The Numerical Validation of the Adomian Decomposition Method for Solving Volterra Integral Equation with Discontinuous Kernels Using the CESTAC Method

The aim of this paper is to present a new method and the tool to validate the numerical results of the Volterra integral equation with discontinuous kernels in linear and non-linear forms obtained from the Adomian decomposition method. Because of disadvantages of the traditional absolute error to show the accuracy of the mathematical methods which is based on the floating point arithmetic, we apply the stochastic arithmetic and new condition to study the efficiency of the method which is based on two successive approximations. Thus the CESTAC method (Controle et Estimation Stochastique des Arrondis de Calculs) and the CADNA (Control of Accuracy and Debugging for Numerical Applications) library are employed. Finding the optimal iteration of the method, optimal approximation and the optimal error are some of advantages of the stochastic arithmetic, the CESTAC method and the CADNA library in comparison with the floating point arithmetic and usual packages. The theorems are proved to show the convergence analysis of the Adomian decomposition method for solving the mentioned problem. Also, the main theorem of the CESTAC method is presented which shows the equality between the number of common significant digits between exact and approximate solutions and two successive approximations.This makes in possible to apply the new termination criterion instead of absolute error. Several examples in both linear and nonlinear cases are solved and the numerical results for the stochastic arithmetic and the floating-point arithmetic are compared to demonstrate the accuracy of the novel method.


Introduction
There are many phenomena in the world that can be modelized in the form of mathematical problems such as HIV infection [1,2], smoking habit [3], computer viruses [4], energy supply-demand model [5] and others [6,7]. They can help us to analyse and predict the phenomena using the mathematical methods, deep learning and big data. Such Volterra models that contain past information are called hereditary systems. There are various applications in economics (Solow models of capital growth of global economy, optimal renovation), many examples from biology (Lotka-Volterra predator-prey, spread of epidemics, e.g., COVID epidemic), and from engineering (mechanical and electrical engineering, material sciences, and other application). Recently, many authors modelized the where y(t) and y n (t) are the exact and approximate solutions. But we have some disadvantages. Condition (1) depends on the value ε and also the exact solution. But we do not know the optimal value of ε and in many cases we do not have the exact solution to compare the results. If we choose the small values of ε we will have extra iterations and if we have the large values then the numerical process will be stopped very soon and we will not be able to produce the accurate results. Thus, in order to show the efficiency od the numerical procedures instead of condition (1) we apply the following termination criterion |y n (t) − y n+1 (t)| = @.0, which depends on two successive approximations y n (t) and y n+1 (t) and in the right hand side we have the informatical zero @.0. It shows that the NCSDs between two successive approximations is zero.
Because of these problems, we introduce the stochastic arithmetic (SA) instead of the FPA. In the SA, we apply the CESTAC method and instead of the absolute error we use the termination criterion based on two successive approximations. So we do not need to have the exact solution. Also, in the right hand side we have the informatical zero @.0 instead of ε. The numerical algorithm will be stopped when the number of common significant digits(NCSDs) of two successive approximations equals zero. Also, the CESTAC method can be implemented on the CADNA library using LINUX operating system that its codes must be written by ADA, FORTRAN or C/C++ codes [38]. Using the CESTAC method and the CADNA library we can find the optimal approximation, error and iteration of the numerical procedure [39,40]. The CESTAC method was studied by Laporet and Vignes for the first time and after that some researchers from LIP6, the computer science laboratory in Sorbonne University in Paris, France (https://www-pequan.lip6.fr/) extended this method by producing the CADNA library [41][42][43][44]. Also, recently this method has been applied to validate the results of the Newton-Cotes integration rule [45], Gaussian integration rule [46], collocation method for solving Fredholm IEs [47], finding the optimal convergence control parameter of the homotopy analysis method [48], solving fuzzy IEs by Sinc-collocation method [49], solving fuzzy numerical integrals [50], finding the optimal regularization parameter for solving first kind IEs [51], solving osmosis model [52,53], solving load leveling problem and solving the VIEs with discontinuous kernel using the homotopy perturbation method and the Taylor-collocation method [17][18][19].
This study applies the ADM for solving the linear and non-linear VIE with discontinuous kernel and validates the numerical results using the CESTAC method and the CADNA library. So we will be able to find the optimal approximation, the optimal error and the optimal iteration of the ADM for solving Equation (4). The uniqueness theorem, the error theorem and the convergence theorem of the ADM are proved. Also, the main theorem of the CESTAC method is discussed. Based on this theorem, we can apply the new termination criterion instead of the absolute error. Several examples are solved and the CESTAC method is applied to validate the results and finding the optimal results of the ADM for solving the mentioned problem.

