Some Application of the Hybrid Methods to Solving Volterra ... - wseas

Advances in Applied and Pure Mathematics

Some application of the hybrid methods to solving Volterra integral equations G.MEHDIYEVA, V.IBRAHIMOV, M.IMANOVA Calculation Mathematics Baku State University Z.Khalilov 23 AZERBAIJAN [email protected] Abstract: - Scientists basically began studies of the numerical solution of the Volterra integral equation after publishing of the known paper of Volterra. To define solution of such equation they used quadrature methods. The first quadrature method for solving linear integral equations with the variable boundaries has been constructed by Volterra. For receiving more exactly results the scientists have constructed the methods with the high order of accuracy. Thus here have considered construction and application of the one hybrid method to solving of the Volterra integral equations. And also constructed concrete here methods with the order of accuracy p ≤ 8 . Which can be applied to solving integral equations if it is known the value of the solution of initial problem in one mesh point. For the illustration of the received results, here are used some model equations.

Key-Words: - Volterra integral equation, hybrid methods, stability and degree, explicit and implicit methods It is known that the equation (1) in the case

1 Introduction

k ( x, s, y ( s )) = b( x, s ) y ( s ) has investigated by the

Let us consider to solving of the next integral equation with the variable bounders:

Volterra. And for the finding of integral equations (1) Volterra proposed the quadrature methods (see [1, p.3]) which has some advantages and

x

y ( x) = f ( x) + ∫ k ( x, s, y ( s ))ds,

disadvantages (see, for example [2] − [6] ). There are many papers of the different authors dedicated to application of the quadrature methods to solving equation (1) (see [5] − [9]). For receiving sufficiently exact results in solving equation (1), the scientists have used some modification of the quadrature method or constructed of the knew methods (see [6] − [18] ).Here for this aim proposed to using hybrid methods. The first time applied the hybrid methods to solving integral and integro-differential equation of Volltera type by Makroglow (see [12] ). For solving equation of type (1) one can be used the following methods :

x0

x0 ≤ s ≤ x ≤ X

(1) In the reference, the equation (1) is called as the nonlinear Volterra integral equation of the second kind Here we suppose that Volterra integral equation (1) has the unique solution determined on the interval

[x0 , X ] ,

for the investigation of the numerical solution of the equation of (1), assume that the kernel K ( x, z , y ) of the integral equation is defined in the domain G = {x 0 ≤ s ≤ x + ε ≤ X , | y |≤ b} , where it has partial derivatives up to some order p, inclusively. But the given sufficiently smooth function f ( x ) has determined on the

k

k

i =0

i =0

i =0

∑ α i y n+i = h∑ β i y n, +i + h∑ γ i y n, +i +ν

interval [x 0 , X ] . For determining the approximate values of the solution of equation (1), we divide the interval [x 0 , X ] into N equal parts by the mesh

(ν

i

< 1; i = 0,1,..., k )

i

(2) This method has applied to solving of the problem (see for example [19] ):

points xi = x 0 + ih (i = 0,1,2,..., N ) .

ISBN: 978-960-474-380-3

k

y , = f ( x, y ), y ( x0 ) = y 0 , x0 ≤ x ≤ X .

352

(3)


The method (2) by depending of its application usually has called as the multistep methods with the constant coefficients or the finite difference methods. Therefore some authors are called the correlation of (2) as the finite difference equation (see for example [20], [21] ) .

y ( x + ih) = y ( x) + ihy ,

+

,

(ih )

,,

2 ( ih ) ,,, ( x) + y ( x) + ... +

2!

p −1

( p − 1)!

y ( p ) ( x ) + O ( h) p

(5)

y , ( x + (i + ν i )h) = y , ( x) + (i + ν i )hy ,, ( x) + ... +

2 APPLICATION OF THE METHOD (2) TO SOLVING OF EGUATION (1)

+

Note that the method (2) in the work [19] has applied to solving of the initial-value problem for the ordinary differential equation of the first order and proof, that if the method (2) is convergence, then its coefficients satisfies the following conditions: A: The coefficients and αi , βi ,γ i

B: The characteristic polynomials k

k

i =0

i =0

i =0

ρ ( λ ) ≡ ∑ α i λi ; σ ( λ ) ≡ ∑ β i λ i ; γ ( λ ) ≡ ∑ γ i λ i + l i

∑ (α y( x + ih) − hβ y ( x + ih) − hγ y ( x + (i + ν )h)) = (7) k

,

i =0

have no common multiple different from constant. C: σ (1) + γ (1) ≠ 0 and p ≥ 1 .

