Jul 16, 2017 - In the present paper, we propose Simpson's quadrature rule to solve the system of linear Volterra Integral Equations of the second kind.
Journal of Computer and Mathematical Sciences, Vol.8(7), 332-339 July 2017 (An International Research Journal), www.compmath-journal.org
ISSN 0976-5727 (Print) ISSN 2319-8133 (Online)
A New Technique for Numerical Solution of System of Volterra Integral Equations of the Second Kind by Simpson’s Quadrature Rule Irin Rahman, M. M. Parvez and Sumita Ghosh Department of Mathematics, Dhaka University of Engineering & Technology, Gazipur Gazipur-1700, BANGLADESH. (Received on: July 15, Accepted: July 16, 2017) ABSTRACT In the present paper, we propose Simpson’s quadrature rule to solve the system of linear Volterra Integral Equations of the second kind. We convert the system of linear Volterra Integral Equation of the second kind to a recurrence relation and an approximation solution of them is obtained. We show that our estimates have a good degree of accuracy. Keywords: Simpson’s quadrature rule, System of linear Volterra Integral Equations of the Second Kind.
1. INTRODUCTION In recent years, there has been a growing interest in the Volterra integral equations arising in various fields of physics and engineering, e.g. potential theory and Dirichlet problems, electrostatics, the particle transport problems of astrophysics and reactor theory, contact problems, diffusion problems, and heat transfer problems. Also, many initial and boundary value problems associated with the ordinary and partial differential equations can be cast into the Volterra integral equations. The problem of finding a numerical solution of Fredholm and Volterra integral equations of the second kinds is one of the oldest problems in the applied Mathematics literature and many computational methods are introduced in this field. In the recent paper, we apply repeated Simpson's quadrature rule to solve the system of linear Volterra integral equations of the second kind. By using this method we convert the system of linear Volterra integral equations of the second kind of a recurrence relation and find an approximate solution for them. 332
Irin Rahman, et al., Comp. & Math. Sci. Vol.8 (7), 332-339 (2017)
2. SIMPSON’S QUADRATURE RULE If N is even, then Simpson’s quadrature rule may be applied to each subinterval ଶ , ଶାଵ , ଶାଶ ; = 0, 1, … … … … … . , − 1 2 individually yielding the approximations ℎ ଶ + 4ଶାଵ + (ଶାଶ ) 3 ே Summing these approximation result in the composite version of Simpson’s quadrature ଶ ℎ
ℎ = + 4 + ℎ + 2 + 2ℎ + 4 + 3ℎ + ⋯ + 2 − 2ℎ 3 + 4 − ℎ + ( ) ே For the entire interval. The error of S(h) is the sum of all individual errors
