Terminal Value Problems of Non-homogeneous Fractional Linear Systems with General Memory Kernels

Terminal value problems of fractional linear systems with non-homegenous terms are investigated in this paper for the first time. They are equivalent to a second kind weakly singular Volterra–Fredholm integral system. Picard’s method is used to obtain a closed form solution. The exact solution is checked to satisfy the terminal value problem. Numerical solutions are provided in comparison with truncated exact ones.


Introduction
Fractional operators hold a feature of memory effects. Fractional differential equations of different types have been employed in many real-world applications and obtained better results. Recently, the function space, definitions and numerical methods of a general fractional calculus were developed in [7,11,13]. The physical meaning of the fractional derivative was provided by continuous time random walk [8].
The general fractional calculus can be reduced to the Caputo [13], Hadamard [13], Katugampola fractional derivatives [12] but also reveals new fractional derivatives of non-convolution types. The general time-fractional Fokker-Planck equation [8] shows this new feature. As a result, it is important to investigate the fractional differential equations with general memory kernels.
Existences of terminal value problems (TVPs) of fractional differential equations [2-4, 10, 14] were investigated and applications were suggested in [6,15,16]. A linear -dimensional TVP for a general fractional system (GFS) is described by is a source function (or external force), A is a × coefficient matrix and y(b) = T ∈ ℝ is a terminal value. Many models can be given in such a form. However, explicit solution of such linear systems are not clear.
The main purpose of this study is to obtain a closed form solution for GFS (1). It is organized as follows: In Sect. 2, preliminaries of general fractional integral and derivative are provided. In Sect. 3, the given TVP for GFS is transformed into a system of weakly singular integral equations. In Sect. 4, a closed form solution is given by use of Picard's method. In Sect. 5, the solution is checked to satisfy the TVP and the numerical solution is given.

Preliminaries
Let us revisit the general fractional integral in X p c (a, b) with the norm.
is a strictly increasing function with g(a) ≥ 0 . The general fractional integral of order > 0 is defined by The semi-group property of general fractional integral can hold The space AC n [a, b] is defined by the use of the derivative, namely: Then we have the definition of the general fractional derivative in the space.
Definition 3 [8,11] ] be a strictly monotone function with g(a) ⩾ 0 and g � (t) > 0 . The left g-Caputo fractional derivative of arbitrary order can be defined as We recall that and The fractional calculus of vector functions is defined by Taking the general fractional integral to both side of Eq. (1) and using Eq. (7), we obtain Let t = b and use the terminal value condition. We get So the system (1) is transformed into a weakly singular Volterra-Fredholm integral system (10)       The famous one and two parameter Mittag-Leffler functions [5,9] can be used to simplify the solution as However, there exists a fractional integral of y(t) in y 0 such that the solution is implicit.
In order to treat this problem, we restart from Eq.

Existence Conditions of Exact Solutions
In order to verify that the obtained solution satisfies the TVP, we need following lemma.

Numerical Versus Exact Solutions
We use a truncation of E and E , to estimate the solution. Let M ∈ ℕ, and By our notation, B M is also an approximation for B. Substituting, this approximations in (22) we obtain (29) The approximate solution for M = 20, 25 and 30 shows the proposed method's convergence. Fig. 1a, b illustrate y 1 (t) and y 2 (t) of the system. We choose the spline parameters c = 1, N = 100 and r = 2 according to [1]. The expected error of spline collocation method is O(h) and we use the step-size h = 0.01 in the example. As we observe in Fig. 1, by increasing M the solution converges and they have a good agreement with the numerical ones. We also given the computational cost in Table 1. Figure 2a, b demonstrate the case of g(t) = ln(t) when other parameters are the same.