Abstract
Many nonlinear phenomena are modeled in terms of differential and integral equations. However, modeling nonlinear phenomena with fractional derivatives provides a better understanding of processes having memory effects. In this paper, we introduce an effective model of iterative fractional partial integro-differential equations (FPIDEs) with memory terms subject to initial conditions in a Banach space. The 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.
1. Introduction
Most of the physical phenomena are modeled in ordinary differential equations (ODEs) and partial differential equations (PDEs). During the last decades, it has been noted that modeling complex phenomena, using fractional derivatives, provides a good fit due to their nonlocal nature. Fractional derivatives are effective tools to formulate processes having memory effects. 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 fields in applied sciences [1–7]. Furthermore, FPIDEs are applicable in sciences and engineering, and many works in FPIDEs have been introduced (see, for example, [3, 8–11]), 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 [12–14]. The study of the theory of the iterative differential equations began with the work of Eder [15] where Eder worked on a solution of an iterative functional differential equation. Moreover, many studies on iterative differential equations have been conducted (see, for example, [16–18]).
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 differential 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 differential equations [19–22]. The SAM, which is also called the Picard iterative solutions method, has been increasingly applied to solve differential equations and integral equations [23, 24]. The SAM provides an approximate solution in a short series convergent with readily determinable terms [25].
The 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].
This paper presents new analytical and numerical solutions of a new model called “iterative fractional partial integro-differential equation” of iterative Volterra-type equation. This 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. The 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.
2. Preliminaries and Definitions
There are various definitions and theorems of fractional calculus available in the literature. This section presents some of these definitions and theorems that are needed in this paper and can be found in [32–36] and among other references cited therein.
Definition 1. Let and . The Riemann–Liouville integral of time fractional order for a function is defined bywhere is the well-known gamma function.
Definition 2. Let and . The Riemann–Liouville time fractional partial derivative of order for a function is defined by
Definition 3. Let and ; then, the Caputo derivative of time fractional order for a function is
Theorem 1. Let and . Then,
Theorem 2. Let , and . Then,
Lemma 1 (Gronwall–Bellman inequality). Let be a nonnegative continuous function on , . If where is an analytic function and is a nonnegative constant, then .
3. 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 is the -th Caputo fractional partial derivative, is a bivariate kernel, and are known analytic functions, and 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 byfor where is of class from for .
Our extension here is that all the components are continuous where can be given as a sum of successive differences in the following form:
Next, if converges, then converges and the solution for (6) is given by
Lemma 2. Let a function satisfy (6) on ; then,
3.1. Existence and Uniqueness
This section presents new results for existence and uniqueness of solution for the proposed model (6).
Theorem 3. Suppose that and . Then, there is a unique solution for equation (6).
Proof. Let be a Banach space with a norm andwhere and .
Before we apply the Banach contraction principle, we need to define an operator asFrom (12), we haveBy similar argument, we obtainThis proves that is a function from to . Next, for , we haveTherefore, we obtainSince which implies that , then by Banach principle, the operator has a unique fixed point. Therefore, equation (6) has a solution.
Theorem 4 (convergence).. If the assumptions of Theorem 3 are proposed, then (7) converges.
Proof. Define the sequence . Then,Since is a function from to , we get :By induction, we have . Since , we get when . Therefore, goes to zero as goes to infinity. For every subsequence of , there exists a subsequence which uniformly converges and the limit must to be a solution of (6). Thus, uniformly goes to a unique solution of (6).
3.2. 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 Theorem 3 holds. Let be two solutions satisfying equation (6) for with the initial approximations and , respectively. Then, the maximum absolute error for a solution series (7) for (6) is estimated to be
Proof. By using Theorem 3, we haveNext, by using Theorem 4, we haveBy using Gronwall–Bellman inequality given by Lemma 1, we getThus, we obtainThis completes the proof of Theorem 5.
3.3. Algorithms for Computer Implementations
In this section, we introduce algorithms for computing numerical results. Algorithm 1 computes the existence conditions given by Theorem 3.
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Further, Algorithm 2 can be used to obtain particular approximate numerical solutions at particular values of the fractional order .
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4. Analytical Solutions for Iterative Volterra FPIDEs
This section introduces solutions for new examples of iterative FPIDEs. These examples are chosen since their solutions are not available in the literature or they have been solved previously some other well-known methods.for .
Example 1. In this example, we solve the following iterative FPIDEs of Volterra type with initial value:
Then, equation (25) is of form (6) with which satisfieswhere for all and .
As the hypotheses of Theorem 3 are satisfied, a unique solution for equation (25) exists.
Next, by using Theorem 4 and assuming that , the first few iterative solutions are
Therefore, 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:
Equation (30) is of form (8) with , which satisfieswhere for all and . As all the hypotheses of Theorem 3 are satisfied, a unique solution for (30) exists.
By using Theorem 4, we obtain a solution of (30) for different values of . We assume that and by using Mathematica software, the first three iterative solutions are obtained as follows:
Therefore, the third order term iterative solution (30) is .
5. Numerical Solutions and Discussion
Table 1 presents numerical solutions for equation (25) through various values of when . Table 2 includes numerical values of the iterative solutions for equation (30) by different values of at . In Figures 1(a) and 1(b), we plot the graphs of first-order iterative solution for (25) using various values of at respectively. We plot the graph of second-order iterative solution for (25) by various values of at in Figures 2(a) and 2(b), respectively. The iterative solution for Example 1 is graphically represented in Figures 3(a) and 3(b) through various points of when , respectively. Figures 4(a) and 4(b) represent the graphs of solution through various values of for a fixed value of when , respectively, for Example 1. The first-order iterative solution for Example 2 is graphically represented in Figures 5(a) and 5(b) through various values of when , respectively. The second order of iterative solution for Example 2 is graphically represented in Figures 6(a) and 6(b) by various values of when , respectively. The third-order iterative solution for Example 2 is graphically represented in Figures 7(a) and 7(b) by various points of at , respectively. The solution for Example 2 is graphically represented in Figures 8(a) and 8(b) using various points of when , respectively. In Figures 9(a) and 9(b), we plot the graphs of the solution through different values of for a fixed value of when , respectively, for Example 2.
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6. Conclusion
In this paper, we introduced a model of FPIDEs. The 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. This extension derives good approximations and reliable techniques to handle iterative FPIDEs. New solutions for Volterra types of iterative FPIDEs were introduced. The numerical solutions were successfully obtained which confirm the presented results.
Data Availability
The datasets used or analyzed during the current study are available from the corresponding author on reasonable request.
Conflicts of Interest
The authors declare that they have no conflicts of interest.