Numerical solution of a Volterra integral equation

This work deals with a generalized nonlinear Volterra integral equation of the second kind, in whose kernel the unknown function occurs with two different arguments. The equation is solved by a collocation approach with piecewise Hermite polynomials. When using polynomials of degree 2 m-1, m e N


INTRODUCTION
Volterra integral equations are a special type of integral equations that involve integration over the past history of the unknown function. They are divided into two groups referred to as the first and the second kind. A linear Volterra equation of the first kind is where f and K are given functions and x is an unknown function to be solved for. Volterra integral equations of the second type, In the classical theory of E. I. Fredholm, X was the Banach space of continuous functions on a closed interval. Another natural situation, which however offers some additional difficulties, is the case X = L2(a, b) functions of square-integrable functions on a (closed) interval. Finally, there is a case where X is a space of number sequences and T is an infinite matrix. In this case, one can see that this is a linear system of equations in infinitely many unknowns and with infinitely many equations.
Tahmasbi et al. proposed a numerical method for solving systems of linear Volterra integral equations of the second kind based on the power series method. This method does not require derivatives and can recover the analytical solution when it is a polynomial. The numerical results showed that the method is simple and effective. they used MATLAB for the numerical computations in their study (Tahmasbi & Fard, 2008). Steinberg applied Gregory's formula to numerically integrate Volterra linear integral equations of the second kind. He used recursive inequalities to derive the order and the asymptotic behavior of the truncation error (Steinberg, 1972).
Salim el al. used the linear spline function of the unknown function at an arbitrary point to numerically solve linear mixed Volterra-Fredholm integral equations of the second kind. they transformed the integral equation into a system of linear equations with respect to the unknown function by integrating. They can obtain an approximate solution by solving this system with a Python code program version 3.9. they also prove theoretical results on the uniqueness and convergence of the method (Salim, Saeed, & Jwamer, 2022).
Fahim et al. numerically solved a Volterra integro-differential equation of parabolic type with memory term and initial boundary value conditions. They used finite difference method and product trapezoidal integration rule for time discretization and sinc-collocation method for space discretization. they focused on the case of a weakly singular kernel. they showed that the method converges exponentially to the solution and provide the convergence analysis. they also gave numerical examples and illustrations to support the method (Fahim, Fariborzi Araghi, Rashidinia, & Jalalvand, 2017). A Volterra integro-differential equation of parabolic type with memory term and initial boundary value conditions solved numerically. Finite difference method and product trapezoidal integration rule have used for time discretization and sinc-collocation method for space discretization. We focus on the case of a weakly singular kernel. The method converges exponentially to the solution have shown and provided the convergence analysis. The numerical examples and illustrations to support the method also have given.
Lema and co-workers studied second kind Volterra integral equations with weakly singular kernels. They defined some suitable function spaces and showed that Euler's method has an asymptotic error expansion. This result enables them to apply some extrapolation methods, which they successfully demonstrate with some numerical examples (Lima & Diogo, 1997).
In this work we describe a method for the numerical solution of the integral equation (Steinberg, 1972): Equations of a similar type occur when evaluating experiments in the Atomic Physics on (Doornenbal, 2012;Zhu, Ma, Chen, & Han, 2016). We generally make the following assumption: To solve (1) we will use a collocation method (see references Delves and Walsh (Golberg, 2013) and Atkinson (Atkinson, 1972)). Hermite functions (see Ciarlet, Schultz and Varga (Ciarlet, Schultz, & Varga, 1967;Jajarmi & Baleanu, 2020)), i.e. functions that (m -1) times are continuously differentiable and piecewise with polynomials of degree 2 m -1. The approximate solution can be calculated step by step as in an initial value problem (Niu, Lin, & Zhang, 2012).
Piecewise polynomials have been used several times to solve the integral equation: The procedure of Tom [5] is a special case of the method explained here. Netravali (Netravali, 1973) solves (3) with twice continuously differentiable cubic splines. De Hoog and Weiss (Diogo, McKee, & Tang, 1994) make an overall collocation approach for (3) with piecewise polynomials that are only continuous. In this essay, all statements and procedures for a scalar integral equation are formulated for reasons of clarity. However, they apply without distinction also for systems of integral equations of type (1) (i.e.[ (Maleknejad & Aghazadeh, 2005)]).

Existence statements
We quote some of the existence statements proved in (Maleknejad & Aghazadeh, 2005) for the integral equation (1). Before that we note that (1) has no solution in general in C (I) if condition (2 iii) is not satisfied. For example, the equation: : LITERATURE REVIEW Theorem 1 : i. Under the assumptions (2) the integral equation (1) has a unique one solution f  C (I). ii. f can be estimated by a suitable c > 0 independent of A: iii. iv.
If for all , one can replace by in the last estimate. v. (iv) When with n > 2, then . The same is true for n = 1 under the additional assumptions: Kx and Kz satisfy \ Lipschitz conditions in a neighborhood of the point (0, 0, g (0), g (0)).
Regarding the last argument.
(v) Under the conditions of (iv) applies In gn and Kn, g and K occur with their derivatives as well as f, ...,f(n-1). In this and the following sections we use the following notations: We cite some facts about piecewise Hermite interpolation.

