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THE APPLICATION DIFFERENCE METHODS TO SOLVING VOLTERRA INTEGRAL EQUATION

Galina Mehdiyeva, доктор, профессор

Ibrahimov V.R., доктор, профессор

Imanova M.N., ph.d.

Бакинский государственный университет, Азербайджан

Участник конференции

UDK519.62/.642

It is known that to find the solution of many problems of natural science, are reduced to solving Volterra integral equations. One of the popular methods for solving such equations is the method of quadrature. However, methods such as quadrature methods have disadvantages . Thus, scientists have recently begun to exploit different techniques that permit the construction of more accurate methods than the known methods. To this end, we offer a number of ways to construct more accurate methods for solving Volterra integral equation. In particular, here construct specific second derivative one step methods with the degree p≤10  using information about the solution by considering the problem at only one point.Given a way for applying second derivative multistep and hybrid method to solving Volterra integral equation.

Keywords: multistep methods, initial value problem, second derivative, numerical solutions.

 

Consider the following nonlinear Volterra integral equation of the second kind:

Integral equation (1) is commonly referred to as the Volterra integral equation of the second kind. This convention is due to the fact that equation (1) in the linear case was thoroughly investigated by Volterra, and at a sufficiently high level, such equations occurrence in practical problems has been studied (see, e.g., [1], [2]). Note that the singular integral equation with a variable boundary in a particular form was studied for the first time by Abel (see, e.g., [1, p.12]). Given that even in the linear case, the exact solution to equation (1) is not always possible; many experts have used approximation methods to solve it (see, e.g., [3] - [7]).

The quadrature method applied to the solution of equation (1) can be written as follows (see [2]):

where Rn is the remainder term of the quadrature method and  are real numbers; these numbers are called the coefficients of the quadrature method. By discarding the remainder term, we obtain the following method:

which is called the quadrature method with a variable boundary.

Although method (3) with  is implicit, when , this method is explicit. As a result of relations (3), each value of the kernel of the integral is calculated n times when the number of values has the quantity n, ​​which may increase the amount of computing functions K(x,s,y). Consequently, at each step the amount of calculation work increases, which is a major disadvantage of quadrature methods. To eliminate this drawback of method (3), some authors have proposed the use of methods such as the following k-step method with constant coefficients (see for example [8] - [11]):

which has been thoroughly studied for finding the numerical solution of the following initial value problem:

Note that the construction of methods of type (4) takes into account some equivalence with the solutions to problem (5) and equation (1).

Here, to solve equation (1), we use the finite-difference method, which can be written as follows.

In particular, for ,from the method of (6) we obtain an ordinary k-step method with constant coefficients and for , we obtain hybrid methods from the method of (6) (see, e.g., [12]- [15]).

Consider the construction of methods of type (6) for  . To this end, we construct a formula for calculating the values ​​of the function y'(x) and y''(x). Suppose that by some methods the solution of the equations is found after taking into account the fact that one identity is obtained in equation (1). Then, from identity (1) we can write:

Finding the values ​​of the quantities y'n+k and y''n+k by means of formulas (7) and (8) is reduced to the calculation of the following integral:

If quadrature formula (3) is applicable to an evaluation of integral (9), we obtain the integral sum of the variable boundaries. For that reason, we give the prominence values yn+k-1, ​​y'n+k-1and y''n+k-1, and we mustfind the relationship between these values ​​with the values of yn+k, ​​y'n+k, y''n+k, which ​​are unknown.

Consider the following equation:

Here, the quantity b'(x) is obtained in the form . Let us take equation (11) into account in equation (10). Then, we have:

Thus, the computation of the integral in the segment [x0,xn+1] using equation (8) has been reduced to calculations in a segment [xn,xn+1] that calculates this integral. Moreover, this reduction was accomplished without increasing the computational work by increasing the values ​​of the variable n. Note that this type of scheme is applied to the determination of the relationship between the variables y'n+1 and y'n.

It is known that for a sufficiently smooth function, the following holds:

  

here  are real numbers.

