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.
The integral equations form an important part of applied mathematics, with links with many theoretical fields, especially with practical fields [
A real function
The RiemannLiouville fractional integral operator of order
For
If
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 (
First, making use of the vector norm, we assume that
Under the conditions that the vector functions
The proof is similar to the one in [
Assuming that the system (
From (
Thus, making use of the hypothesis, we obtain
If now the difference between (
Under the condition that
Let
Since
In this case, the nonlinear Volterra fractional integral equations of the second kind considered here are
In analogy with what was done in Section
As in Section
Under the conditions that
From (
Now, applying the norm and Lipchitz condition, we arrive at the following inequality:
We will now present the uniqueness of this solution. To achieve this, we assume that (
We consider the general form of the Volterra fractional integral equation as
where
In numerical analysis, Simpson’s rule is a method for numerical integration, the numerical approximation of definite integrals [
To use Simpson’s rule here, we let
For the rest of the paper,
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
This section is devoted to the discussion underpinning the convergence of the wellknown Simpson’s
Simpson’s
In this section, we present some numerical examples of solutions of the Volterra fractional integral equations via the socalled Simpson’s
Let us consider the following Volterra fractional integral equation for which the order is half:
The exact solution of this equation is given as
Numerical errors corresponding to the value of

 

0.001  0.01  0.02 0.03  0.04  0.05  
Error (1) 
Error (2) 
Error (3) 
Error (5) 
Error (6) 

0  0.01  0.02  0.03  0.04  0.05 
0.1  0.00216833  0.00433666  0.00650499  0.00867332  0.0108417 
0.2  0.003822808  0.00765615  0.0114842  0.0153123  0.0191404 
0.3  0.0113064  0.0226128  0.339192  0.0452256  0.0565321 
0.4  0.0211693  0.0423387  0.063508  0.0846773  0.105847 
0.5  0.0344335  0.0688669  0.1033  0.137734  0.172167 
0.6  0.0524239  0.104848  0.157272  0.209695  0.262119 
0.7  0.0769263  0.157272  0.230779  0.307705  0.384631 
0.8  0.110371  0.220743  0.331114  0.441485  0.551856 
0.9  0.156078  0.312156  0.468234  0.624312  0.780391 
1  0.218586  0.437173  0.655759  0.874345  1.09293 
The approximate solutions have been depicted in Figures
Comparison of exact solution and approximate solution for
Comparison of exact solution and the approximate solution for
Comparison of the exact and approximate solution for
The existence and the uniqueness of the Volterra fractional integral equations second kind were examined in this work. The numerical method called the Simpson
The authors declare no conflict of interests.
Abdon Atangana wrote the first draft and Necdet Bildik corrected the revised form; the both authors read the revised and submitted the paper.