Existence and Numerical Solution of the Volterra Fractional Integral Equations of the Second Kind

Abstract

This work presents the possible generalization of the Volterra integral equation second kind to the concept of fractional integral. Using the Picard method, we present the existence and the uniqueness of the solution of the generalized integral equation. The numerical solution is obtained via the Simpson 3/8 rule method. The convergence of this scheme is presented together with numerical results.

1. Introduction

The integral equations form an important part of applied mathematics, with links with many theoretical fields, especially with practical fields [1–3]. The Volterra integral [1–3] equations were introduced by Vito Volterra and then studied by Traian Lalescu in his 1908 thesis. Volterra integral equations find application in demography, the study of viscoelastic materials, and in insurance mathematics through the renewal equation. Fredholm equations [4] arise naturally in the theory of signal processing, most notably as the famous spectral concentration problem popularized by David Slepian [4]. They also commonly arise in linear forward modeling and inverse problems. Throughout the last decade, physicists and mathematicians have paid attention to the concept of fractional calculus [5–9]. Actually, real problems in scientific fields such as groundwater problems, physics, mechanics, chemistry, and biology are described by partial differential equations or integral equations. Many scholars have shown with great success the applications of fractional calculus to groundwater pollution and groundwater flow problems [5–9], acoustic wave problems [10], and others [11–14]. There are also several iteration methods for solving fractional integral equations like homotopy decomposition method [15–17], variational iteration method [18–20], Adomian decomposition method [21, 22], and others [23, 24]. But in this work, we will make use of the numerical method called the Simpson 3/8 rule. The general equation under analysis here is given as 𝐹(𝑡)(𝑡)+1=𝐺Γ(𝛼)^𝑡_𝑎(𝑡−𝜏)^𝛼−1[𝐾(𝑡,𝜏)𝐹(𝜏)]𝑑𝜏,0<𝜏≤𝑡≤𝑇;𝛼≥0. (1) Here, 𝐹(𝑡)=𝑓₁(𝑡),𝑓₂(𝑡),𝑓₃(𝑡),𝑓₄(𝑡),…𝑓_𝑛(𝑡)^𝑇,𝐺(𝑡)=𝑔₁(𝑡),𝑔₂(𝑡),𝑔₃(𝑡),𝑔₄(𝑡),…𝑔_𝑛(𝑡)^𝑇,𝐾((𝜏=𝐾𝑡,𝜏,𝐹))₁𝑡,𝜏,𝑓₁(𝜏),𝑓₂(𝜏),𝑓₃(𝜏),𝑓₄(𝜏),…𝑓_𝑛(𝜏𝐾₂)𝑡,𝜏,𝑓₁(𝜏),𝑓₂(𝜏),𝑓₃(𝜏),𝑓₄(𝜏),…𝑓_𝑛(𝜏⋮𝐾_𝑛)𝑡,𝜏,𝑓₁(𝜏),𝑓₂(𝜏),𝑓₃(𝜏),𝑓₄(𝜏),…𝑓_𝑛(𝜏) (2) For the rest of this paper, we assume that 𝑎<𝜏≤𝑡≤𝑇<∞. In this paper, system (1) can be linear or nonlinear.

2. Basic Information about the Fractional Calculus

Definition 1. A real function 𝑓(𝑥), 𝑥>0, is said to be in the space 𝐶_−𝜇, 𝜇∈ℝ, if there exists a real number 𝑝>𝜇, such that 𝑓(𝑥)=𝑥^𝑝ℎ(𝑥), where ℎ(𝑥)∈𝐶[0,∞), and it is said to be in space 𝐶^𝑚_𝜇 if 𝑓^(𝑚)∈𝐶_𝜇, 𝑚∈ℕ.

Definition 2. The Riemann-Liouville fractional integral operator of order 𝛼≥0, of a function 𝑓∈𝐶_𝜇, 𝜇≥−1, is defined as 𝐽^𝛼𝑓(𝑥)=1Γ(𝛼)^𝑥₀(𝑥−𝑡)^𝛼−1𝑓(𝑡)𝐽𝑑𝑡,𝛼>0,𝑥>0,₀𝑓(𝑥)=𝑓(𝑥). (3) Properties of the operator can be found in [25–29]; we mention only the following.

For 𝑓∈𝐶_𝜇, 𝜇≥−1, 𝛼,𝛽≥0 and 𝛾>−1, 𝐽^𝛼𝐽^𝛽𝑓(𝑥)=𝐽^𝛼+𝛽𝑓(𝑥),𝐽^𝛼𝐽^𝛽𝑓(𝑥)=𝐽^𝛽𝐽^𝛼𝑓(𝑥),𝐽^𝛼𝑥^𝛾=Γ𝛾+1Γ𝑥𝛼+𝛾+1^𝛼+𝛾. (4) The fractional derivative of 𝑓(𝑥) in the Caputo sense is defined as ^𝑐_𝑎𝐷^𝛼_𝑥𝑓(𝑥)=𝐽^𝑚−𝛼𝐷^𝑚𝑓(𝑥)=1Γ(𝑚−𝛼)^𝑥_𝑎(𝑥−𝑡)^{𝑚−𝛼−1}𝑓^𝑚(𝑡)𝑑𝑡,𝑚−1<𝛼≤𝑚,𝑚∈ℕ,𝑥>0,𝑓∈𝐶^𝜇₋₁. (5) Also, we need here two of its basic properties.

