Generalized squared remainder minimization method for solving multi-term fractional differential equations

Mir Sajjad Hashemi , Mustafa Incb,c,1 , Somayeh Hajikhah Department of Mathematics, Basic Science Faculty, University of Bonab, P.O. Box 55517-61167, Bonab, Iran hashemi_math396@yahoo.com; so.hajikhah@gmail.com Department of Mathematics, Firat University, 23119 Elazig, Turkey minc@firat.edu.tr Department of Medical Research, China Medical University Hospital, China Medical University, Taichung, Taiwan


Introduction
Fractional integration and differentiation are generalizations of integer-order calculus to noninteger ones. It is demonstrated in literature that fractional calculus can play a justifiable and beneficial role in the modeling of various phenomena e.g. science and engineering [2,3,6,25,29,30,[35][36][37]42]. The classical differential operators are defined as local operators, whereas the fractional differential operators are nonlocal. This significant property makes studying fractional differential equations an active area of research. There are a several number of definitions of fractional derivatives involving the kernel of the special functions, such as Mittag-Leffler function, Prabhakar function and so on. Some important ones are, Riemann-Liouville and Caputo fractional derivatives in [14,41], Caputo-Fabrizio fractional derivative in [8,13], and Atangana and Baleanu suggested another version of fractional-order derivative, which uses the generalized Mittag-Leffler function with strong memory as nonlocal and nonsingular kernel in [5].
Multi-point fractional differential equations appear in different types of visco-elastic damping [40]. The multi-point boundary conditions in the fractional differential equations can be understood as the controllers at the end points dissipate or add energy according to censors located at intermediate points. Model equations proposed so far are almost always linear. Therefore, we concentrate on the following multi-point fractional differential equation: where γ j , j = 1, . . . , k+1 are real constant coefficients and 0 < β 1 < β 2 < · · · < β k < α.
Here, we consider this equation with most usual fractional derivative in Caputo sense. Equation (1) permits us to describe the model more accurately than the classical integer equation. The nonlocality of Caputo fractional derivative means that the next state of a system depends not only upon its current state but also upon all of its historical states. Bhrawy et al. [11] applied the spectral algorithm based on generalized Laguerre tau (GLT) method with generalized Laguerre-Gauss (GQ) and generalized Laguerre-Gauss-Radau (GRQ) quadrature methods for Eq. (1). The shifted Chebyshev spectral tau (SCT) method based on the integrals of shifted Chebyshev polynomials is utilized to construct the approximate solutions of such problems [12]. Spectral tau method combined with the shifted Chebyshev polynomials are proposed to solve the Eq. (1) in [15]. Three alternative decomposition approaches are introduced for the approximate solution of Eq. (1) by Ford et al. [16]. More discussion about the approximate solution of Eq. (1) can be found in [27,34,38].
The paper is organized as follows: In Section 2, we briefly discuss some necessary definitions and mathematical preliminaries of fractional calculus, which will be needed in the forthcoming sections. The third section deals with a generalization of squared remainder minimization (GSRM) method for the multi-point fractional differential equations. Section 4 is devoted to the study of convergence analysis for the proposed method. Some illustrative examples are investigated in Section 5. Conclusion remarks provide our final section.

Preliminaries and notations
In this section, we review some preliminaries and properties of well-known fractional derivatives and integrals for the purpose of acquainting with sufficient fractional calculus theory. Among various definitions of fractional derivatives e.g. 13,19], Atangana-Baleanu [7,17,18] and conformable fractional derivative [20,21,26], we introduce two most commonly used definitions, namely, the Riemann-Liouville and Caputo derivatives [9,14,33]. Various analytical and numerical methods are utilized to consider the fractional differential equations e.g. [1, 10, 22-24, 31, 32, 39]. http://www.journals.vu.lt/nonlinear-analysis Definition 1. Let α ∈ R + . The operator J α is called the Riemann-Liouville fractional integral operator. Now, we discuss the interchange of the Riemann-Liouville fractional integration and limit operation, which is useful in the convergence analysis of further discussed method.
is called the Riemann-Liouville fractional differential operator, where the ceiling function α denotes the smallest integer greater than or equal to α.
is called the Caputo fractional differential operator.
For the Caputo derivative, we have

