Abstract
The numerical method for solution of the weakly regular scalar Volterra integral equation of the 1st kind is proposed. The kernels of such equations have jump discontinuities on the continuous curves which starts at the origin. The midrectangular quadrature rule is employed for the numerical method construction. The accuracy of proposed numerical method is
Numerical Solution of Weakly Regular Volterra Integral Equations of the First Kind
Denis Sidorov, Aleksandr Tynda and Ildar Muftahov
Mathematics Subject Classification 2010: 65R20, 45D05.
1 Introduction
This article deals with the following linear weakly regular Volterra integral equation (VIE) of the first kind
(1.1) 
where kernel is defined as follows
have continuous derivatives (w.r.t. ) for increase at least in the small neighborhood
Such integral equations are in the core of many mathematical models in physics, economics and ecology. The theory of integral models of evolving systems was initiated in the early works of L. Kantorovich, V. Glushkov and R. Solow in the mid20th Century. Such theory employs the VIEs of the first kind where bounds of the integration interval can be functions of time. It is to be noted that conventional Glushkov integral model of evolving systems is the special case of the VIE (1.1) where all the functions are zeros except of .
We stress here that the VIE (1.1) is the illposed problem. Such weakly regular equations have been introduced in [7]. It is to be noted that solution of the equation (1.1) may contain some arbitrary constants and can be unbounded as goes to Indeed, if
(1.2) 
then equation (1.1) has the solution reamains free parameter. Numerical solution of the VIE (1.1) based on combinations of the left and right rectangle rules has been discussed by E.V. Markova and D.N. Sidorov in [3]. where
In this paper for such VIE with jump discontinuous kernels we propose numerical method and discuss the analytical algorithm for construction of the continuous solutions in the following form:
(1.3) 
Coefficients are constructed as polynomials on powers of and they may depend on certain number of arbitrary constants. defines the necessary smoothness of the functions
Let us make the following notation and brifly outline the main results. Here readers may refer to the papers [1, 2].
Theorem 1.1.
Let us outline the following conditions.
A. Exists polynomial ,
, ,
where
such as for
the following estimates hold: , ,
,
.
B. For fixed ,
C. Exists such as
Theorem 1.2.
(Regularization) Let be the known function such as the condition C is true for . Then eq. (1) has the solution where is unique and can be constructed by means of the successive approximations method from the equation where
Theorem 1.3.
(Relaxed Sufficient Condition for Existence & Uniqueness) Let conditions B and C be satisfied, and for . Then eq. (1.1) has unique solution in Moreover, for polynomial is an th order asymptotic approximation of such solution.
The paper is organized as follows. In section 2 we propose the numerical method for solution of the VIE (1.1). In section 3 we demonstrate the efficiency of proposed numerical method on synthetic data. As footnote, we outline the final conclusions in section 4.
2 Numerical method
In this section we propose the generic numerical method for weakly singular Volterra integral equations (1.1) based on the midrectangular quadrature rule. The accuracy of proposed numerical method is Section 3 illustrates concepts and results of proposed numerical method on synthetic data.
For numerical solution of the equation (1.1) on the interval we introduce the following mesh (the mesh can be nonuniform)
(2.1) 
Let us search for the approximate solution of the equation (1.1) as follows
(2.2) 
with coefficients are under determination.
In order to find we differentiate both sides of the equation (1.1) wrt :
Therefore
(2.3) 
Here we assume that conditions of the Theorem 1.1 are satisfied and
Let’s make the notation . In order to define the coefficient we rewrite the equation in :
(2.4) 
It is to be noted that the lengths of all the segments of integration in (2.4) are less or equal to and an approximate solution is then application of the midrectangular quadrature rule yields
(2.5) 
The mesh point of the mesh (2.1) which coincide with we denote as , i.e. . Obviously for , . It is to be noted that are not always coincide with any mesh point. Here is used as index of the segment , such as (or its righthand side).
Let us now assume the coefficients be known. Equation (1.1) defined in as
we can rewrite as follows: where

If then

If then
Remark 1.
The number of terms in each line of the last formula depends on an array , defined using the input data: functions , and fixed (for specific ) mesh.
Each integral term we approximate using the midrectangular quadrature rule, e.g.
Moreover, on those intervals where the desired function has been already determined, we select (i.e. ).
On the rest of the intervales an unknown value appears in the last terms. We explicitly define it and proceed in the loop for . The number of these terms is determined from the initial data analysis. The accuracy of the numerical method is .
3 Numerical illustrations
Let us consider three examples. In all cases the uniform mesh is used.
Example 3.1.
exact solution is . Tab. 1 demonstrates the errors for various steps .
0.13034091293670258  
0.07804538180930365  
0.03989003750757547  
0.01975354947865071  
0.010027923872257816  
0.005083865773485741  
0.0025693182974464435  
0.001288983987251413  
0.0006500302042695694 
Example 3.2.
exact solution is . Tab. 2 demonstrates the errors for various steps .
0.13718808476353672  
0.07408554651043886  
0.04531351578371812  
0.022111520501482573  
0.011079518173630731  
0.005492567505257284  
0.0027453216364392574  
0.0014125244842944085  
0.00077170109943836 
Example 3.3.
exact solution is . Tab. 3 demonstrates the errors for various steps .
1.2810138805937967  
0.7064105257311724  
0.3172969521937503  
0.16990268475221626  
0.11787087222029413  
0.07940422358498633  
0.06518995509284764  
0.06004828109245386  
0.046102790104048275 
4 Conclusion
In this article we addressed the novel class of weakly regular linear Volterra integral equations of the first kind first introduced in [7]. We outlined the main results for this class of equation previously derived. The main contribution of this paper is a generic numerical method designed for solution of such weakly regular equation. The numerical method employes the midpoint quadrature rule and enjoy the the order of accuracy. The illustrative examples demonstrate the efficiency of proposed method. As footnote let us outline that proposed approach enable construction of the 2nd order accurate numerical method. This improvement will be done in our further works.
References
 [1] D.N. Sidorov, On parametric families of solutions of Volterra integral equations of the first kind with piecewise smooth kernel, Differential Equations, Vol. 49, N.2, 2013, 210–216.
 [2] D.N. Sidorov, Solution to systems of Volterra integral equations of the first kind with piecewise continuous kernels, Russian Mathematics, Vol. 57, 2013, 62–72.
 [3] E.V. Markova, D.N. Sidorov, On one integral Volterra model of developing dynamical systems, Automation and Remote Control, Vol. 75, No. 3, 2014, 413–421.
 [4] Boikov I.V., Tynda A.N., Approximate solution of nonlinear integral equations of developing systems theory. Differential Equations, Vol.39, 9, 2003, 12141223.
 [5] A.N. Tynda, Numerical algorithms of optimal complexity for weakly singular Volterra integral equations, Comp. Meth. Appl. Math., Vol.6 (2006) No. 4, 436–442.
 [6] A.N. Tynda, Numerical methods for 2D weakly singular Volterra integral equations of the second kind. PAMM, Volume 7 (2007), Issue 1.
 [7] D.N. Sidorov, Volterra Equations of the First kind with Discontinuous Kernels in the Theory of Evolving Systems Control. Stud. Inform. Univ., Vol. 9 (2011), 135–146