Lemma 3. If 𝑚−1<𝛼≤𝑚, 𝑚∈ℕ, and 𝑓∈𝐶^𝑚_𝜇, 𝜇≥−1, then ^𝑐_𝑎𝐷^𝛼_𝑥𝐽^𝛼𝑓(𝑥)(𝑥),𝐽=𝑓_𝛼𝐷^𝛼₀𝑓(𝑥)=𝑓(𝑥)−^𝑚−1_𝑘=0𝑓^(𝑘)0⁺𝑥^𝑘𝑘!,𝑥>0. (6)

3. Existence and Uniqueness Analysis

The analysis of the existence and the uniqueness analysis are important aspects that must be investigated before the presentation of the solution. One of the most common techniques used to achieve this is the fixed point theorem technique. To prove the existence and uniqueness of the solution of the system (1), we make use of the method of successive approximation, also called the Picard method [30]. This consists of simple iterations. Before we start this proof, we will assume the following.

First, making use of the vector norm, we assume that ‖𝐹(𝑡)‖=max_{1≤𝑖≤𝑛}𝑓_𝑖(𝑡,‖𝐾()𝑠,𝑡)‖=max^𝑛_{1≤𝑖≤𝑛𝑘}𝑘_𝑖,𝑗(𝑠,𝑡). (7) In this method, we assume that the following iteration can be used to provide a series solution of the problem under investigation: 𝐹_𝑛(𝑡)(𝑡)+1=𝐺Γ(𝛼)^𝑡_𝑎(𝑡−𝜏)^𝛼−1𝐾(𝑡,𝜏)𝐹_𝑛−1(𝜏)𝑑𝜏. (8) It also assumes that the initial component of the series solution is given as 𝐹₀(𝑡)=𝐺(𝑡). (9) Let us in addition put the difference between the consecutive components as 𝛿_𝑛(𝑡)=𝐹_𝑛(𝑡)−𝐹_𝑛−1(𝑡). (10) It is very easy to see that 𝐹_𝑛(𝑡)=^𝑛_𝑖=0𝛿_𝑖(𝑡). (11)

3.1. Existence and Uniqueness of the Linear Volterra Fractional Integral Equations of the Second Kind

Theorem 4 (see [31]). Under the conditions that the vector functions 𝐺(𝑡)and 𝐾(𝑠,𝑡) are continuous 0≤𝑎<𝜏≤𝑡≤𝑇<∞, then, the system of Volterra fractional integral equations of the second kind (1) has a unique continuous solution for0≤𝑎<𝑡≤𝑇<∞.
The proof is similar to the one in [31].

Theorem 5. Assuming that the system (1) has a unique solution, say 𝐹(𝑡) in 𝑎<𝑡≤𝑇, such that 𝐾(𝑡,𝜏)𝐹(𝜏) is absolutely fractionally integrable, and if in addition ‖𝐺(𝑡)‖<𝑔(𝑡),‖𝐾(𝑡,𝑠)‖<𝑘(𝑡,𝑠), (12) providing that 𝑘 and 𝑔 are continuous functions, then it is possible to find a function, say 𝑓(𝑡), such that ‖𝐹(𝑡)‖<𝑓(𝑡), (13) where 𝑓(𝑡) is the continuous function solutions of 𝑓(𝑡)(𝑡)+1=𝑔Γ(𝛼)^𝑡_𝑎(𝑡−𝜏)^𝛼−1𝐾(𝑡,𝜏)𝑓(𝜏)𝑑𝜏. (14)

Proof. From (1), applying the vector norm on both sides, we obtain the following: ‖𝐹(𝑡=𝐺(𝑡)+1)‖Γ(𝛼)^𝑡_𝑎(𝑡−𝜏)^1−𝛼[𝐾(𝑡,𝜏)𝐹(𝜏)]𝑑𝜏. (15) Now, making use of the inequality triangular, we obtain the following: ‖𝐹(𝑡≤‖𝐺(𝑡+1)‖)‖Γ(𝛼)^𝑡_𝑎(𝑡−𝜏)^𝛼−1[𝐾()𝐹(𝜏,‖𝐹(𝑡≤‖𝐺(𝑡+1𝑡,𝜏)]𝑑𝜏)‖)‖Γ(𝛼)^𝑡_𝑎(𝑡−𝜏)^𝛼−1[‖𝐾(𝑡,𝜏)‖‖𝐹(𝜏)‖]𝑑𝜏. (16)
Thus, making use of the hypothesis, we obtain ‖𝐹(𝑡(𝑡)+1)‖<𝑔Γ(𝛼)^𝑡_𝑎(𝑡−𝜏)^𝛼−1[𝑘(𝑡,𝜏)‖𝐹(𝜏)‖]𝑑𝜏. (17)
If now the difference between (14) and (17) gives 𝑓(𝑡)−‖𝐹(𝑡>1)‖Γ(𝛼)×^𝑡_𝑎(𝑡−𝜏)^𝛼−1×𝑘(𝑡,𝜏)𝑓(𝑡)−‖𝐹(𝜏)‖𝑑𝜏, (18) since 𝑓(𝑎)−‖𝐹(𝑎)‖>0 and also 𝑘(𝑡,𝜏) is a continuous positive function, it is then true to conclude that 𝑓(𝑡)−‖𝐹(𝑡)‖>0,for𝑡≤𝑇, (19) which concludes the proof.

