An Efficient Numerical Algorithm for Solving Fractional Higher-Order Nonlinear Integrodifferential Equations

This paper is devoted to both theoretical and numerical study of boundary value problems for higher-order nonlinear fractional integrodifferential equations. Existence and uniqueness results for the considered problem are provided and proved. The numerical method of solution for the problem is based on a conjugate collocation and spline approach combined with shooting method. Some numerical examples are discussed to demonstrate the efficiency and the accuracy of the proposed algorithm.


Introduction
Within the context of fractional calculus, it is argued that anticipated sort of memory is being carried out from past states to current states; see, for example, the recent work of Agarwal et al. [1].Therefore, in recent years, several phenomena in physics, chemistry, life sciences, geophysics and earth sciences, and fluid dynamics have been extensively investigated through mathematical models involving fractional calculus; see, for example, the works by Podlubny [2], Mainardi [3], and Kilbas et al. [4].
In this paper we consider a class of boundary value problems for nonlinear integrodifferential equations of the form L :=    () + ∫ where  = ⌈⌉ is the smallest integer greater than or equal to .It is well-known that many mathematical models in engineering and other disciplines in science involve integrodifferential equations of fractional order, for example, problems in modeling of turbulent aerodynamic phenomena, continuum under viscoelastic situations, certain population dynamics problems, and heat transfer in composite materials with certain properties.Account for similar issues is given in [5][6][7][8] and the references therein.
The rest of the paper is organized as follows: some definitions and preliminary results are presented in Section 2. In Section 3, some relevant theoretical results such as the existence and uniqueness of the considered problem are presented.The numerical method of solution is presented in Section 4. In Section 5, numerical examples are discussed to demonstrate the efficiency and the rapid convergence of the present algorithm.

Definitions and Preliminary Results
This section presents some definitions and preliminary results that will be extensively used in this study.We first introduce the Riemann-Liouville definition of fractional derivative operator.Definition 1.The left sided Riemann-Liouville fractional integral operator of order  is defined by where  ∈  1 (, ) and  ∈ R + .

Analytical Results
The existence and uniqueness of the exact solution to problem (1) subject to the boundary conditions (2a) and (2b) are discussed herein.Since we are using the shooting method which requires converting the boundary value problem to initial value problem, we will discuss in the next theorem the existence of the solution to (1) subject to Theorem 4 (existence).Assume that ℎ ∈ [0, 1],  ∈ [0, 1] 2 , and  ∈ [0, 1] × R. Then for any  > 0 and there exists  : [0, ] → R solving the initial value problem (1) and (6).
Proof.Taking the Riemann-Liouville functional operator of both sides of (1) and applying Lemma 3, one obtains Let B = { ∈ [0, ] : ‖ − ∑ 3 =0 ( () (0)/!)‖ ∞ < }.Obviously, B is a closed subset of the Banach space of all continuous functions on [0, ] equipped with the Chebyshev norm.Moreover, since () = ∑ 3 =0 ( () (0)/!)  for  ∈ [0, ] is in B then B ̸ = .Define the operator L on B by The equation under consideration can be written as Our aim is to show that (10) has a fixed point in To achieve our target, we need to show that L is a self-mapping on B. For any  ∈ B and  ∈ [0, ], we have Therefore, Based on Banach's fixed point theorem, it follows that the proof is complete.
It is easy to see that the solution produced by Theorem 4 depends on ,  2 , and  3 .To prove the existence of solution to problem ( 1)-(2b), it is enough to force the solution to satisfy the condition (2b).In this case, we can determine values of  2 and  3 so that the solution of ( 1)-(2b) depends on  only.

Method of Solution
The following is a brief derivation of the numerical algorithm used to solve problem (1) subject to (2a) and (2b).It is based on conjugate collocation approach with multiple shooting method.It consists of three main steps: (1) Collocation step.

Collocation and Spline Methods.
For the sake of simplicity, we discuss the solution of (1) as initial value problem with where  2 and  3 are unknown constants which will be determined later.

Conclusion and Future Work
In this paper, we have solved special class of higher-order nonlinear integrodifferential equations of order 4 subject to  boundary conditions.The method of solution is based on conjugate collocation and spline technique with shooting method.The numerical results for given examples demonstrate the efficiency and accuracy of the present method.It should be noted that applying Theorem 5 causes two difficulties.Firstly, the Lipschitz constant for (, ) with respect to  is not easy to find and, secondly, the necessary condition (0 < /Γ() < 1) does not satisfy several cases of the problem (1)-(2b).Therefore, a further discussion on weaker necessary conditions should be followed in the future work to cover wider range of problem (1)-(2b).

Figure 2 :
Figure 2: Computed absolute error between the present numerical solution and the exact one for Example 1.

Figure 4 :
Figure 4: Computed absolute error between the present numerical solution and the exact one for Example 2.

Example 3 .𝑦 2 (𝑎 2 Figure 5 𝑢 2 (
Figure5shows the graph of the approximate solution.Since the actual solution for this problem is unknown, we may measure the error of the approximation by using the residual function, (), defined by  () :=    () + ∫  0

Figure 5 :
Figure 5: The exact and approximate solutions for Example 3.