where is Riemann-Liouville fractional order integral (Oldham and Spanier 1974), To accomplish the goal of numerically solving the multi-order fractional differential equation in (1) using HFs, we first derive a formula, by means of generalized one-shot operational matrices, expressing explicitly the fractional order integral of HFs in terms of HFs themselves and in the second step we make use of the derived generalized one-shot operational matrices to transform (1) into a system of algebraic equations whose solution is the approximate solution of (1). The remaining paper is structured as follows. Section 2 introduces new orthogonal hybrid functions. The generalized one-shot operational matrices are derived in Section 3. Based on the results of Section 3, a new numerical algorithm is proposed in Section 4. In Section 5, we test our numerical algorithm by applying it to a set of test problems consisting of linear and nonlinear multi-order fractional differential equations. Section 6 presents the significant inferences drawn from the present work. (Deb et al. 2012(Deb et al. , 2016 Let us consider the time function,   t f , of Lebesgue measure, which is defined on the interval,

Orthogonal hybrid functions
Using (5) to (7), the actual function,   t f , can be approximated piecewise as shown below.
The first order integral of     t S m is estimated in the orthogonal HF domain as follows. where In the same way, the right-handed triangular function vector,     t T m is integrated once and the result is expanded into orthogonal HF series where Employing (9) and (10),  are called the complementary one-shot operational matrices of first order integration and acting as a first order integrator in the orthogonal HF domain.
Orthogonal and a few operational properties of HFs are listed in Appendix 1.

Generalized one-shot operational matrices for fractional integration.
The main objective of this section is to derive an approximation using orthogonal HFs for the , n is an integer, and where

Proof
The proof is given in Appendix 2.
) of the right-handed triangular function vector, , is expressed by means of orthogonal HFs

Proof
Using Equation (8) in the definition of the Riemann-Liouville fractional order integral of   From Theorems 3.1 and 3.2, we can get the expression in (14). This completes the proof. Remark. The generalized HF approximation in (14) yields the HF estimate for the first order

Example 3.1
We now test the viability of the generalized one-shot operational matrices based HF approximation

The algorithm for solving multi-order FDEs (HFM)
We now explain the procedure of finding the approximate solution of the multi-order FDE by the application of the generalized one-shot operational matrices derived in the previous section.
Let us recall the general form of the multi-order FDE.
For the sake of simplicity, we consider the homogeneous initial conditions     0 Equation (17) becomes, Estimating the variable coefficients   t b k and   t b 0 in the HF domain, From (23) and (24), * . Equating the coefficients of the SHF vector,     t S m , and the TF vector, From (22), we get the piecewise linear approximation for   t y in the orthogonal HF domain.
Solving the system of algebraic equations in (27) and (28) produces the approximation for   t y in the orthogonal HF domain. As the formula in (14) works well for arbitrary order  , the developed numerical algorithm is generic in the sense that it can be applied to classical (integer) and fractional (non-integer) multi-order differential equations.

Numerical examples
In this section, we implement the numerical algorithm developed in the former section on linear and nonlinear multi-order FDEs with constant and variable coefficients.
The exact solution of (27) is   3 t t y  .
We consider the following three cases.
The multi-order FDE in (29) with above parameters is solved using our numerical algorithm (HFM) with the step size of 0.1. The absolute error between the exact solution and the piecewise linear HF solution is computed and plotted in Figure 1. The  -norm of the absolute error is compared in Table 2 with the maximal absolute error produced in (Ford and Connolly 2009) [25] (see Example 1 and Table 1 The maximal absolute error attained by our numerical algorithm (HFM1) is compared (Table 4) with that got in (Shiralashetti and Deshi 2016) (see Example 3 and Table 6 in (Shiralashetti and Deshi 2016)), (Ford and Connolly 2009) (see Example 3 and Table 3 Table 3. Figure 1 shows the absolute error (on logarithmic scale) obtained via HFM in case 2.
The exact solution for (32) is   The given multi-order FDE is solved using HFM for the following two cases. (33) (34) Figure 2 gives the absolute error obtained (respective computation times are provided in Table 3) via our numerical algorithm for two cases. In (  The linear multi-order FDE is The given linear non-homogenous multi-order FDE has the analytical solution   .
We consider the following two cases. (36) The accuracy of the HF approximate solution obtained in both cases (shown in Figure 3, Table 6 and the respective elapsed times are given in Table 3)

Example 5.4
The nonlinear multi-order FDE is The exact solution for the following five cases is   3 (39) (40) Using the step size of 1/10, the nonlinear non-homogenous multi-order FDE in (38) is solved for the above two cases and the corresponding absolute errors are compared in Figure 4(a). Table 7 presents the maximal absolute errors gained via our algorithm and those obtained by ET and ER methods in ( Case 4   Table 10 Error analysis of Example 5.5.

Method
Step subject to the initial conditions   0 The piecewise linear HF approximate solution of (43) is obtained by using the numerical algorithm for the following cases. (45) As shown in Table 10 and  Table 11.    The linear multi-order FDE with variable coefficients is The given problem possesses the closed form solution    Figure 6 displays the absolute error of HF solution for the following cases.  T  T  T  T  T  S  T  T  T  T  T  S  m  T  S  T  S  m  T  T  m  T  S  m  T  T  m  T  ) is expanded into orthogonal HFs using the following expression.