Theorem 6. Under the condition that 𝐺(𝑡), 𝐾(𝑡,𝑠), Δ𝐺(𝑡), and Δ𝐾(𝑡,𝑠) are smooth functions and bounded one has ‖𝐾(‖(‖𝐺(𝑡‖(𝑡𝑡,𝜏)‖≤𝑘,Δ𝐾𝑡,𝜏)‖≤Δ𝑘,)‖≤𝑔,Δ𝐺)‖≤Δ𝑔. (20)
Let 𝐹_{𝑒𝑥𝑎𝑐𝑡}(𝑡) be the exact solution of 𝐹_{𝑒𝑥𝑎𝑐𝑡}(𝑡)(𝑡)(𝑡)+1=𝐺+Δ𝐺Γ(𝛼)^𝑡_𝑎(𝑡−𝜏)^𝛼−1×𝐾()(𝐹𝑑𝜏(𝑡,𝜏−Δ𝐾𝑡,𝜏))_{𝑒𝑥𝑎𝑐𝑡}(𝜏); (21) then, 𝐹_{𝑒𝑥𝑎𝑐𝑡}(𝑡)(𝑡≤𝐸−𝐹)Δ𝑔+Δ𝑘𝑡𝑔+Δ𝑔_𝛼((𝑘+Δ𝑘)𝑡)×𝐸_𝛼𝑘^−𝛼𝑡=𝑂Δ𝑔+𝑂(Δ𝑘), (22) with 𝐹_{𝑒𝑥𝑎𝑐𝑡} is the solution of system (1).

Proof. Since 𝐺(𝑡), 𝐾(𝑡,𝑠) are smooth functions and bounded, using Theorem 4, there exists a positive smooth function 𝑓(𝑡) such that 𝑓(𝑡)(𝑡)+1=𝑔Γ(𝛼)^𝑡_𝑎(𝑡−𝜏)^𝛼−1𝑘𝑓(𝜏)𝑑𝜏. (23) For simplicity, we chose 𝑎=0; then, 𝑓(𝑡)(𝑡)+𝑘=𝑔Γ(𝛼)^𝑡₀(𝑡−𝜏)^𝛼−1𝑓(𝜏)𝑑𝜏. (24) Using the methodology of the homotopy decomposition method, we arrive at the following exact solution: 𝑓(𝑡)=𝑔𝐸_𝛼𝑘^−𝛼𝑡, (25) where 𝐸_𝛼𝑘^−𝛼𝑡=^∞_𝑛=0𝑘^𝑛𝑡^𝑛𝛼Γ[1+𝑛𝛼] (26) known as the Mittag-Leffler function. Therefore, ‖𝐹(𝑡)‖<𝑔𝐸_𝛼𝑘^−𝛼𝑡. (27) With the above in hand, it is very easy to show that 𝐹_exact(𝑡)≤𝑔+Δ𝑔𝐸_𝛼(𝑘+Δ𝑘)^−𝛼𝑡. (28) Since 𝐹_exact(𝑡) is the approximate solution of system (1), then it follows that 𝐹_exact(𝑡)(𝑡)+1≈𝐺Γ(𝛼)^𝑡₀(𝑡−𝜏)^𝛼−1𝐾(𝑡,𝜏)𝐹_exact(𝜏)𝑑𝜏. (29) So the error in the approximation can be represented as 𝑅(𝑡)(𝑡)+1=𝐺Γ(𝛼)^𝑡₀(𝑡−𝜏)^𝛼−1×𝐾(𝑡,𝜏)𝐹_exact(𝜏)𝑑𝜏−𝐹_exact(𝑡). (30) Now replacing 𝐹_exact(𝑡) as in (21), we obtain 𝑅(𝑡)(𝑡)+1=−Δ𝐺Γ(𝛼)^𝑡₀(𝑡−𝜏)^𝛼−1Δ𝐾(𝑡,𝜏)𝐹_exact(𝜏)𝑑𝜏. (31) Then, the difference between the exact solution and the approximate solution can be obtained as 𝐹(𝑡)−𝐹_exact(𝑡)=𝑅(𝑡)+^𝑇₀1Γ(𝛼)×(𝑡−𝜏)^𝛼−1[()×𝐹(𝑡)Δ𝐾𝑡,𝜏−𝐹_exact(𝜏)𝑑𝜏 (32) so that 𝐹(𝑡)−𝐹_exact(𝑡𝐹(𝑡≤)<Δ𝑔+▵𝑘𝑡max(‖)‖)Δ𝑔+Δ𝑘𝑡𝑔+Δ𝑔×𝐸_𝛼((𝑘+Δ𝑘)𝑡)𝐸_𝛼𝑘^−𝛼𝑡=𝑂Δ𝑔+𝑂(Δ𝑘), (33) which completes the proof.

3.2. Existence and Uniqueness of the Nonlinear Volterra Fractional Integral Equations of the Second Kind

In this case, the nonlinear Volterra fractional integral equations of the second kind considered here are 𝐹(𝑡)(𝑡)+1=𝐺Γ(𝛼)^𝑡_𝑎(𝑡−𝜏)^𝛼−1[𝐾(𝑡,𝜏,𝐹(𝜏))]𝑑𝜏,0<𝜏≤𝑡≤𝑇;𝛼≥0. (34)𝐹(𝑡), 𝐺(𝑡), and 𝐾(𝑡,𝜏,𝐹(𝜏)) have the same form as in the previous subsection; also the norm used in the previous section is maintained.