y n +i

and

of the function y(x). It is known that for the application of multistep method to solving of some problems. The values yi (i=0,1,…,k-1) must be known. Therefore if are known the values

y n, + i +ν i

k

i =0

i

k

∑ iα i = ∑ (β i + γ i ) ,

(8)

i =0

k

k

i =0

i =0

∑ (i l −1 β i + (i + ν i ) l −1 γ i ) = l ∑ i l α i

we the (2) the

(l = 2,3,..., p )

Proof. We first prove that if method (2) has the degree p , then the coefficients α i , β i , γ i , l i (i = 0,1,2,..., k ) will satisfy the system of nonlinear algebraic equations given in (8). Taking into account that the method (2) has the degree p , then by using Taylor expression (4)-(6) in the left hand-side of asymptotic equality (8). We have:

2 ( ih ) ,, y( x + ih) = y( x) + ihy ( x) + y ( x) + ... + ,

2!

(4)

p!

ISBN: 978-960-474-380-3

,

i

∑ α i = 0;

and

(i=0,1,…,k) of the function y(x),then can be applied the method (2) to solving of equation (1).So for the contraction type method to solving of the equation (1),considers to following expansion of Taylor:

p ( ih ) ( p ) y ( x) + O(h) p +1 . +

i

then p is called the degree of the method (2). For the definition of the coefficients of the method (2) one can be used the next lemma. Lemma. Yet y(x) be a sufficiently smooth function, and assume that condition A,B and C are holds. For the method (2) to have the degree p , satisfies the following conditions of its coefficients are necessary and sufficient:

y n+i +ν i

y n, + i

i

= O(h) p +1 , h → 0

Here we, assume that the conditions are holds. Note that using of the method (2) to solving equation (1) we must determine the values

( p − 1)!

(6)

y ( p ) ( x ) + O ( h) p

here x = x0 + nh is the fixed point. As mentioned above, the aim of this work is the construction of the methods with the high order of the methods with the high order of accuracy. It is known that in the theory of multistep methods the notion “order of accuracy” is replace to the notion of the degree of the multistep method. The degree of the method (2) can be defined as the following: Definition 1. If for the sufficiently smooth function y(x) and for the integer value quantity p the following is holds:

li , (i = 0,1,2,..., k ) are the real numbers; moreover, αk ≠ 0 . k

((i + ν i )h) p−1

353


k k   ∑ α i  y ( x) + h∑ (iα i − βi − γ i )y′( x) + i =0  i =0  2 k  i + h2 ∑  α i − iβi − (i + li )γ i  y′′( x) +... + i =0  2 

y n +1 = y n + h( y ' n + l0 + y ' n +1+ l1 ) / 2.

(12) Here l1 = −l 0 ; l0 = (3 − 3 ) / 6, 1 + l1 = (3 + 3 ) / 6 . Remark that for applying the hybrid method (12) to solving some problems it is needed known of the values y n +1 2 + 3 6 , y 1 and y n +1 2− 3 6 . Note, that

(9)

k  ip i p −1 (i + l ) p −1  βi − i γ i y ( p ) ( x) = O(h p +1 ), h → 0. + h p ∑  α i − ( p − 1)! ( p − 1)!  i = 0  p! If take in account that the method (2) has the degree p , then we obtain the following: k

k

∑ α y( x) + h∑ (iα i

i =0

i =0

k

i

n+

these variables are independent from y n+1 , because that method (12) is explicit. But there is, exist implicit hybrid methods of type (12). For example, consider the following method:

− β i − γ i ) y ′( x) +

2

i + h 2 ∑ ( − iβ i − (i + l i )γ i ) y ′′( x) + ... i =0 2

y n +1 = y n + h(3 y ' n +1 / 3 + y ' n +1 ) / 4.

It is known that 1, x, x 2 ,..., x p forms a linearly independent system; therefore, equality (10) is equivalent to the following:

∑α i =0

k

∑( i =0

= 0,

i

k

∑ (iα i =0

i

− β i − γ i ) = 0,....,

β 0 + β 1 + γ 0 + γ 1 = 1, β 1 + l 0 γ 0 + l1γ 1 = 1 / 2,

β 1 + l 02 γ 0 + l12 γ 1 = 1 / 3, β1 + l 03γ 0 + l13γ 1 = 1 / 4,

(11)

(i + l i ) p −1 i i p −1 αi − βi − γ i ) = 0. p! ( p − 1)! ( p − 1)! p

β1 + l 04 γ 0 + l14 γ 1 = 1 / 5, β1 + l 05γ 0 + l15γ 1 = 1 / 6.