Stochastic Arithmetic and the CESTAC Method
The CESTAC method is based on a probabilistic approach of the round-off error propagation which can help us to replace the FPA by a random arithmetic. The parallel implementation is one of the good aspect of this method. Applying this method, k runs of the computer program can be done in parallel. Thus, a new arithmetic that we call the SA is defined. For definitions and properties of the SA please see [54]. In order to apply the CESTAC method, we should substitute the SA instead of the FPA. Thus we will be able to run each arithmetical operation k times synchronously before running the next operation. All of this process should be done using the CADNA library. During the run, the CADNA library can be found the NCSDs of each results and if the result is zero then the CADNA library will be stopped by showing the informatical zero @.0. Thus each result can be appeared as a random variable.
If we produce the representable values by computer and collect them in B, then S * ∈ B can be written for s * ∈ R with α mantissa bits of the binary FPA as where ρ, 2 −α φ and E are sign, missing segment of the mantissa and the binary exponent of the result, respectively. Also, we know that for α = 24, 53, the numerical results can be produced in single and double precisions [39,40]. By assuming φ as a casual variable that uniformly distributed on [−1, 1], we will be able to make perturbation on last mantissa bit of s * . Then the mean (µ) and the standard deviation (σ) values can be produced for results of S * which have important role to identify the precision of S * . If we repeat the process for k times, we will have the quasi Gaussian distribution on S * i , i = 1, · · · , k and we will have equality between µ and the exact s * . Algorithm 1, shows the process step by step, where τ δ is the value of T distribution as the confidence interval is 1 − δ, with k − 1 freedom degree [40,[42][43][44].
Algorithm 1: Algorithm of the CESTAC method.
Step 1-Produce k samples of S * in the form of Φ = S * 1 , S * 2 , ..., S * k by making perturbation on the last bit of mantissa.
In order to apply the CESTAC method we do not need to apply the mentioned algorithm directly by the usual softwares such as MATLAB, Mathematica, Maple and others. This method can be implemented using the CADNA library that we need to write the CADNA codes using C, C++, FORTRAN or ADA codes [38], then the CESTAC method can be done automatically on the numerical procedures.
Applying the CESTAC method and the CADNA library we have the following advantages than the mathematical methods based on the FPA: • Generally, the FPA depends the absolute error that we need to have the exact solution but in the CESTAC method we do not need to the exact solution. • In some cases, the absolute error depends on the positive small value ε that we do not know its optimal value. In the CESTAC method we do not need to have this value. • In the CESTAC method the algorithm will be stopped in the optimal iteration but in the FPA, the extra iterations can be produced without improving the accuracy of results. • In the FPA, the numerical algorithm can be stopped very soon before producing the accurate results. • In the CESTAC method, we will be able to identify the optimal values such as optimal iteration, approximation and error but in the FPA we can not do it.
The following codes are the sample codes of the CADNA library:

Main Idea
Consider the following second kind nonlinear VIE with discontinuous kernel where and β j (0) = 0 [15,20]. Also, ∀t ∈ J = [0, T] we assume that x(t) is bounded and k j (t, τ) is discontinuous along continuous curves β j (t), j = 0, 1, · · · , m such that |k j (t, τ)| < M j , ∀0 ≤ τ ≤ t ≤ T and the nonlinear term F(y) satisfies in the Lipschitz continuous such The ADM assumes that the unknown function y(t) can be constructed by an infinite series of the form and the Adomian polynomials [55] can be obtained in the following form: where P n = ∑ n i=0 y i (t) shows the partial sum. Then we have Also, the nonlinear term F(y) can be decomposed by an infinite series of polynomials given by where which is called the Adomian polynomials.
The following theorems show the uniqueness, convergence and error of the method. The well known contraction mapping principle is applied to prove them. Lemma 1. If we apply the ADM for solving Equation (4), the obtained solution will be unique whenever Proof. See [55].
Proof. Let (C[J], . ) be the Banach space of all continuous functions on J such that f (t) = max ∀t∈J | f (t)|. Let {P n } be the sequence of partial sums where P n and P m are arbitrary partial sums with n ≥ m. We should prove that {P n } is a Cauchy sequence in the Banach space: Using Equation (6), we can write ∑ n−1 i=m A i = F(P n−1 ) − F(P m−1 ) and then and for n = m + 1 we have Applying the triangle inequality we have Since 0 < η < 1 we can write 1 − η n−m < 1 and we have We know that x(t) is bounded and |y 1 | < ∞. So, P n − P m converges to zero, as m approaches infinity. It can shows that P n is a Cauchy sequence in C[J] and the series converges.

Theorem 2.
If we apply the series solution (5) for solving Equation (4), the maximum absolute error truncation can be obtained as follows where k = max ∀t∈J |F(x(t))|.
Proof. Applying inequality (10) and Theorem 1 lead to If n approaches ∞ then s n will approach to y(t) and Finally, the maximum error on J can be obtained as

Remark 1.
Introducing an auxiliary parameter and differentiating with respect to it for calculating the initial approximations was effectively employed and in other nonlinear problems. Here readers may refer to [56].
Theorem 3. Let y(t) and y n (t) be the exact and numerical solutions of problem (4) which y n (t) is obtained by using the ADM. We have C y n (t),y n+1 (t) C y n (t),y(t) , where C y n (t),y(t) shows the NCSDs of y n (t), w(t) and C y n (t),y n+1 (t) is the NCSDs of two successive iterations y n (t), y n+1 (t).
Theorem 3 shows that when n increases, the NCSDs between two sequential results obtained from the algorithm is almost equal to the NCSDs of the n-th iteration and the exact solution at the given point t which means that for an optimal index like n = n o pt, when y n (t) − y n+1 (t) = @.0 then y n (t) − y(t) = @.0.

