Numerical Analysis of Iterative Fractional Partial Integro-Differential Equations

Many nonlinear phenomena are modeled in terms of di­erential and integral equations. However, modeling nonlinear phenomena with fractional derivatives provides a better understanding of processes having memory e­ects. In this paper, we introduce an e­ective model of iterative fractional partial integro-di­erential equations (FPIDEs) with memory terms subject to initial conditions in a Banach space. e convergence, existence, uniqueness, and error analysis are introduced as new theorems. Moreover, an extension of the successive approximations method (SAM) is established to solve FPIDEs in sense of Caputo fractional derivative. Furthermore, new results of stability analysis of solution are also shown.


Introduction
Most of the physical phenomena are modeled in ordinary di erential equations (ODEs) and partial di erential equations (PDEs). During the last decades, it has been noted that modeling complex phenomena, using fractional derivatives, provides a good t due to their nonlocal nature. Fractional derivatives are e ective tools to formulate processes having memory e ects. Furthermore, fractional PDEs, which are considered the generalization of PDEs with fractional-order derivatives, have been widely used in many areas of sciences and engineering, and they have been the topics of many workshops and conferences due to their essential uses applied in numerous diverse and widespread elds in applied sciences [1][2][3][4][5][6][7]. Furthermore, FPIDEs are applicable in sciences and engineering, and many works in FPIDEs have been introduced (see, for example, [3,[8][9][10][11]), while studying iterative FPIDEs is very rare and currently an active area of research due to their particular applications in neural networks. However, iterative FPIDEs are useful tools for modeling the memory properties of various materials and processes, with a nonlinear relationship to time, such as anomalous diffusion, an elasticity theory, solids mechanic, and other applications [12][13][14].
e study of the theory of the iterative di erential equations began with the work of Eder [15] where Eder worked on a solution of an iterative functional di erential equation. Moreover, many studies on iterative di erential equations have been conducted (see, for example, [16][17][18]).
In many physical systems described as models in terms of initial and boundary value problems, it is essential to develop techniques based on various types of successive approximations constructed explicitly in analytic forms. Several analytical and numerical methods for solving di erential and integral equations are available in the literature. One of the powerful methods is the successive approximations method (SAM) which was introduced in 1891 by E. Picard, and it has been used to prove the existence and uniqueness of solutions of di erential equations [19][20][21][22]. e SAM, which is also called the Picard iterative solutions method, has been increasingly applied to solve di erential equations and integral equations [23,24]. e SAM provides an approximate solution in a short series convergent with readily determinable terms [25]. e existence and uniqueness of solutions are proved with initial conditions for various types of iterative differential equations or iterative integro-differential in some works available in the literature, for example, the exact analytical solution for an iterative nonlinear differential equation was given in [26] where the authors studied a second-order nonlinear iterated differential equation, analytic solutions for an iterative differential equation were given in [27] where the authors studied an iterative functional differential equation, Yang and Zhang introduced solutions for iterative differential equations [28], and Zhang et al. [29] introduced the existence of wavefront solutions for an integral differential equation in a nonlinear nonlocal neuronal network. However, few works have been introduced for the stability analysis of solutions for iterative fractional integro-differential equations [30,31].
is paper presents new analytical and numerical solutions of a new model called "iterative fractional partial integro-differential equation" of iterative Volterra-type equation.
is model is solved by using the method of successive approximations. Moreover, the primary advances applied in this paper are very effective with applications of a Banach space and Gronwall-Bellman integral inequality in sense of Caputo derivative. e rest of the paper is organized as follows. Section 2 gives the preliminaries. Section 3 presents the description of the method of successive approximations, existence, uniqueness, convergence, and error analysis of the solution for the proposed model. Section 4 introduces solutions for two types of iterative FPIDEs. Numerical results and discussion are given in Section 5.

Preliminaries and Definitions
ere are various definitions and theorems of fractional calculus available in the literature.
is section presents some of these definitions and theorems that are needed in this paper and can be found in [32][33][34][35][36] and among other references cited therein.
e Riemann-Liouville integral of time fractional order α for a function u is defined by where Γ is the well-known gamma function.
e Riemann-Liouville time fractional partial derivative of order α for a function u is defined by Definition 3. Let u(x, t): R × (0, ∞) ⟶ R and n − 1 < α < n ∈ N; then, the Caputo derivative of time fractional order α for a function u is Theorem 2 Let α, t ∈ R, t > 0, and n − 1 < α < n ∈ N. en,

Description of the Numerical Scheme
In this section, we introduce an effective model of an iterative fractional partial integro-differential equation with memory term subject to initial value conditions of the following form: where D α t is the α-th Caputo fractional partial derivative, K(x, t) is a bivariate kernel, f(x, t) and f k (x) are known analytic functions, and u(x, t) is the unknown function to be determined.
To find the solution for the iterative fractional partial integro-differential equation (6), we introduce an extension of the SAM as follows. We assume that (6) has an approximate solution given by for Our extension here is that all the components u n (x, t) are continuous where u n can be given as a sum of successive differences in the following form: converges, then u n (x, t) converges and the solution for (6) is given by