In analogy with what was done in Section 3.1, we define the iteration formula as 𝐹_𝑛(𝑡)(𝑡)+1=𝐺Γ(𝛼)^𝑡_𝑎(𝑡−𝜏)^𝛼−1[𝐾(𝑡,𝜏,𝐹(𝜏))]𝑑𝜏, (35) with initial component 𝐹₀(𝑡)=𝐺(𝑡). (36) Similarly, the difference between the consecutive terms is given as 𝐹_𝑛(𝑡)−𝐹_𝑛−1(𝑡)=1Γ(𝛼)^𝑡_𝑎(𝑡−𝜏)^𝛼−1𝐾𝑡,𝜏,𝐹_𝑛−1(𝜏)−𝐾𝑡,𝜏,𝐹_𝑛−2(𝜏)𝑑𝜏. (37) We will perhaps recall that in case the kernel 𝐾 satisfies the Lipchitz conditions, we have the following inequality: 𝐾𝑡,𝜏,𝐹₁−𝐾𝑡,𝜏,𝐹₂≤𝐻𝐹₁−𝐹₂, (38) with of course 𝐻 being a real positive number not depending on the parameters 𝑡, 𝜏, 𝐹₁, and 𝐹₂.

As in Section 3.1, we put 𝛿_𝑛(𝑡)=𝐹_𝑛(𝑡)−𝐹_𝑛−1(𝑡). (39) Again, we have that 𝐹_𝑛(𝑡)=^𝑛_𝑖=0𝛿_𝑖(𝑡). (40) Then, 𝛿_𝑛(𝑡=1)Γ(𝛼)^𝑡_𝑎(𝑡−𝜏)^𝛼−1𝐾𝑡,𝜏,𝐹_𝑛−1(𝜏)−𝐾𝑡,𝜏,𝐹_𝑛−2(𝜏)𝛿𝑑𝜏_𝑛(𝑡≤1)Γ(𝛼)^𝑡₀(𝑡−𝜏)^𝛼−1×𝐾𝑡,𝜏,𝐹_𝑛−1(𝜏)−𝐾𝑡,𝜏,𝐹_𝑛−2(𝜏)𝑑𝜏. (41) In case the Lipchitz condition is satisfied by the kernel, we have the following inequality:𝛿_𝑛(𝑡≤1)Γ(𝛼)^𝑡_𝑎(𝑡−𝜏)^𝛼−1×𝐾𝑡,𝜏,𝐹_𝑛−1(𝜏)𝑑𝜏,−𝐾𝑡,𝜏,𝐹_𝑛−2(𝜏)𝛿_𝑛(𝑡≤1)Γ(𝛼)^𝑡_𝑎(𝑡−𝜏)^𝛼−1𝛿_𝑛−1𝑑𝜏. (42)(43) We will then present the following theorem.

Theorem 7. Under the conditions that 𝐺(𝑡), 𝐾(𝑡,𝜏,𝐹) are continuous in 0<𝜏≤𝑡≤𝑇<∞,  −∞<𝐹<∞ and the kernel satisfies the Lipchitz condition, that is, ‖𝐾(𝑡,𝜏,𝐹₁)−𝐾(𝑡,𝜏,𝐹₂)‖≤𝐻‖𝐹₁−𝐹₂‖, then, (34) has a unique solution.

Proof. From (43), it follows that 𝛿_𝑛(𝑡)≤max_{0<𝑡≤𝑇}‖𝐺(𝑡𝐻)‖^−𝛼𝑡^𝑛𝛼Γ();1+𝑛𝛼 (44) therefore, 𝐹(𝑡)=^𝑛_𝑖=0𝛿_𝑛(𝑡) (45) exists and is a continuous function. However, to prove that the above function is the solution of the system of the nonlinear Volterra fractional integral equations (34) of the second kind, we let 𝐹(𝑡)=𝐹_𝑛(𝑡)−𝑃_𝑛(𝑡). (46) Now, using (36), we have the following equation: 𝐹(𝑡)−𝐹_𝑛(𝑡)(𝑡)+1=𝐺Γ(𝛼)^𝑡₀(𝑡−𝜏)^𝛼−1𝐾𝑡,𝜏,𝐹(𝜏)−𝑃_𝑛(𝜏)𝑑𝜏. (47) It follows that 𝐹(𝑡)(𝑡)−1−𝐺Γ(𝛼)^𝑡₀(𝑡−𝜏)^𝛼−1𝐾(𝑡,𝜏,𝐹(𝜏))𝑑𝜏=𝑃_𝑛(𝑡)+1Γ(𝛼)^𝑡₀(𝑡−𝜏)^𝛼−1×𝐾𝑡,𝜏,𝐹(𝜏)−𝑃_𝑛(𝜏)−𝐾(𝑡,𝜏,𝐹(𝜏))𝑑𝜏). (48)
Now, applying the norm and Lipchitz condition, we arrive at the following inequality: 𝐹(𝑡)(𝑡)−1−𝐺Γ(𝛼)^𝑡₀(𝑡−𝜏)^𝛼−1𝐾((𝜏≤𝑃𝑡,𝜏,𝐹))𝑑𝜏_𝑛(𝑡)+𝐻𝑡𝑃_𝑛−1(𝑡). (49) Applying the limit on both sides of the above inequality when 𝑛 tends to infinity, the right-hand side tends to zero; then, 𝐹(𝑡) in (45) satisfies 𝐹(𝑡)(𝑡)+1=𝐺Γ(𝛼)^𝑡₀(𝑡−𝜏)^𝛼−1𝐾(𝑡,𝜏,𝐹(𝜏))𝑑𝜏, (50) and indeed it is the solution of (34).
We will now present the uniqueness of this solution. To achieve this, we assume that (34) has another solution, say 𝐹₁(𝑡); then, 𝐹(𝑡)−𝐹₁(𝑡)=1Γ(𝛼)^𝑡₀(𝑡−𝜏)^𝛼−1×[𝐾(𝑡,𝜏,𝐹(𝜏))−𝐾𝑡,𝜏,𝐹₁(𝜏)𝑑𝜏. (51) Applying the norm and making use of the Lipchitz condition of the kernel, we arrive at 𝐹(𝑡)−𝐹₁(𝑡)≤𝐻^𝑡₀(𝑡−𝜏)^𝛼−1𝐹(𝜏)−𝐹₁(𝜏)𝑑𝜏, (52) but ‖𝐹(𝑡)−𝐹₁(𝑡)‖≤𝐷; then, 𝐹(𝑡)−𝐹₁(𝑡𝐻)≤𝐷^−𝛼𝑡^𝑛𝛼Γ(),1+𝑛𝛼 (53) for any 𝑛; then, 𝐹(𝑡)=𝐹₁(𝑡). (54)