Hermite -Functions
In this and the following sections, we use the following designations (Weisner, 1959): We cite some facts about the staggered Hermite interpolation (Imai & Aoki, 2004). (i) Let f ∈ Cm_ 1 (I), then there exists exactly one function Pm (. ;f, h) Cm_ 1 (I), which is in the subintervals [xn-1, xn], n = l (1)N, with a polynomial yore degreeless than or equal to 2 m -1 and which satisfies the interpolation condition filled.
(ii) Let f ∈ Cm_ 1 (I), then then for j=0 (1) 2 m -1 applies He error estimate in the form given can be found in Kansy (Kansy, 1973;Levesley & Luo, 2003).Very general error statements even for lower smoothness properties off can be found in the work of Swartz and Varga (Swartz & Varga, 1972).

METHODOLOGY
General nonlinear Volterra integral methods of the second kind, where an unknown function kernel appears with two different arguments.

RESEARCH RESULT AND DISCUSSION
As an approximation for the solution f of the integral equation (1) (Dzyadyk, 1995), we determine a Hermite function Pm(x;u,h) at the vertices xn, n=0(1)N. The components of the vector u=(unj) then form approximations for the function and derivative values of f at the points of intersection: We call the vector u = u (m, h) the discrete solution of (1).
To calculate Pm(x; u,h), we first formally insert this Hermite function intoformally into (1). Then we replace the integral by one that fits the subintervals [xn-1, xn], n = l (1)N, matching the summed quadrature formula. By inserting of m arguments xnj, j = 0 (1) m -1, per subinterval [xn-1, xn]we then obtain m.N equations for the coefficients unj ("collocation"). The starting coefficients Uoo.... ,uo, m-1 are calculated directly from (1). For each partial interval a (generally nonlinear) m x m equation system is obtained. To obtain a procedure of order 2m we must use Hermite functions of degree 2 m -1 and integration formulae of order 2 m.
The summed quadrature formula should be constructed as follows. For h>0, φ∈ C2m ([0, h]) let a quadrature formula be given by: d1 are the parameters of the points, Wl the weights.Practically, one will only use formulae with Wl > 0. Let Eo(φ) be deductible from: To approximate the integral Formosa Journal of Applied Sciences (FJAS) Vol. 2, No. 5 2023: 823-836 829 this formula becomes summation: The following then applies as: Suitable quadrature formulae in this sense are the Gauss-formulae with m points or the Lobatto formulae with m + 1 points. Other formulas require more function evaluations.
We can now define the procedure of order 2 m with N subintervals.

Remarks:
Practically , the methods for m = 1, 2, 3, possibly m = 4. The choice of the parameters 0iis not quite arbitrary. For stability reasons they are , in dependence of m , subject to certain restrictions . Normally one will choose θm-1=1. For the first derivatives of f in zero one gets: The unknowns Uno, ... , Un,m-1 occur on the left side, in the argument Pm(y;u,h) at the last subinterval and in the argument Pm(a(xn,j,y);u,h) at the subintervals where a(xnj, y) > xn-1.
The m×m system of equations (5) can be solved by Newton's method. Initial values for the iterations are Un-1,0, ... , Un,m-1 The functional matrixfor the Newton method can be calculated exactly or also be formed by numerical differentiation. If the constants θj are chosen in such a way that the stability condition (12) is satisfied, the equations (5) for a sufficiently small h are uniquely sable.

The stability condition
In this section, we want to discuss the stability condition (Burton & Zhang, 2004): This condition restricts the choice of the intermediate points , for which the condition 0 < θ0 <... < θm-1 was set. The following example shows that such a restriction is useful.
If one applies to the integral equation: our method with m = 1 and θ0 ∈ (0, 1], one obtains for the zeros un of the exact solution f ( n h)= 1 the recursion: so un= 1 for all n ∈ No. The same recursion with initial value , has the solution For θ0 ∈ (0, 1], the perturbation with h→0, where n →∞ is stable for all limits, the discretisation is not stable.
Before we give the practically important cases of the number θ, we quote a theorem about the existence of stable discretisation.
In the cases m = 1, 2, 3 admissible θ can be determined directly from condition (12). m = 1; The stability condition is fulfilled if θ0 > 1/2. m = 2; The stability condition is fulfilled if 1/2< θ0 < 1 and θ1 = 1. m = 3: The stability range was determined by numerical calculation of p (R -1 Q) (see. [6]). The result is shon in Fig. 1 In the practical calculations, good experience has been made with the following values for θ.

With
Meis shows, under suitable Lipschitz conditions on F and K, that (13) has an unambiguous, continuous solution. Furthermore, under certain conditions the equation of the first kind: with g ∈ C (I), other conditions as above, to a system of two equations (13).
The numerical method described above has been extended by Maskos [-9]. in such a way that it can handle systems of equations of type (13) and the equation (14) to such systems. The numerical results confirm the expectation that the convergence statements can also be applied to the more general procedure.

Numerical Results
The effectiveness of the procedure is to be demonstrated by means of two examples. The calculations were carried out inhouse computer in the university. The machine accuracyis about 14 decimal. For θ, the values given at the end of section 5were used. The Gaug formulae were used for squaring. Example 1: (system for 2 equations of type (1)):