Uncertain inverse problem for fractional dynamical systems using perturbed collage theorem

Highlights

• To Solve Inverse problem related to a class of fractional differential equations under interval uncertainty based on the perturbed collage theorem.

• To improve an approximation of interval valued function based on Schauder basis in the complete metric space.

• To obtain the best choice of the parameter in the inverse problem using an optimization problem.

Abstract

In this paper, we intend to obtain some numerical approximations of inverse problem by means of fractional differential equations under interval uncertainty. For this purpose, using perturbed Collage theorem, we achieve the approximations with the help of minimization procedure. We discuss the Picard operator and the existence with the uniqueness of solutions of interval fractional differential equations (IFDEs) using contraction mappings. Some modifications of existing results on the approximations of interval-valued functions using Schauder basis in the metric space are derived. Then, the fractional version of approximation in the Banach space is obtained to compute the numerical solution. In fact, we also employ Collage-based method for solving IFDEs with inverse problem. Finally, illustrative examples are solved in details to show that the results are in good agreement with the exact solutions for different values of fractional order derivative under generalized differentiability.

Introduction

In this research, we aim to provide an approximation approach to solve a class of inverse problems for differential equations under uncertainty based on the ”Collage approach”. Indeed, we consider three concepts simultaneously in the current study. (1) Fractional order derivative; (2) Interval uncertainty; (3) Inverse problem. By taking into consideration the above conceptions, we motivated to confine ourselves on providing new and novel results in this report and for the first time in the literature. In this regards, through the following questions and answers, we explain our contribution and motivation of this study.

Question 1 Why fractional order derivative?

Due to influential aspects of fractional calculus in describing complex dynamics of unsentimental systems, specially in the systems with having memory effect on physical structures such as in signal processing, bioengineering, control process, economic and etc., we preferred to employ this interesting operator to model the system[1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22]. On the other hand, to model the real phenomena, we apply the Caputo-type fractional differentiability of order α (0 < α ≤ 1) because of its advantages on initial conditions which take the same form as that for integer-order differential equations and makes the model closer to the behaviour of ordinary system. Also, fractional-based system can be considered as a model placed among the integer-order systems. Therefore, we expect that if the order of differentiability approaches to a natural number, then the model becomes as a ordinary case. But, how? with some smooth behaviour or not? the solution of this question is critical. Since, the mathematical modeling based on the fractional calculus can be only considered as a routine generalization of ordinary differential equations system with prior prediction about the behaviour of solution or may be find some extraordinary situations which were not appeared in the usual dynamical systems!

Question 2 Why interval uncertainty?

Due to the lack of information, missing data, existing several types of errors in the modeling, considering the uncertainty is inevitable! In this letter, we apply the basic one which is called ”interval uncertainty”. Firstly, because of natural perception of this type of uncertainty as a generalization of real numbers. Second, computations based on intervals are well-understood because of existing several real-world applications. In the recent years, the scientists found the applicability of this significant notion that measures uncertainties in the mathematical modeling with uncertain parameters. Therefore, a number of researches have been done in this regards to analyze the mathematical systems based on the interval parameters and study the existence and uniqueness of the interval solutions of the interval differential equations. As a matter of fact, interval arithmetic is a branch of fuzzy sets that deals with the intervals from the first step of modeling or numerical algorithm that can reduce the complexity and computational difficulties compared with fuzzy systems when describing a system’s states and dynamics. In fact, these kinds of modeling have been considered in several research works in terms of theoretical and computational approaches with suitable applications and interpretation [23], [24], [25], [26], [27], [28], [29], [30], [31], [32], [33].

Question 3 Why inverse problem?

Inverse problems based on the use of contraction maps in ”fractal-based” approximation, was stated by Kunze and Vrscay [35] as an interesting idea which is mentioned as follows: Suppose that (z, d_z) stated a complete metric space and Con(z) is the set of contraction maps on z. If z ∈ Z be as a target, we are looking for to approximate it. The question is that: How can we (if it is possible) find a mapping T_δ ∈ Con(z) such that $d_z(z,{\overline{z}}_{\delta })<\delta$ for a fixed point ${\tilde{z}}_{\delta }\in Z$?

This topic in the fractional dynamical system becomes as the following expressions:

For a given curve z(t) (as a solution), we aim to to find some suitable fractional derivative-based system ${}^{C}z{}^{\left(\alpha \right)}\left(t\right)=f(t,x)$ such that z(t) becomes closely as much as possible, while the field f should be considered in some prescribed limitations [34], [35], [36], [37], [38], [39], [40], [41], [42], [43], [44], [45], [46].

Question Question 1, Question 2 and 3 and the related explanations focus on the fact that, for real modeling using dynamical system under real data (even missing data, lack of information, optimistic/pessimistic situations, sensitivity analysis) in the numerical framework, with combination of optimization problem, we should consider the inverse problem for a class of interval fractional differential equations (IFDEs) and the numerical method by employing Collage theorem. To best of knowledge of the authors, one paper published about inverse problem for solving a class of fractional differential equation [47] and also one research paper on interval uncertainty of inverse integral problem [48]. Therefore, the current paper is the first attempt for solving inverse problem to handle both uncertainty and fractional differentiability.

The paper is sorted out as pursues: in Section 2, some essential concepts of fractional calulus including definition of Caputo-type derivative and interval arithmetic features with the inverse problem in the classical form are recalled. Section 3 provides the main part of results including the Picard operator and inverse problem for IFDEs. Besides, the inverse problem using Collage theorem for solving IFDEs is developed in this section. In Section 4, an example is presented to validate the proposed approach in details. Different values of fractional-order derivatives are experienced under both types of interval differentiability to depict the strength of the presented technique for finding the approximate solution of the IFDEs. At last, some end comments are attracted in the last section.