Collocation Method for the Numerical Solution of Multi-Order Fractional Di ﬀ erential Equations

This study presents a collocation approach for the numerical integration of multi-order fractional di ﬀ erential equations with initial conditions in the Caputo sense. The problem was transformed from its integral form into a system of linear algebraic equations. Using matrix inversion, the algebraic equations are solved and their solutions are substituted into the approximate equation to give the numerical results. The e ﬀ ectiveness and precision of the method were illustrated with the use of numerical examples


Introduction
In the fields of mathematics, physics, chemistry, and engineering, differential and integral equations involving fractions are of the utmost significance. The use of functional equations, such as ordinary and partial differential equations, is typical when applying mathematics to the modeling of problems arising in the real world. In the early 1900s, Italian mathematician Vito Volterra came up with a whole new sort of equation that came to be known as integro-differential equations in order to investigate the phenomenon of population expansion. In these types of equations, one or more derivatives of the function whose value is unknown is placed under the integral sign. Integro-differential equations can be found in a variety of mathematical formulations of physical phenomena. Additionally, these equations can be found in the modeling of certain phenomena in the fields of science and engineering. For instance, the equations of kinetics that support the kinetic theory of rarefied gases, plasma, radiation transmission, and coagulation are some examples. [1]. Some of the numerical solution of fractional differential equations developed in the literature include: Perturbed collocation method [2], Adomian decompositions method by [3][4][5], Collocation method by [6][7][8][9], Chebyshev-Gelerkin method [10], Bernoulli matrix method [11], Differential transform method [12], Pseudospectral method [13], Bernstein Polynomials method [14,15], the Mellin transform approach [16]. [17] utilized a numerical approach based on the boubaker polynomial to generate approximate numerical solutions to the multi-order fractional differential equations. Their decision was to use an operational matrix for fractional integration based on boubakar polynomi-1 Ajileye & James / J. Nig. Soc. Phys. Sci. 5 (2023) 1075 2 als. Collocation approach for the computational solution of fredholm-volterra fractional order of integro-differential equations was presented by [18]. They solved the problem by first obtaining the linear integral form of it and then transforming it into a system of linear algebraic equations by making use of conventional collocation points.
In this research, the collocation method is utilized to solve multi-order fractional differential equations of the form subject to the initial condition where y(x) is the unknown function, D α j and D β are the Caputo's derivative, h(x) is the force known -prior. q j (x) is the known function, a j and λ j are known constants.

Basic Definitions
In this section, we present certain definitions and fundamental ideas of fractional calculus for the purpose of the formulation of the problem that has been presented. Definition 2.1: The Caputo derivative with order α > 0 of the where m − 1 ≤ α ≤ m, m ∈ N, x > 0 Definition 2.2: Let (a n ) , n ≥ 0 be a sequence of real numbers. The power series in x with coefficients a n is an expression where φ(x) = [1 x x 2 · · · x N ], A = [a 0 a 1 · · · a N ] T then y(x, n) = x n A, n = 0(1)N, n ∈ Z + Definition 2.3: Standard Collocation Method (SCM). This method is used to determine the desired collocation points within an interval. i.e [a,b] and is given by Definition 2.4: Let y(x) be a continuous function, then where m − 1 < β < 1 Definition 2.5: Let p(s) be an integrable function, then Definition 2.6: The Riemann -Liouville derivative of order α > 0 with n − 1 < α < n of the power function f (t) = t p−α is given by

Mathematical Background
In this part, we create a collocation approach for numerically solving multi-order fractional differential equations utilizing power series polynomials as the basis function. Lemma (3.1) (Integral form) Let y(x) be a solution to (1) subject to (2), the integral form is using (6) on equation (10) gives applying equations (3) and (7) to equation (11) gives Substituting equation (4) into equation (12) gives

Method of Solution
Collocating at x i in equation (13) gives where Simplifying equation (14) gives Factorizing the values of A from equation (15) gives Equation (16) can be in the form where and Multiplying both sides of equation (17)

Convergence Analysis
In this section, we establish the convergence of the method by substituting the approximate solution into equation (3.0) Subtracting (9) from (29) gives

Numerical Examples
In this section, we considered two numerical examples to evaluate the effectiveness and clarity of the method. A MAPLE 18 program is used to perform the computations. Let y n (x) and y(x) be the approximate and exact solutions respectively. Error N = |y n (x) − y(x)| .
with this condition y (0) = y(0) = 0 and exact solution y( Using N = 4 for illustration, and applying equation (6) gives where   Using N = 4 for illustration, Applying equation (6) gives where where

Discussion of Results
In this section, we discuss the numerical results obtained by applying the derived numerical method to the solved examples. We observed from the result obtained for example 1 as shown in Table 1 that the approximate solution at N=4 gives y 4 (x) = 1.365574320288940 × 10 −14 − 2.546407529280260 × 10 −12 x+ 1.000000001121990x 2 − 1.000000001716120x 3 + 6.387779194483300 × 10 −10 x 4 . The numerical result almost converges to the exact solution and produces extremely small errors. This demonstrated that our method outperformed the proposed method by Uwaheren et al (2020).
The results of the numerical example 2 in Table 2 shows the approximate solution at N=4 as y 4 (x) = 1.428190898877800 × 10 −12 − 2.777014174171200 × 10 −10 x + 1.0000001121990x 2 − 1.0000001716120x 3 + 6.387779194483300 × 10 −10 x 4 . The numerical result converge to the exact solution and give better result than the method proposed by Uwaheren et al (2020) at the same value of N. This shows that the numerical method developed is consistent and converges faster.

Conclusion
In this paper, a new numerical method was developed for solving multi-order fractional differential equations with initial conditions using collocation method. The numerical method derived is consis-tent, efficient and reliable and easy to compute. Maple code was used to implement the developed method.
Solved numerical examples show that the method is reliable and suitable for these kind of problems. We also compare our absolute errors with Uwaheren et al. (2020) as shown in Tables 1  and 2. Hence, we safely conclude that our method is preferable to the existing methods.