4. Numerical Method to Solve the Volterra Fractional Integral Equations

We consider the general form of the Volterra fractional integral equation as 𝐹(𝑡)(𝑡)+1=𝐺Γ(𝛼)^𝑡_𝑎(𝑡−𝜏)^𝛼−1×[𝐾(𝑡,𝜏,𝐹(𝜏))]𝑑𝜏,0<𝜏≤𝑡≤𝑇;𝛼≥0, (55)

where 𝐺(𝑡) is a known function, 𝐹(𝑡) is an unknown function to be determined, and the kernel is 𝐾(𝑡,𝑦,𝑓).

In numerical analysis, Simpson’s rule is a method for numerical integration, the numerical approximation of definite integrals [32]. Simpson’s rule also corresponds to the 3-point Newton-Cotes quadrature rule. The method is credited to the mathematician Thomas Simpson (1710–1761) of Leicestershire, England. Simpson’s rule is a staple of scientific data analysis and engineering. It is widely used, for example, by naval architects, to numerically integrate hull offsets and cross-sectional areas to determine volumes and centroids of ships or lifeboats [33].

4.1. Application of the Simpson 3/8 to Volterra Fractional Integral Equation

To use Simpson’s rule here, we let 0=𝑎<𝑡₁<𝑡₂<𝑡₃⋯<𝑡_𝑛 be a possible division of [0,𝑏], with step size 𝑥_𝑖=𝑖𝑘 for 𝑖=0,1,…𝑁. We construct a block by block method that is the system of Volterra fractional integral equation (1) for 𝑞>1 simultaneous equations is then a set of 𝑞 simultaneous value of the function 𝐹. Without loss of generality, we consider 𝑞=6.