ℎ = () − ℎ = −
And we conclude that ℎ = For further information, see [10]
ಿ ିଵ మ
ଶ
ℎହ ସ ƺ ƺ ∈ ଶ , ଶାଶ 90
ୀ ି ସ ସ − ℎ ƺ ; ƺ ∈ ଵ଼
( , )
3. METHOD OF SOLUTION A linear system of Volterra integral equation of the second kind is considered in the following way, + ଵ , ଵ = ଵ ଵ ௧ ଶ + ଶ , ଶ = ଶ ௧
, < < , (1) . . . ௧ + , = where , , are known functions with , ≠ 0 = 1,2, … . and = 1,2, … . are unknown functions. Let the interval , be finite and partitioned by N equally spaced points. = ; ே = ; 333
Irin Rahman, et al., Comp. & Math. Sci. Vol.8 (7), 332-339 (2017)
= + ℎ ; = 1,2, … ே The approximations of system of VIE (1) in the even nodes ଶ where = 1,2, … … , are మ ଵ ଶ + ଵ , ଵ = ଵ ଶ
ଶ
௧
௧మ ଶ ଶ + ଶ , ଶ = ଶ ଶ . given by (2) . . ௧మ ଶ + , = ଶ
Instead of a system of VIE (2), consider the following equation:
ଵ ଶ + ଵ ଶ , ଵ = ଵ (ଶ ) ୀ ௧మ ିଵ ௧మశమ ଶ ଶ + ଶ ଶ , ଶ = ଶ (ଶ ) ିଵ ௧మశమ
మ . . . ିଵ ௧మశమ ଶ + ଶ , = (ଶ ) ୀ ௧మ
ୀ ௧
(3)
And applying repeated Simpson’s quadrature rule, we have
ℎ ଵ ଶ + ଵ ଶ,ଶ ଵ ଶ + 4ଵ ଶ,ଶାଵ ଵ ଶାଵ + ଵ ଶ,ଶାଶ ଵ ଶାଶ = ଵ ଶ 3 ୀ ିଵ ℎ ଶ ଶ + ଶ ଶ,ଶ ଶ ଶ + 4ଶ ଶ,ଶାଵ ଶ ଶାଵ + ଶ ଶ,ଶାଶ ଶ ଶାଶ = ଶ ଶ 3 ୀ . . . ିଵ ℎ ଶ + ଶ,ଶ ଶ + 4 ଶ,ଶାଵ ଶାଵ + ଶ,ଶାଶ ଶାଶ = ଶ 3 ିଵ
We set ଶାଵ = ୀ
௬ మ ା௬ మశమ ଶ
for = 1,2, … … …
334
Irin Rahman, et al., Comp. & Math. Sci. Vol.8 (7), 332-339 (2017)
ଵ ଶ + ଵ ଶାଶ ℎ + ଵ ଶ,ଶାଶ ଵ ଶାଶ = ଵ ଶ ଵ ଶ + ଵ ଶ,ଶ ଵ ଶ + 4ଵ ଶ,ଶାଵ 3 2 ୀ ିଵ ଶ + ଶ ଶାଶ ℎ ଶ ଶ + ଶ ଶ,ଶ ଶ ଶ + 4ଶ ଶ,ଶାଵ ଶ + ଶ ଶ,ଶାଶ ଶ ଶାଶ = ଶ ଶ 3 2 ୀ . . . ିଵ + ଶାଶ ℎ ଶ + ଶ,ଶ ଶ + 4 ଶ,ଶାଵ ଶ + ଶ,ଶାଶ ଶାଶ = ଶ 3 2 ିଵ
ୀ
Then we have
ℎ ℎ + (ଵ ଶ,ଶ + 2ଵ ଶ,ଶାଵ )ଵ ଶ + (2ଵ ଶ,ଶିଵ + ଵ ଶ,ଶ ) ଵ ଶ = ଵ ଶ ଵ ଶ 3 3 ୀ ୀଵ ିଵ ℎ ℎ ଶ ଶ + (ଶ ଶ,ଶ + 2ଶ ଶ,ଶାଵ )ଶ ଶ + (2ଶ ଶ,ଶିଵ + ଶ ଶ,ଶ ) ଶ ଶ = ଶ ଶ 3 3 ୀଵ ୀ . . . ିଵ ℎ ℎ ଶ + ( ଶ,ଶ + 2 ଶ,ଶାଵ ) ଶ + (2 ଶ,ଶିଵ + ଶ,ଶ ) ଶ = ଶ 3 3 ିଵ
By expanding every row of above system for = 1,2, … … … . ଶ , we have ୀଵ
ୀ
ே
ଵ ଶ − ∑ିଵ ୀଵ ଵ ଶ,ଶିଵ + ଵ ଶ,ଶ + ଵ ଶ,ଶାଵ ଵ ଶ − ଷ ( ଵ ଶ, + 2 ଵ ଶ,ଵ )ଵ ଷ = ଵ ଶ 1 + 2 ଵ ଶ,ଶିଵ + ଵ ଶ,ଶ
ଷ ଶ ିଵ ∑ − + ଶ ଶ ଶ ଶ,ଶିଵ ଶ ଶ,ଶ + ଶ ଶ,ଶାଵ ଶ ଶ − ଷ ( ଶ ଶ, + 2 ଶ ଶ,ଵ )ଶ ୀଵ ଷ ଶ ଶ = 1 + 2 ଶ ଶ,ଶିଵ + ଶ ଶ,ଶ
(4) ଷ . . . ଶ ିଵ ∑ − + ଶ ଶ,ଶିଵ ଶ,ଶ + ଶ,ଶାଵ ଶ − ଷ ( ଶ, + 2 ଶ,ଵ ) ୀଵ ଷ ଶ = 1 + 2 ଶ,ଶିଵ + ଶ,ଶ
ଷ ଶ
Now taking the initial value = in system (1) we obtain
= ଵ = ଵ = ଵ () ଵ = ଶ = ଶ = ଶ () ଶ . (5) . . = = = () 335
Irin Rahman, et al., Comp. & Math. Sci. Vol.8 (7), 332-339 (2017)
The computation of ଶ , for = 1,2, … … … ଶ will facilitate the recurrence relation (4) and the above mentioned relation. ே
4. NUMERICAL EXAMPLES Consider the system of Volterra integral equation of the second kind
ଵ = 1 + 2 − () ௧
ଶ =
+1+
௧
௧
ିሺ௧ି௦ሻ
ଷ = + 2
௧
௧ି௦
0 ≤ ≤ 1
()
ଵ = " + ଶ = + 2
! ଵ ଷ = ହ 2 − " +
The exact solution is
ଷ௧