Numerical Results
In this section, we apply the ADM for solving the mentioned examples. The numerical results are obtained based on the FPA and the SA. In the FPA, the numerical algorithm depend on the value ε. Also, the number of iterations for different values of ε are obtained. It is obvious that for small values of ε the algorithm can not be stopped and we will have many iterations without improving the accuracy of the results. Also, for large values of ε, the algorithm will be stopped very soon without providing the accurate results. In the SA and applying the CESTAC method and the CADNA library we can find the optimal results and the optimal iteration and error of the ADM for solving the VIEs in linear and nonlinear forms with discontinuous kernels. Clearly, we can see that applying the CESTAC method, CADNA library and the novel termination criterion (2) is better and applicable than the FPA and the stopping condition (1).

Example 1. Consider the following linear VIE with discontinuous kernel
and the exact solution is y(t) = sin t + cos t.
In Table 1 the numerical results are obtained using the ADM based on the FPA for ε = 10 −5 and the algorithm is stopped at n = 6. Also, in Table 2, the number of iterations for various ε are shown. It is obvious that for large and small values of ε the accurate results can not be found. But in Table 3, the results are obtained based on the SA using the CESTAC method and the CADNA library. We do not have ε in this table. The algorithm is stopped at n opt = 7 and it shows the optimal iteration of the ADM for solving this problem. Also, the optimal error is 0.1 × 10 −4 and the optimal approximation is y n opt = 1.25085.
and the exact solution is y(t) = exp t. The numerical results are obtained based on the FPA for ε = 10 −5 and demonstrated in Table 4. Also, the number of iterations for different values of ε are presented in Table 5. The numerical results based on the SA are demonstrated in Table 6. Using this table, the optimal iteration, the optimal approximation and the optimal error can be found that they are n opt = 6, y n opt = 1.10516 and E n opt = 0.8 × 10 −6 .
(2 − 3t 2 − 2 cos(t) + 2t sin(t)) + 1 2 (−t − sin(t) + 2 cos(t) sin(t)), and the exact solution is y(t) = sin t. In Table 7, the numerical results are obtained from the CESTAC method and the CADNA library. We can find that the optima iteration for solving this example using the ADM is n opt = 6, the optimal approximation is y n opt = 0.198821 and the optimal error is 0.15 × 10 −4 . The informatical zero @.0, shows that the NCSDs between y n+1 (t), y n (t) are almost equal to the NCSDs between y n+1 (t) and y(t). In Table 8 , and the exact solution is y(t) = t 2 . The numerical results of the CESTAC method are presented in Table 9. For finding these results we applied the termination criterion (2) which depends on two successive approximations. We should note that the third column in this table is only for comparison between results and generally we do not need to have the exact solution in the CESTAC method. Based on this table we can find the optimal iteration n opt = 3, the optimal approximation y n opt = 0.16 and the optimal error E n opt = 0.1 × 10 −5 . In Table 10, we can find the number of iterations of the ADM for solving this example based on the FPA.
where the exact solution is (solution with fluctuations), and the discontinuous kernel is where p is a parameter. The direct problem (calculating x(t)): Nonuniform grid of nodes, identical to t and τ, is given by where N is the number of nodes. The right hand-side x(t) is calculated numerically using the trapezoidal formula on grids (18) according to Algorithm 2.

The inverse problem (solving y(t) of VIE):
Algorithm 3: Algorithm of the recurrent solution.

Conclusions
The CESTAC method is among applicable and important methods to validate the numerical results based on the SA. For this aim, instead of usual applications such as Mathematica, Maple and Matlab the CADNA library should be run. Using this method and the CADNA library we can find the optimal iteration, the optimal approximation and the optimal error of numerical procedures. We introduced the stopping condition based on this method that it is independent from the exact solution. Also, using this condition we do not have the disadvantages of the traditional absolute error. Several theorems were proved to show the convergence of the ADM for solving linear and nonlinear VIE with discontinuous kernel. The main theorem of the CESTAC method was presented. Based on theorem we can apply the termination criterion (2) instead of (1). Several examples were solved by the ADM and the numerical results were validated using the CESTAC method. Also, we compared the results with numerical results obtained from the FPA.

Acknowledgments:
The authors are thankful to three anonymous reviewers for their careful reading of manuscript and their insightful comments and suggestions.

Conflicts of Interest:
The authors declare no conflict of interest.