For the rest of the paper, (𝑡−𝜏)^𝛼−1𝐾[𝑡,𝜏𝐹(𝜏)]=𝐾_𝛼(𝑡,𝜏,𝐹(𝜏)) (56) will be called the fractional kernel. Having the fractional kernel in hand, system (1) can be rewritten as follows: 𝐹(𝑡)(𝑡)+1=𝐺Γ(𝛼)^𝑡₀𝐾_𝛼(𝑡,𝜏,𝐹(𝜏))𝑑𝜏. (57) Now, if we set 𝑡=𝑡_3𝑛+1 in the previous equation, we obtain 𝐹_1,3𝑛+1(𝑡)=𝑔₁𝑡_3𝑛+1+1Γ(𝛼)^𝑡3𝑛+1₀𝐾_𝛼(1,1)𝑡_3𝑛+1,𝜏,𝑓₁(𝜏+1)𝑑𝜏Γ(𝛼)^𝑡3𝑛₀𝐾_𝛼(1,2)𝑡_3𝑛+1,𝜏,𝑓₂(𝜏+1)𝑑𝜏Γ(𝛼)^𝑡3𝑛+1_𝑡3𝑛𝐾_𝛼(1,1)𝑡_3𝑛+1,𝜏,𝑓₁(𝜏+1)𝑑𝜏Γ(𝛼)^𝑡3𝑛+1_𝑡3𝑛𝐾_𝛼(1,2)𝑡_3𝑛+1,𝜏,𝑓₂(𝜏𝐹)𝑑𝜏,_2,3𝑛+1(𝑡)=𝑔₂𝑡_3𝑛+1+1Γ(𝛼)^𝑡3𝑛+1₀𝐾_𝛼(2,1)𝑡_3𝑛+1,𝜏,𝑓₂(𝜏+1)𝑑𝜏Γ(𝛼)^𝑡3𝑛₀𝐾_𝛼(2,2)𝑡_3𝑛+1,𝜏,𝑓₂(𝜏+1)𝑑𝜏Γ(𝛼)^𝑡3𝑛+1_𝑡3𝑛𝐾_𝛼(2,1)𝑡_3𝑛+1,𝜏,𝑓₂(𝜏+1)𝑑𝜏Γ(𝛼)^𝑡3𝑛+1_𝑡3𝑛𝐾_𝛼(2,2)𝑡_3𝑛+1,𝜏,𝑓₂(𝜏)𝑑𝜏. (58)(59) From here if one integrates over the interval [0,𝑡_3𝑛], we can apply Simpson’s 3/8 rule, and also by integrating over [𝑡_3𝑛,𝑡_3𝑛+1], one can calculate it by using a cubic interpolation. Then, we can have the following: 𝐹_1,3𝑛+1𝑡_3𝑛+1=𝑔₁𝑡_3𝑛+1+3ℎ(𝛼)𝐾8Γ^𝛼(1,1)𝑡_3𝑛+1,𝑡₀,𝐹_1,0+3𝐾_𝛼(1,1)𝑡_3𝑛+1,𝑡₁,𝐹_1,1+3𝐾_𝛼(1,1)𝑡_3𝑛+1,𝑡₂,𝐹_1,2+2𝐾_𝛼(1,1)𝑡_3𝑛+1,𝑡₃,𝐹_1,3+⋯+𝐾_𝛼(1,1)𝑡_3𝑛+1,𝑡_3𝑛,𝐹_1,3𝑛+3ℎ(𝛼)𝐾8Γ^𝛼(2,1)𝑡_3𝑛+1,𝑡₀,𝐹_2,0+3𝐾_𝛼(2,1)𝑡_3𝑛+1,𝑡₁,𝐹_2,1+3𝐾_𝛼(2,1)𝑡_3𝑛+1,𝑡₂,𝐹_2,2+2𝐾_𝛼(2,1)𝑡_3𝑛+1,𝑡₃,𝐹_2,3+⋯+𝐾_𝛼(2,1)𝑡_3𝑛+1,𝑡_3𝑛,𝐹_2,3𝑛+ℎ(𝛼)𝐾8Γ^𝛼(1,1)𝑡_3𝑛+1,𝑡_3𝑛,𝐹_1,3𝑛+3𝐾_𝛼(1,1)×𝑡_3𝑛+1,𝑡_3𝑛+1/3,40𝐹81^1,3𝑛−20𝐹27^1,3𝑛+1−8𝐹27^1,3𝑛+1+5𝐹81^1,3𝑛+3+3𝐾_𝛼(1,1)×𝑡_3𝑛+1,𝑡_3𝑛+2/3,14𝐹81_1,3𝑛+1−7𝐹27^1,3𝑛+2+4𝐹81^1,3𝑛+3+𝐾_𝛼(1,1)𝑡_3𝑛+1,𝑡_3𝑛+1,𝐹_1,3𝑛+ℎ(𝛼)𝐾8Γ^𝛼(1,2)𝑡_3𝑛+1,𝑡_3𝑛,𝐹_2,3𝑛+3𝐾_𝛼(1,2)𝑡_3𝑛+1,𝑡_3𝑛+1/3,40𝐹81_2,3𝑛−20𝐹27^2,3𝑛+1−8𝐹27^2,3𝑛+2+5𝐹81^2,3𝑛+3+3𝐾_𝛼(1,2)𝑡_3𝑛+1,𝑡_3𝑛+2/3,14𝐹81^2,3𝑛+1−7𝐹27^2,3𝑛+2+4𝐹81^2,3𝑛+3+𝐾_𝛼(1,2)𝑡_3𝑛+1,𝑡_3𝑛+1,𝐹_2,3𝑛. (60) In