This article focuses on the following weakly regular Volterra equations of the first kind
where jump discontinuous kernels are defined as follows
where , functions have continuous derivatives with respect to for functions increase in a small neighborhood denotes closure of set
Such weakly regular Volterra equations of the first kind with piecewise continuous kernels were first classified and generalised by Sidorov and Lorenzi  and extensively studied by many authors during the last decade. Here readers may refer to monograph  and references therein. Volterra operator equations of the first kind were studied by Sidorov and Sidorov , sufficient conditions for existence of unique solution are obtained. Tynda et al  employed direct quadrature methods for solution of equations (1) both in linear and nonlinear cases. Muftahov and Sidorov  considered the numerical solution of nonlinear systems of such equations. Aghaei et al  applied Legendre polynomials approximation method for solution of solution of linear Volterra integral equations with piecewise continuous kernels. The numerical solution of the first kind Volterra convolution integral equations of the first kind with broad class of piecewise smooth kernels was considered by Davies and Duncan . They employed the cubic convolution spline method and proved a stability bound. Such Volterra models enjoys applications in modeling various dynamical processes including storage systems [13, 14]. Generalized quadratures were employed by Sizikov and Sidorov  for solving singular Volterra integral equations of Abel type in application to infrared tomography. The numerical solution of the second-kind Volterra integral equation with weakly singular kernel is considered in the piecewise polynomial collocation space by Linag and Brunner . For conventional review of Volterra integral equations theory readers may refer e.g. to monographs by Brunner  and by Apartsyn . Some studies of the Volterra integral equations of the first kind have led to the paradoxes as noted by Tynda in zbMATH .
In this paper, we also implement the CESTAC method and apply the CADNA library to find the numerical validation of the spline-collocation method to solve the problem (1
). The priority of this strategy is to find the optimal step, accuracy and error of the numerical method. The paper is organised as following. In Section 2, the spline-collocation method is presented. Also the convergence of the method and smoothness of the solutions are studied. The use of Floating Point Arithmetic (FPA) is discussed in Section 3. The CESTAC method and its principle theorem can be found in Section 4. Using the this theorem we will show, how can we replace the conditions (14) with (15). The numerical results are illustrated in Section 5. Also in this section the comparative study between the results of the stochastic arithmetic (SA) and the FPA can be found.
2 Polynomial spline-collocation method
In this section, we construct a numerical method for solving problem (1) – (2), based on the approximation of the exact solution by continuous local splines. First of all, within the framework of conditions (2), we replace the original equation (1) of the first kind with an equivalent equation of the second kind. To do this, we apply the standard technique of differentiating the equation:
Let us rewrite this equation in operator form
2.1 Numerical scheme
Let us introduce the partition of the interval with grid points . The introduced grid of nodes is not necessarily uniform and depends on the smoothness properties of the exact solution. Denote by the segments Let then
be additional nodes distributed in a certain way over the segment . We denote by an operator, putting to the function
in correspondence an interpolation polynomial of degreefor constructed on knots Let then be a local spline, defined on and composed of polynomials
We look for an approximate solution of (4) as a spline with unknown values , at the knots of the grid.
The grid (5) depends on the considered class of functions to which the exact solutions belong and will be specified below.
The values in each segment are determined step-by-step by the spline-collocation technique from the systems of linear equations
Here is an operator of projection on the set of the local splines of the form ; is a new right part of equation (3) including the integrals over segments processed at the previous steps (in these domains, the spline parameters are already known):
2.2 Convergence substantiation
where is a dense set in and are the sets of corresponding local splines.
Since the homogenous Volterra integral equation has only the trivial solution, the operator is injective. Hence, the operator has the bounded inverse operator For all sufficiently large we have the estimates
Let us show that the last estimate holds for all sufficiently large . Since and is a dense set in , we have
where as . Therefore, starting with sufficiently large .
Thus, the operators are exist and uniformly bounded and equation (8) has a unique solution for all sufficiently large . Taking into account that as for all , we apply the projection operator both to the left and the right parts of equation (7): Subtracting this equation from (8), we obtain and
Thus, the accuracy of the approximate solution obtained via projective method (8) is determined by the accuracy of the approximation of functions from by the local splines.
2.3 Smooth solutions
In this chapter, we describe in more detail the projection method (6) for the case of smooth input functions. Namely, let the functions and in (1)-(2) satisfy additional smoothness conditions. Let , (continuously differentiable on each variable). The exact solution of the equation (1) in this case belongs to .
We introduce the uniform partition of the interval with grid points . Denote by the segments Let and where are the roots of the Legendre polynomials of degree .
We denote by an operator, putting to the function in correspondence an interpolation polynomial of degree for constructed on knots Let then be a local spline, defined on and composed of polynomials We look for an approximate solution of the equation (1) as a spline with unknown coefficients Let us describe the process of definition . At first we find the coefficients from the system of equations
Here for integrals calculation in (10) we employ compound Gaussian quadrature rule with points constructed on the auxiliary mesh linked to the lines of the kernel discontinuities for each specific value of . Note also that all values of the unknown function in intermediate points are computed with help of interpolation. The resulting system of linear algebraic equations is then solved by the Jordan-Gauss method.
In order to determine the coefficients of the local spline on the segment we represent (3) in the following form
The error of approximation of the exact solution by the polynomials constructed in this way at each step of the process can be estimated by the following inequality (here readers may refer to )
Taking into account the general estimate (9) of the error of the method, we obtain the error estimate in this case ()
Boikov and Tynda [18, 19] established that such type numerical methods for Volterra integral equations are also optimal with respect to complexity and accuracy order. Thus, an effective projective method for solving equations of the form (1) is proposed.