We now will prove that if the coefficients of the method (2) are the solution of the nonlinear system (8), then its degree is equal to p . Indeed, if we used the system of equalities of (11) into equality (9), then we obtain the asymptotic equality of (7). From this asymptotic equality it follows that method (2) must have the degree p . It is easy to determine that for the chosen values l i = 0 (i = 0,1,..., k ) , the system (8) is linear and coincides with the known systems used for defining the coefficients of the multistep method with constant coefficients. Subject

By solving this system of algebraic equations, one can constructed some hybrid methods. One of that has the solving form:

y n +1 = y n + h( y ' n +1 + y ' n ) / 12 + + 5h( y ' n +1 / 2 −

5 / 10

+ y ' n +1 / 2 +

5 / 10

) / 12, ( p = 6).

(14)

To apply hybrid methods to solving of some problems, we should know the values of y n +1 / 2− 5 / 10 and y n +1 / 2+

5 / 10

, and the accuracy of these values

should have order at least O (h 6 ) . Note that hybrid method (14) is implicit and that when applying it to solving of equation (1), is used a predictor-corrector scheme containing only one explicit method. Therefore, we consider the construction of an explicit method which in one variant has the following form:

+ l1 + ... + l k ≠ 0 , the system (8) is nonlinear. This system contains p + 1

to the conditions from l 0

unknowns and is equations in 4k + 4 homogeneous; it must possess the zero solution, and for system (8) has a non-zero solution, suppose that the condition 4k + 4 > p + 1 is holds. Hence, we obtain that there are methods of type (2) with the order p ≤ 4k + 2 . Consider the construction methods of type (2) for

y n +1 = y n + hy ' n / 9 + h((16 + 6 ) y ' n + ( 6− + (16 − 6 ) y ' n + ( 6+

6 ) / 10

+ (15)

) / 36. 6 ) / 10

This method is explicit and has the degree p = 5 . Note that the methods (12) and (15) are explicit, whereas method (14) is implicit. The application of explicit hybrid methods requires some additional auxiliary formulas. Here let us consider to

β = 0(i = 0,1,..., k ) and suppose that k = 1 . i

Note that the method with the degree p = 4 can be written as follows:

ISBN: 978-960-474-380-3

(13)

This method is implicit and has the degree p = 3 . Now, let us consider to construction of the method of type (2) for the case k = 1 . In this case, assuming that α 1 = −α 0 = 1 , the system (15) takes the following form:

(10)

k (i + l i ) p −1 ip i p −1 + h p ∑ ( αi − βi − γ i ) y ( p ) ( x) = 0. ( p − 1)! ( p − 1)! i = 0 p!

k

2

354


construction an algorithm for using of the method (12). Algorithm 1. Applies method (12) to the solution of problem (1). Step 1. Calculate y n+l and y n+1−l by with the

0.50 0.60 0.70 0.80 0.90 1.00

following block method:

y n +l = y n + f n +l − f n + hlk ( x n , x n , y n ) lh yˆ n +l = y n + f n +l − f n + (2k ( x n , x n , y n ) + 4 + k ( x n , x n , y n ) + k ( x n , x n , y n )) hl y n +l = y n + f n +l − f n + (5k (x n , x n , y n ) + 12 + 3k ( x n , x n +l , y n +l ) + 5k ( x n +l , x n +l , y n +l )) −

0.1E-16 0.1E-16 0.1E-16 0.1E-16 0.1E-16 0.2E-16

Given the widespread use of integral equations is investigated numerical solution of integral equations with variable abroad. To this end, proposed the use of hybrid methods which benefit has been proven, both theoretical and practical way. Sake of objectivity, we note that the hybrid methods have some disadvantages in finding values of the function at points of intermediate type y (x)

lh k ( x n + 2lh, y n + lh(k ( x n , x n +l , yˆ n +l ) + 12 + k ( x n +l , x n +l , yˆ n +l )) l := (3 − 3 ) / 6

0.24E-05 0.32E-05 0.40E-05 0.48E-05 0.57E-05 0.66E-05

3 Conclusion

−

Repeat this scheme for

0.14E-09 0.18E-09 0.21E-09 0.25E-09 0.30E-09 0.34E-09

x n +l = x n + lh ( l < 1) .

To

remedy

this

shortcoming, this proposed scheme is one type of forecast-correction. Thus, we find that the hybrid methods are promising because they are more accurate, and it seems to us that the region of stability for the hybrid methods in some cases will be wider.

and

1 − l = (3 + 3 ) / 6 Step 2. Calculate yn +1 according to method (12). Here, we compute the values of the quantities y n +l

y

and n +1− l to within O(h 4 ) , which suffices for this algorithm. For the illustration of the results received here, we have considered to application of the method (12) to solving the following equations: 2 1. y ( x) = 1 + x

4 Acknowledgment The authors wish to express their thanks to academician Ali Abbasov for his suggestion to investigate the computational aspects of our problem and for his frequent valuable suggestion. This work was supported by the Science Development Foundation of Azerbaijan (Grand EIF2011-1(3)-82/27/1). We are also grateful to the referees whose useful suggestions greatly improve the quality of this paper.

x

2

+ ∫ y ( s )ds the exact solution is 0

y ( x ) = 2e − x − 1 . x

x

∫

2. y ( x) = e − x + e −( x − s ) y 2 ( s )ds the exact solution 0

is y ( x) = 1 .