a similar way, if one let 𝑥=𝑥_3𝑛+2, we get the following: 𝐹_1,3𝑛+2(𝑡)=𝑔₁𝑡_3𝑛+2+1Γ(𝛼)^𝑡3𝑛+2₀𝐾_𝛼(1,1)×𝑡_3𝑛+2,𝜏,𝑓₁(𝜏+1)𝑑𝜏Γ(𝛼)^𝑡3𝑛₀𝐾_𝛼(1,2)×𝑡_3𝑛+2,𝜏,𝑓₂(𝜏+1)𝑑𝜏Γ(𝛼)^𝑡3𝑛+2_𝑡3𝑛𝐾_𝛼(1,1)×𝑡_3𝑛+1,𝜏,𝑓₁(𝜏+1)𝑑𝜏Γ(𝛼)^𝑡3𝑛+2_𝑡3𝑛𝐾_𝛼(1,2)×𝑡_3𝑛+2,𝜏,𝑓₂(𝜏𝐹)𝑑𝜏,_2,3𝑛+2(𝑡)=𝑔₂𝑡_3𝑛+1+1Γ(𝛼)^𝑡3𝑛+2₀𝐾_𝛼(2,1)×𝑡_3𝑛+2,𝜏,𝑓₂(𝜏+1)𝑑𝜏Γ(𝛼)^𝑡3𝑛₀𝐾_𝛼(2,2)×𝑡_3𝑛+2,𝜏,𝑓₂(𝜏+1)𝑑𝜏Γ(𝛼)^𝑡3𝑛+2_𝑡3𝑛𝐾_𝛼(2,1)×𝑡_3𝑛+2,𝜏,𝑓₂(𝜏+1)𝑑𝜏Γ(𝛼)^𝑡3𝑛+2_𝑡3𝑛𝐾_𝛼(2,2)×𝑡_3𝑛+2,𝜏,𝑓₂(𝜏)𝑑𝜏. (61) From here if one integrates over the interval [0,𝑡_3𝑛], we can apply Simpson’s 3/8 rule, and also by integrating over [𝑡_3𝑛,𝑡_3𝑛+2], one can calculate it by using a cubic interpolation. Then, we can have the following: 𝐹_1,3𝑛+2𝑡_3𝑛+2=𝑔₁𝑡_3𝑛+2+3ℎ(𝛼)𝐾8Γ^𝛼(1,1)𝑡_3𝑛+2,𝑡₀,𝐹_1,0+3𝐾_𝛼(1,1)𝑡_3𝑛+1,𝑡₁,𝐹_1,1+3𝐾_𝛼(1,1)𝑡_3𝑛+2,𝑡₂,𝐹_1,2+2𝐾_𝛼(1,1)𝑡_3𝑛+2,𝑡₃,𝐹_1,3+⋯+𝐾_𝛼(1,1)𝑡_3𝑛+2,𝑡_3𝑛,𝐹_1,3𝑛+3ℎ(𝛼)𝐾8Γ^𝛼(2,1)𝑡_3𝑛+2,𝑡₀,𝐹_2,0+3𝐾_𝛼(2,1)𝑡_3𝑛+2,𝑡₁,𝐹_2,1+3𝐾_𝛼(2,1)𝑡_3𝑛+2,𝑡₂,𝐹_2,2+2𝐾_𝛼(2,1)𝑡_3𝑛+2,𝑡₃,𝐹_2,3+⋯+𝐾_𝛼(2,1)𝑡_3𝑛+1,𝑡_3𝑛,𝐹_2,3𝑛+ℎ(𝛼)𝐾4Γ^𝛼(1,1)𝑡_3𝑛+2,𝑡_3𝑛,𝐹_1,3𝑛+3𝐾_𝛼(1,1)×𝑡_3𝑛+2,𝑡_3𝑛+2/3,14𝐹81^1,3𝑛+28𝐹27^1,3𝑛+2−7𝐹27^1,3𝑛+2+4𝐹81^1,3𝑛+3,+3𝐾_𝛼(1,1)×𝑡_3𝑛+1,𝑡_3𝑛+2/35,−𝐹81_1,3𝑛+20𝐹27^1,3𝑛+1+10𝐹27^1,3𝑛+2−4𝐹81^1,3𝑛+3+𝐾_𝛼(1,1)𝑡_3𝑛+2,𝑡_3𝑛+1,𝐹_1,3𝑛+ℎ(𝛼)𝐾8Γ^𝛼(1,2)𝑡_3𝑛+2,𝑡_3𝑛,𝐹_2,3𝑛+3𝐾_𝛼(1,2)×𝑡_3𝑛+1,𝑡_3𝑛+2/3,40𝐹81^2,3𝑛−20𝐹27^2,3𝑛+2−8𝐹27^2,3𝑛+2+5𝐹81^2,3𝑛+3+3𝐾_𝛼(1,2)×𝑡_3𝑛+1,𝑡_3𝑛+2/3,14𝐹81^2,3𝑛+1−7𝐹27^2,3𝑛+2+4𝐹81^2,3𝑛+3+𝐾_𝛼(1,2)𝑡_3𝑛+2,𝑡_3𝑛+2,𝐹_2,3𝑛+3. (62) Using the similar formulas, we can obtain 𝐹_1,3𝑛+3 and 𝐹_2,3𝑛+3. Now, from (58) to (62), we formed a system of six equations with normally six unknowns for 𝑛=1,2,….