The GSRM method
For problem (1), we specify the linear operator Then there exist unknown constants c 0 , c 1 , . . . , c n such that Obviously, the approximate solution u n (x) needs to satisfy the following conditions: Substituting (3) into Eq. (2) concludes where For any , then the nth-order remaining terms can be given by Remark 2. If the linearly independent functions are supposed as then Eq. (5) becomes http://www.journals.vu.lt/nonlinear-analysis The important point to note here is the minimization problem in the GSRM method by the fact (4). That is, the problem is minimizing (Ou n )(x) in such a way that To do this, we introduce the real functions Therefore in order to find the unknown vector C = (c 0 , c 1 , . . . , c n ), we have to solve the minimization problem Remark 3. If we set polynomials (6) as basis functions, then constrains in minimization problem (7) become We use the Lagrange-multiplier method [28] to minimize problem (7). By using this method, we solve the following system of algebraic equations 2 : where Λ T = (λ 0 , λ 1 , . . . , λ α −1 ) and I T (C) = (I 0 (C), I 1 (C), . . . , I α −1 (C)). Equivalently, we can define the vector For simplicity of notations, if we use Therefore, the linear system of equations (8) becomes or, in the abstract form, where 2 ω 0 , ω 0 2 ω 0 , ω 1 · · · 2 ω 0 , ω n 2 ω 1 , ω 0 2 ω 1 , ω 1 · · · 2 ω 1 , ω n . . . . . . . . . . . .

Convergence analysis
This section is a discussion about the convergence of the GSRM method for the multipoint fractional differential equations of the form (1).
Theorem 1. Suppose that u(x) is the exact solution of fractional differential equation (1) defined on [x 0 , x f ], and u n (x) is the corresponding approximate solution of problem given by the GSRM method. If there exists a polynomial p n (x) ∈ P n [x 0 , x f ] such that for Proof. One can easily conclude that for any n ∈ N, we have Moreover, from the continuity of norms and p n (x) → u(x) we get From Eqs.

Numerical results
In this section, we present the results of the GSRM method on five test problems. We perform our computations using Maple 18 software with 30 digits.
Example 1. Consider the equation with initial condition u(0) = 0. Exact solution of this equation is given by u(x) = x 4 − x 3 /2. Figure 1 shows the absolute errors for various α by the GSRM method with polynomial bases as (6) and n = 4. In Table 1, the absolute errors obtained by present method are compared with GLT method [11] with N = 50 and SCT method [12] with N = 64. From this table it may be concluded that the GSRM method is more accurate than the mentioned approaches. The CPU time used in this example is 0.842, and condition number of coefficient matrix in (9) w.r.t. α = 0.01 is 2.61.
Example 2. Let us consider the following fractional equation: Example 3. Consider the Bagley-Torvik equation By solving the resultant system, we get c 0 = c 1 = 0 and c 2 = 1. Therefore we obtain the exact solution for this example by using the GSRM method. The best maximum absolute errors by GLT(GQ) and GLT(GRQ) reported in [11] with initial conditions u(0) = u (0) = 0. Exact solution of this equation is given by u(x) = x 3 . Numerical results will not be presented since the exact solution is achieved by choosing n = 3.
Regarding Example 4 and in [16], the best result is attained with 512 steps, and the maximum absolute errors are 6.93E − 05, 1.18E − 04 and 3.10E − 06 by using method 1, method 2 and method 3, respectively. Obtained results by GLT(GQ) and GLT(GRQ) methods [11] with N = 64 are 1.43E − 05 and 1.80E − 05, respectively. Moreover, in [38], the absolute error 1.86E − 09 is reported by HWCM method, and 3.39E − 13 is reported by SCT method in [15]. In [27], the maximum absolute error by the Haar wavelet operational matrix method is 1.12E − 02, and reported error in [38] is 2.91E − 03, w.r.t. N = 64. Whereas by the GSRM method, we obtain 4.99E − 28. The CPU time used in this example is 0.811s, and condition number of coefficient matrix in (9) is 849.54.

Conclusion
In the present paper, the squared remainder minimization method is developed to the multi-term fractional differential equations. A minimization problem is manifested and it considered by the Lagrange-multiplier method. Convergence of the GSRM method is theoretically proved. Five test problems are investigated. For some of the given examples, exact solutions by the present method are extracted. Accuracy and reliability of the GSRM method is revealed by the reported figures and tables.