3 Using the Floating Point Arithmetic
In general form when floating point arithmetic (FPA) is employed it is necessary to have the exact and approximate solutions and and also small value to use the following conditions
But the main problem is that the exact solution and optimal are unknowns. Thus by putting small values instead of , the approximate solution will not be accurate and for large values we will have many extra iterations. In order to avoid these problems, the CESTAC method and the CADNA library will be utilized . In this novel method, the accuracy will be obtained using successive iterations and and the following condition
We apply this condition to control the accuracy of the method and avoid extra iterations by using number of common significant digits (NCSDs) between and . in Eq. (15) displays the informatical zero which can be produced only in the CESTAC method by the CADNA library. It shows that the NCSDs of two successive approximations and approximate and exact solutions are almost equal to zero. Vignes and La Porte  have presented the method for the first time in 1974. In  Vignes has described the CESTAC method to evaluate the numerical results of some computational methods. Some conditions of the CESTAC method, applying different tools to write the CADNA codes  and also some properties of the SA have been studied by Chesneaux. All the CESTAC evaluations should be accomplished using the CADNA library. Handling this scheme, the optimal results, step and error of the method can be recognized. Lamotte et al. has implemented the CESTAC method using C and C++ codes. Jézéquel et al have discussed the new version of the CADNA library using Fortran programs. Recently applying this method to control the accuracy of the Taylor expansion method to solve the generalized Abel’s integral equation , mathematical model of Malaria infection [21, 23], nonlinear fractional order model of COVID-19 , solving nonlinear shallow water wave equation , Adomian decomposition method, homotopy perturbation method and Taylor-collocation method for solving Volterra integral equation [25, 28, 29, 30], dynamical control of the reverse osmosis system [26, 27], solving integrals using the numerical methods have been done. Moreover the CESTAC method has been used to find the optimal convergence control parameter of the homotopy analysis method in both fuzzy and crisp forms [31, 32].
4 CESTAC Method
The CESTAC method is a powerful and applicable tool to validate the numerical results of numerical procedures. It should be applied based on the SA. Let be a set of reproduced values by computer. For real value , we can find a member of set such as with mantissa bits of the binary FPA as where the sign showed by , the missing segment of the mantissa presented by and the binary exponent of the result displayed by . Replaying 24 and 53 instead of , the results can be found by single and double accuracies. Assuming
as a stochastic variable and having uniformly distribution on, we will be able to make perturbation on the last mantissa bit of . Thus for the obtained results of
, the mean and standard deviation valuesand can be found. Doing the mentioned scheme -times samples of can be produced as Thus the mean and standard deviation can be found as follows
Using the mentioned computations the NCSDs of and can be generated using the following relation where is the value of
distribution as the confidence interval is, with freedom degree. Showing , the process stopped if we have or
In this method, the mathematical softwares Mathematica, Maple or MATLAB must be replaced by the CADNA library. This library should be implemented on the LINUX operating system and we all codes should be compiled by C, C++, FORTRAN or ADA. The main benefit of the method is to find the optimal results, step size and error of the method.
For the NCSDs can be defined as
Assume that and are the exact and approximate solutions of Eq. (1). The NCSDs of two successive approximations are almost equal to the NCSDs of exact and approximate solutions and we have
Proof: Applying Definition 1 and Eq. (13) for to iterations and we can write
We know therefore we get And finally we have
Repeating the process for exact and approximate solutions, the following relation can be obtained
5 Numerical results
To illustrate the effectiveness of the suggested polynomial spline-collocation method, we present the results for two test equations.
Example 1. Consider the following integral equation
where the right side of the equation was chosen so that the exact solution was . The following designations are used in the tables below: is the number of segments of the main partition, is the parameter responsible for the order of the spline, .
Clearly, Tables 1 and 2 depend on the exact solution. For Table 3 we apply the CESTAC method and the results are obtained and the optimal results are Table 4 is obtained using the spline-collocation method and the FPA for the same value of r. It is obvious that for the algorithm will be stopped at and for we have . For small values of we will need to provide many extra iterations without improving the accuracy. Tables 5 and 6 present the results for using the CESTAC method and we have It means that we do not need to produce a smaller partition and we can stop at the specified value of . Thus according to the obtained results, is enough and we do not need to find more results.
Example 2. Consider the following integral equation
where the right side of the equation was chosen so that the exact solution was .
All calculations were performed in the Maple system with parameter Digits:=30; (the number of digits that Maple uses when making calculations with software floating-point numbers). As we can see from Tables 7 and 8, the practical error of the method corresponds to the theoretical estimate (13). All the results of Tables 9 and 10, are obtained using the CESTAC method. For we get and for we have According to the results for large values of r the results are more accurate. Tables 11 and 12 are obtained for the spline-collocation method using the FPA. Comparing the results of the FPA and the SA, we can introduce the CESTAC method as a good tool to control the accuracy and the step size of the spline-collocation method for solving the mentioned problem.
5.1 Stability experiments
To illustrate the stability of suggested numerical method, we introduced a random error in calculating the values of free term of the equations (20) and (21). The range of introduced random errors is .
Depending on the , the following results are obtained.
From the results proposed in the tables 13 and 14 it is possible to judge the continuous dependence of the solution on the initial data and conclude about the stability of the numerical method. This result is not surprising: despite the fact that the initial equation is an equation of the first kind, it is solved in such spaces in which the problem is well-posed. The case of an ill-posedness (when noisy initial data may lead to instability and regularization will be required) will be studied in future works.
We have applied the spline-collocation method for solving the Volterra integral equations of the first kind with discontinuous kernel. The convergence of the method and the smoothness of the solution have been discussed. Using the CESTAC method we have tried to control the accuracy and step size of the method. The principle theorem of the CESTAC method will help us to apply the condition (15) instead of (14). Thus we will be able to find the optimal results, optimal error and optimal step of the method.
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