In particular, we do have six simultaneous equations for each step. Our next concern in this work is to show the convergence analysis of Simpson’s 3/8 rule for solving the Volterra fractional integral equations, and this will be presented in the next section.

5. Convergence Analysis of Simpson’s 3/8 Rule for Solving the Volterra Fractional Integral Equations

This section is devoted to the discussion underpinning the convergence of the well-known Simpson’s 3/8 rule to approximate the Volterra fractional equation of second kind. There are also other numerical methods to deal with these equations [32, 34–36]. To achieve this, and without loss of generality, we assume that the error in approximating the solution of the Volterra fractional equation of second kind via Simpson’s 3/8 rule is 𝑅_1,3𝑛+1 for the first approximation in (60); the rest can be obtained similarly; then, 𝑅_1,3𝑛+1=𝐹_1,3𝑛+1−𝑓₁𝑥_3𝑛+1≤1Γ(𝛼)ℎ^3𝑚_𝑘=0𝜓_𝑘𝐾_𝛼(1,1)×𝑡_3𝑛+1,𝑡_𝑘,𝐹_1,𝑘+ℎ^3𝑚_𝑘=0𝜓_𝑘𝐾_𝛼(1,2)×𝑡_3𝑛+1,𝑡_𝑘,𝐹_2,𝑘+ℎ(𝛼)×𝐾8Γ_𝛼(1,1)𝑡_3𝑛+1,𝑡_3𝑛,𝐹_1,3𝑛+3𝐾_𝛼(1,1)×𝑡_3𝑛+1,𝑡_3𝑛+1/3,40𝐹81_1,3𝑛−20𝐹27^1,3𝑛+1−8𝐹27^1,3𝑛+2+5𝐹81^1,3𝑛+3+3𝐾_𝛼(1,1)×𝑡_3𝑛+1,𝑡_3𝑛+2/3,14𝐹81_1,3𝑛+1−7𝐹27^1,3𝑛+2+4𝐹81^1,3𝑛+3+𝐾_𝛼(1,1)𝑡_3𝑛+1,𝑡_3𝑛+1,𝐹_1,3𝑛+ℎ(𝛼)×𝐾8Γ_𝛼(1,2)𝑡_3𝑛+1,𝑡_3𝑛,𝐹_2,3𝑛+3𝐾_𝛼(1,2)×𝑡_3𝑛+1,𝑡_3𝑛+1/3,40𝐹81_2,3𝑛−20𝐹27^2,3𝑛+1−8𝐹27^2,3𝑛+2+5𝐹81^2,3𝑛+3+3𝐾_𝛼(1,2)×𝑡_3𝑛+1,𝑡_3𝑛+2/3,14𝐹81^2,3𝑛+1−7𝐹27^2,3𝑛+2+4𝐹81^2,3𝑛+3+𝐾_𝛼(1,2)𝑡_3𝑛+1,𝑡_3𝑛+1,𝐹_2,3𝑛+ℎ(𝛼)×𝐾8Γ_𝛼(1,2)𝑡_3𝑛+2,𝑡_3𝑛,𝐹_2,3𝑛+3𝐾_𝛼(1,2)×𝑡_3𝑛+1,𝑡_3𝑛+2/3,40𝐹81_2,3𝑛−20𝐹27^2,3𝑛+2−8𝐹27^2,3𝑛+2+5𝐹81^2,3𝑛+3+3𝐾_𝛼(1,2)×𝑡_3𝑛+1,𝑡_3𝑛+2/3,14𝐹81^2,3𝑛+1−7𝐹27^2,3𝑛+2+4𝐹81^2,3𝑛+3+𝐾_𝛼(1,2)𝑡_3𝑛+2,𝑡_3𝑛+2,𝐹_2,3𝑛+3+ℎ8𝐾_1,2×𝑥_3𝑛+1,𝑥_3𝑛+1,𝐹_2,3𝑛+1−²_𝑙=10^3𝑛+1𝐾_1,𝑙𝑡_3𝑛+1,𝜏,𝑓_𝑙(𝜏)𝑑𝜏. (63) Now, employing the Lipchitz condition for the fractional kernel function, we can arrive at the following: 𝑅_1,3𝑛+1≤ℎΓ(𝛼)𝑎₁^3𝑛_𝑘=0𝑅_1,𝑘+ℎΓ(𝛼)𝑎₂×^3𝑛_𝑘=0𝑅_2,𝑘+ℎΓ(𝛼)𝑎₃𝑅_1,3𝑛+1+ℎΓ(𝛼)𝑎₄𝑅_2,3𝑛+1+ℎΓ(𝛼)𝑎₅𝑅_1,3𝑛+2+ℎΓ(𝛼)𝑎₇𝑅_1,3𝑛+3+ℎΓ(𝛼)𝑎₈𝑅_2,3𝑛+3+1Γ(𝛼)𝑤_1,3𝑛+1+𝑤_2,3𝑛+1+𝑤_1,3𝑛+2+𝑤_2,3𝑛+2. (64) Here, it is important to recall that 𝑤_𝑘,3𝑛+1, 𝑤_𝑘,3𝑛+2(𝑘=1,2) are the errors of integration rule. In addition, without loss of generality, we assume that 𝑅_𝑘,𝑖∞=max_𝑘=1,2max_{𝑖=3𝑛+1,3𝑛+2,3𝑛+3}𝑅_𝑘,𝑖=𝑅_1,3𝑛+1; (65) thus, by letting 𝐴=max_𝑚[|𝑤_1,𝑚|,|𝑤_2,𝑚|], consequently 𝑅_𝑘,𝑖∞≤ℎΓ(𝛼)𝑎^3𝑛_𝑛=0𝑅_1,𝑛,𝑅_2,𝑛+6ℎΓ(𝛼)𝑎^′𝑅_𝑘,𝑖∞+4𝐴. (66) Now by rearranging, we obtain the following inequality: 𝑅_𝑘,𝑖∞≤(ℎ/Γ(𝛼))𝑎1−(6ℎ/Γ(𝛼))𝑎^′×^3𝑛_𝑛=0𝑅_1,𝑛,𝑅_2,𝑛+4𝐴1−(6ℎ/Γ(𝛼))𝑎^′. (67) However, making use of the so-called Gronwall inequality, we arrive at 𝑅_𝑘,𝑖∞≤4𝐴1−(6ℎ/Γ(𝛼))𝑎^′Expℎ𝑎1−(6ℎ/Γ(𝛼))𝑎^′𝑥_𝑛. (68) For the fractional kernel function 𝐾_𝛼 and 𝐹 with at least fourth-order derivatives, we have 𝐴=𝑂(ℎ⁴) and then ‖𝑅_𝑘,𝑖‖_∞=𝑂(ℎ⁴). Therefore, we can state the following theorem.

Theorem 8. Simpson’s 3/8 rule for solving the Volterra fractional integral equations of second kind is convergent and its order of convergence is at least four.

6. Numerical Solutions

In this section, we present some numerical examples of solutions of the Volterra fractional integral equations via the so-called Simpson’s 3/8 rule.

Example 9. Let us consider the following Volterra fractional integral equation for which the order is half: 𝑓(𝑥)=2√𝑥−^𝑥₀𝑓(𝑡)√𝑥−𝑡𝑑𝑥=𝑥,0≤𝑡<𝑥<1. (69)
The exact solution of this equation is given as 𝑓(𝑥)=1−exp[𝜋𝑥]er𝑓√𝑥𝜋. (70) Using the Simpson 3/8, we obtained the following numerical values indicated in Table 1.

Table 1 

Numerical errors corresponding to the value of ℎ.

The approximate solutions have been depicted in Figures 1, 2, and 3. Figure 1 shows the comparison of the exact and approximate solutions for ℎ=0.001, Figure 2 shows the comparison for ℎ=0.03, and Figure 3 shows the comparison for ℎ=0.04. The numerical solution shows that the method is very efficient and accurate.

Figure 1 

Comparison of exact solution and approximate solution for ℎ=0.001.

Figure 2 

Comparison of exact solution and the approximate solution for ℎ=0.03.

Figure 3 

Comparison of the exact and approximate solution for ℎ=0.04.

7. Conclusion

The existence and the uniqueness of the Volterra fractional integral equations second kind were examined in this work. The numerical method called the Simpson 3/8 rule method was used to present the numerical solution of these equations. We presented the convergence analysis of this numerical scheme.

Conflict of Interests

The authors declare no conflict of interests.

Author’s Contribution

Abdon Atangana wrote the first draft and Necdet Bildik corrected the revised form; the both authors read the revised and submitted the paper.