References: V.Volterra. Theory of functional and of integral and x 2 + y s ds ( 1 ( ) integro-differensial equations, Dover publications. the exact solution 3. y ( x) = 2 Ing, New York, 304. + s ( 1 ) 0 [2] Mehdiyeva G.Yu., Imanova M.N., Ibrahimov V.R. is y ( x) = x . On one application of forward jumping methods. Applied Numerical Mathematics, Volume 72, October 2013, p. 234–245. The obtained results, place in the following table. [3] A.F.Verlan, V.S. Sizikov Integral equations: methods, algorithms, programs. Kiev, Naukova Step Variabl Example Example Example Dumka, 1986. size 1 2 3 e x [4] G. Mehdiyeva, V.Ibrahimov, M.Imanova On the h = 0,02 0.10 0.26E-10 0.15E-06 0.1E-16 construction test equations and its Applying to solving Volterra integral equation, Methematical 0.20 0.54E-10 0.54E-06 0.1E-16 methods for information science and economics, 0.30 0.83E-10 0.10E-05 0.1E-16 Montreux, Switzerland, 2012, pp. 109-114 0.40 0.11E-10 0.17E-05 0.1E-16 [1]

∫

ISBN: 978-960-474-380-3

355


[5]

[6]

[7]

[8]

[9]

[10]

[11]

[12]

[13] [14]

[15]

[16]

[17]

[18]

Manzhirov, A.V., & Polyanin, A.D. (2000). [19] G.Mehdiyeva, M.Imanova, V.Ibrahimov Handbook of Integral Equations: Methods of Application of the hybrid methods to solving solutions. Moscow: Publishing House of the Volterra integro-differential equations World "Factorial Press", 384. Academy of Science, engineering and Technology, Paris, 2011, 1197-1201. Mehdiyeva G., Imanova M., Ibrahimov V. On a Research of Hybrid Methods. Numerical Analysis [20] N.S. Bakhvalov Numerical methods, M.Nauka, and Its Applications, Springer, 2013, p. 395-402. 1973. Atkinson, K.E. (1992). A survey of numerical [21] Ryaben'kii V.S., A.F. Filippov The stability of methods for solving nonlinear integral equations. J. difference equations. - Moscow: Gostekhizdat, of integral equations and applications, No 1, v.4, 151956. - 171. 46. Ch.Lubich, “Runge-Kutta theory for Volterra and Abel integral equations of the second kind” Mathematics of computation volume 41, number 163, July 1983, p. 87-102. G.Mehdiyeva, M.Imanova, V.Ibrahimov The Application Difference Methods to Solving Volterra Integral Equation. Pensee Journal, Paris, Vol. 75, Issue. 111, 2013. p. 393-400. 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. G.Mehdiyeva, V.Ibrahimov, Bayramova N. Construction of a second derivative one-step method of fourth order accuracy and its application to the solution of volterra integral equation of second kind. The Third international Conference “Problems of Cybernetics and Informatics”, Baku, Azerbaijan, 2010, p. 300-303. Makroglou A.A. Block - by-block method for the numerical solution of Volterra delay integrodifferential equations, Computing 3, 1983, 30, №1, p.49-62. Krasnoselskiy, M.A. (1969). The approximate solution of operator equations. М., Nauka, 210. Mamedov, Y.D., & Ashirov, S.A. (1980). Methods of successive approximations for the solution of operator equations. Ashgabat, 120. G. Mehdiyeva, V.Ibrahimov, M.Imanova Application of A Second Derivative Multi-Step Method to Numerical Solution of Volterra Integral Equation of Second Kind, Pak.j.stat.oper.res. Vol.VIII No.2 2012, 245-258. M.N. Imanova One the multistep method of numerical solution for Volterra integral equation. Transactions issue mathematics and mechanics series of physical -technical and mathematical science, ХХВI, 2006,№1, pp.95-104. Ya.D.Mamedov, VA Musayev. Investigation of solutions of nonlinear operator equations of Fredholm-Voltaire. Dokl t.284, № 6, 1985. Polishuk Ye. M. Vito Volterra. Leningrad, Nauka, 1977, 114p.

ISBN: 978-960-474-380-3

356