Existence and Uniqueness.
is section presents new results for existence and uniqueness of solution for the proposed model (6).
en, there is a unique solution for equation (6).
}. Before we apply the Banach contraction principle, we need to define an operator P: B ⟶ B as From (12), we have By similar argument, we obtain Journal of Mathematics is proves that P is a function from S(ρ) to S(ρ). Next, for u, v ∈ S(ρ), we have erefore, we obtain Since M < Γ(α + 1)/T α+1 k T − 1 which implies that T α+1 k T (M + 1)/Γ(α + 1) < 1, then by Banach principle, the operator P has a unique fixed point. erefore, equation (6) has a solution. □ Theorem 4 (convergence).. If the assumptions of eorem 3 are proposed, then (7) converges.
Since u 0 is a function from [0, T] to [0, T], we get Journal of Mathematics By induction, we have S k ≤ T k . Since |u 0 + T α (N + T 3 k T )/Γ(α + 1)| ≤ T, we get T < 1 when u 0 ≥ 0. erefore, S k goes to zero as k goes to infinity. For every subsequence u kj of S k , there exists a subsequence s kj which uniformly converges and the limit must to be a solution of (6). us, S k uniformly goes to a unique solution of (6).

Error Analysis.
In this section, we evaluate the maximum absolute error of the proposed method for the solution series (7) for (6).

Theorem 5.
Suppose that the hypothesis of eorem 3 holds. Let u n and s n be two solutions satisfying equation (6) for 0 ≤ x, t ≤ T, M > 0 with the initial approximations u n (x, t) and s n (x, t), respectively. en, the maximum absolute error for a solution series (7) for (6) is estimated to be Proof. By using eorem 3, we have Journal of Mathematics 5 Next, by using eorem 4, we have By using Gronwall-Bellman inequality given by Lemma 1, we get us, we obtain is completes the proof of eorem 5. Example 1. In this example, we solve the following iterative FPIDEs of Volterra type with initial value: 6 Journal of Mathematics en, equation (25) is of form (6) with T � 0.75, N � 0, k T � 1 which satisfies where 0 < M < Γ(α + 1)/T α+1 k T − 1 � Γ(α + 1)/0.75 α+1 |cos(x/2)| − 1 < 1 for all 0 ≤ x ≤ 0.75 and 0 < α < 1.
As the hypotheses of eorem 3 are satisfied, a unique solution for equation (25) exists.
ALGORITHM 1: e computation of the existence conditions. Compute ALGORITHM 2: e computation of the numerical solutions.

Journal of Mathematics
. (29) erefore, the approximate solution of (25) is obtained by Example 2. In this example, we solve the following iterative FPIDEs of Volterra type with initial value: where 0 < M < Γ(α + 1)/T α+1 k T − 1 < 1 for all x ∈ [0, 0.75] and 0 < α < 1. As all the hypotheses of eorem 3 are satisfied, a unique solution for (30) exists. By using eorem 4, we obtain a solution of (30) for different values of α. We assume that u 0 (x, t) � u(x, 0) � 0 and by using Mathematica software, the first three iterative solutions are obtained as follows: 8 Journal of Mathematics erefore, the third order term iterative solution (30) is Table 1 presents numerical solutions for equation (25) through various values of x, t when α � 0.5, 1. Table 2 includes numerical values of the iterative solutions for equation (30) by different values of x, t at α � 0.5, 1. In Figures 1(a) and 1(b), we plot the graphs of first-order iterative solution for (25) using various values of x, t at α � 0.5, 1 respectively. We plot the graph of second-order iterative solution for (25) Table 1: Numerical values of the iterative solution when q 1 � q 2 � 0.5, 1 and α � β � 0.5 for Example 2.  Table 2: Numerical values of the iterative solutions when q 1 � q 2 � 0.5, 1 and α � β � 0.5 for Example 1.

Conclusion
In this paper, we introduced a model of FPIDEs. e proposed model is iterative with fractional derivative, which can be used in neural networks and help us to describe how the input data can be accessed. For instance, for subdiffusion in the porous media, fractional-order derivatives determine the decaying rate of the breakthrough curve for long-term observations. Moreover, new results on the local existence, uniqueness, and stability analysis of the solution for the proposed model were introduced. Furthermore, we extended the method of successive approximations to solve FPIDEs with memory terms subject to initial conditions in a Banach space. is extension derives good approximations and reliable techniques to handle iterative FPIDEs. New solutions for Volterra types of iterative FPIDEs were introduced.
e numerical solutions were successfully obtained which confirm the presented results.
Data Availability e datasets used or analyzed during the current study are available from the corresponding author on reasonable request.

Conflicts of Interest
e authors declare that they have no conflicts of interest.