Then, by using formula (13), the interpolation polynomial of Logranzha can be employed to calculate the value and an integral sum can be used to replace the integrals in equation (12). Thus, we have:

Depending on the needs of the user, the coefficients in formulas (14) and (15) can be given different meanings. The formulas for the calculation of y''n+k can be constructed such that in methods of type (14) and (15), the terms of type K'x(x,s,y) and K'x2(x,s,y) do not participate. For this purpose, equations (7) and (8) can be rewritten as follows:

Here,  are real numbers, but R1 and R2 are the remainder terms. By replacing the integrals in formulas (16) and (17) with an integral sum and discarding the remaining terms, we obtain methods to calculate the values of the quantities y'n+k and y''n+k. However, in this case, the coefficients impose the following additional conditions:

We set 

z(x)=y(x)-∫(x) 

and in the following difference method

but the equation (14) rewrite as follows:

                 (22)

   Note that to calculating the integrals in equations (10) - (12) one can use different schemes, and as the result obtain different versions of the method (23). For example, the following scheme

However, in this case the question how to find a "gold middle" is arises that is, how to choose a suitable formula for construction methods with the desired characteristic. Here, as such method we propose the method (23) on the basis that on the following considerations ,  solving of the integral equation (1) is equivalent to solving the following initial value problem:

Indeed, if in (7) take into account , we obtain the problem (24).

In this case, by using the indefinite coefficients one can be obtained a system of algebraic equations for the determine coefficients of the method (23). It should be noted that in the study of the method (20), one of the main issues is to determine its order of accuracy, because the value k is the order of finite difference method (20) and is known. Therefore, in the [16] found a connection between the order k and degree p for the stable methods of the type (20), which has the following form:

p≤2k+2.

The validity of (25) in more general form proved in [17] by another scheme. Thus obtained that for the construction of stable methods with the order of accuracy p>2k+2, it is necessary to use some modification of the method (20). Method (6) is one of the modifications of the method (20), which is constructed on the intersection of the methods of the type (20) with the hybrid methods.

Remark that by solving the received system in [18] one can construct methods with the degree p≤10 for k=1. One of them is the following:

Conclusion. As we have observed, there are different ways to solve integral equations. Recently, increasing numbers of scientists solve these equations using multistep methods with constant coefficients. Among the most popular methods are implicit methods. However, in the work [12], Makroglou showed that by using hybrid methods, one can construct more accurate methods than implicit methods. It is known that second derivative multistep methods are more accurate than the known multistep methods, and their applications to specific problems are easier than is the case with hybrid methods. Based on this fact, the second derivative finite-difference methods in this paper are applied to solve equation (1). In this study, we examined a method derived from formula (6) as a special case. In [15], the application of method (6) to solving equation (1) for  was investigated, and the defined region of stability for some methods of type (6) was also investigated.

The authors express their thanks to the academician Ali Abbasov for his suggestion to investigate the computational aspects of our problem. This work was supported by the Science Development Foundation of Azerbaijan (Grand EIF-2011-1(3)-82/27/1).

 

References:

  1. E.M. Polishuk.Vito Volterra. Leningrad, Nauka, 1977, 114p.
  2. V.Volterra. Theory of functionals and integral and integro-differential equations. M., Nauka, 1982, 304p.
  3. Krasnoselskiy M.A.. The approximate solution of operator equations. М.: Nauka,  1969,  210 p.
  4. Y.D. Mamedov, Ashirov S.A. Methods of successive approximations for the solution of operator equations. Ashgabat, 1980, 120 p.
  5. A.F. Verlan, V.S. Sizikov. Integral equations: methods, algorithms, programs. Kiev, Naukovo Dumka, 1986, 384 p.
  6. Manzhirov A.V. Polyanin A.D. Handbook of Integral Equations: Methods of solutions. Moscow: Publishing House of the "Factorial Press", 2000, 384 p.
  7. Atkinson K.E. A survey of numerical methods for solving nonlinear integral equations. J. of integral equations and applications, 1992, No 1, v.4, p.15-46.
  8. Ch. Lubich. Runqe-Kutta theory for Volterra and Abel Inteqral Equations of the Second Kind. Mathematics of computation, volume 41, number 163, July 1983, p. 87-102.
  9. Brunner H. The solution of Volterra integral equation of the first kind by piecewise polynomials. J.Math . and Appl., 1973, No12, p.295-302.
  10. G. Mehdiyeva, M.Imanova On an application of the finite-difference method. Bulletin of the University, 2008, №2, p.73-78.
  11. Mehdiyeva G.Yu., Imanova M.N., Ibrahimov V.R. An application of the finite-difference method. Bulletin of the University,2009,№3, pp.101-108.
  12. Makroglou A. Hybrid methods in the numerical solution of Volterra integro-differential equations. Journal of Numerical Analysis 2, 1982, pp.21-35.
  13. H. Brunner. Imlicit Runge-Kutta Methods of Optimal oreder for Volterra integro-differential equation. Methematics of computation, Volume 42, Number 165, January 1984, pp. 95-109.
  14. Mehdiyeva G.Yu., Imanova M.N., Ibrahimov V.R. On one generalization of hybrid methods.  Proceedings of the 4th international conference on approximation methods and numerical modeling in environment and natural resources Saidia, Morocco, may 23-26, 2011, P. 543-547.
  15. G.Yu. Mehdiyeva, V.R. Ibrahimov, M.N. Imanova On the construction test equation and its applying to solving Volterra integral equation. Mathematical methods for information science and economics, proceedings of the 17th WSEAS International conference an Applied math. (AMATH 12), pp. 109-114.
  16. G. Dahlquist, Stability and Error Bounds in the Numerical Integration of Ordinary Differential Equations. 85 S. Stockholm 1959. K. Tekniska Högskolans Handlingar, No130, 87pp.
  17. V.R.Ibrahimov On a relation between order and degree for stable forward jumping formula. Zh. Vychis. Mat. , № 7, 1990, p.1045-1056.
  18. G.Yu. Mehdiyeva, V.R. Ibrahimov, M.N. Imanova  General hybrid method in the numerical solution for ODE of first and second order. Recent Advances in Engineering Mechanics, Structures and Urban Planning, Mathematics and Computers in Science and Engineering Series, Cambridge, UK, Febrary 20-22, 2013, p.175-180.
Комментарии: 3

AMIRALI,Gabil

The contribution THE APPLICATION DIFFERENCE METHODS TO SOLVING VOLTERRA INTEGRAL EQUATION by Prof. Mehdiyeva Galina et. al. is very suitable for presentation at the conference. With best regards, Prof. Gabil M. AMIRALIYEV

AMIRALI,Gabil

The contribution THE APPLICATION DIFFERENCE METHODS TO SOLVING VOLTERRA INTEGRAL EQUATION by Prof. Mehdiyeva Galina et al. is very suitable for presentation at the conference. With best regards, Prof. Gabil M. AMIRALIYEV

Симонян Геворг Саркисович

Уважаемая Галина Юрьевна, приветствую Ваше участие в конференции. Хорошая статья. С уважением -Геворг Саркисович.
Комментарии: 3

AMIRALI,Gabil

The contribution THE APPLICATION DIFFERENCE METHODS TO SOLVING VOLTERRA INTEGRAL EQUATION by Prof. Mehdiyeva Galina et. al. is very suitable for presentation at the conference. With best regards, Prof. Gabil M. AMIRALIYEV

AMIRALI,Gabil

The contribution THE APPLICATION DIFFERENCE METHODS TO SOLVING VOLTERRA INTEGRAL EQUATION by Prof. Mehdiyeva Galina et al. is very suitable for presentation at the conference. With best regards, Prof. Gabil M. AMIRALIYEV

Симонян Геворг Саркисович

Уважаемая Галина Юрьевна, приветствую Ваше участие в конференции. Хорошая статья. С уважением -Геворг Саркисович.
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