And numerical approximations by using repeated Simpson’s quadrature rule with the figure are shown in the following tables: Table 01: Exact and approximate solution of example for h=0.05 Node
Exact
Approximate
Solution
Solution
ݕଵ ()ݐ
ݕଶ ()ݐ
0.0
1.000000000000
2.000000000000
ݕଵ ()ݐ
ݕଶ ()ݐ
1.000000000000
2.000000000000
000
000
0.1
1.094837581924
2.100000000000
0.110904295118
000
000
1.094920793613
2.100000021001
0.111348905940
854
000
0.2
1.178735908636
2.200000000000
327
006
523
018
0.247878176827
1.178901506884
2.200000042350
0.249058956929
303 0.3
1.250856695786
000
878
592
166
904
2.300000000000
0.419061407070
1.251103031688
2.300000064045
0.421422368013
0.4
946
000
805
495
927
042
1.310479336311
2.400000000000
0.635578522670
1.310803953713
2.400000086088
0.639787397943
0.5
536
000
193
040
808
493
1.357008100494
2.500000000000
0.912591517131
1.357407760589
2.500000108478
0.919642247684
0.6
576
000
220
041
807
724
1.389978088304
2.600000000000
1.270719359258
1.390448802039
2.600000131215
1.282078926564
714
000
667
289
926
233
t
ݕଷ ()ݐ 0
336
ݕଷ ()ݐ 0
Irin Rahman, et al., Comp. & Math. Sci. Vol.8 (7), 332-339 (2017) 0.7
1.409059874522
2.700000000000
1.737952619951
1.409596942497
2.700000154300
1.755770729178
180
000
708
748
164
693
0.8
1.414062800246
2.800000000000
2.352236370618
1.414660859710
2.800000177731
2.379644011296
688
000
697
975
520
994
0.9
1.404936877898
2.900000000000
3.164955115171
1.405589956363
2.900000201509
3.206488554188
148
000
428
188
996
789
1.0
1.381773290676
3.000000000000
4.245635317387
1.382474865624
3.000000225635
4.307836790024
036
000
064
624
591
433
Table 02: Calculating error by comparing the results of exact and approximate solution for h=0.05 Node t 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Error calculation, | | | , , |; 1,2,3. |ଵ | |ଶ | |ଷ | 0 0 0 0.000083211688152 0.000000021001523 0.000444610821691 0.000165598248289 0.000000042350166 0.001180780102026 0.000246335901449 0.000000064045927 0.002360960942237 0.000324617401504 0.000000086088808 0.004208875273300 0.000399660094465 0.000000108478807 0.007050730553504 0.000470714734575 0.000000131215926 0.011359567305566 0.000537067975568 0.000000154300164 0.017828109226985 0.000598059464287 0.000000177731520 0.027407640678297 0.000653078465040 0.000000201509996 0.031533441017361 0.000701574948588 0.000000225635591 0.062201472637369
Figure: Table 01
337
Irin Rahman, et al., Comp. & Math. Sci. Vol.8 (7), 332-339 (2017) Table 03: Exact and approximate solution of example for h=0.01 Node
t 0.0 0.02 0.04 0.06 0.08 0.1
Exact Solution
ݕଵ ()ݐ 1.00000000000 0000 1.01979867335 9911 1.03918944084 7612 1.05816454641 4649 1.07671640027 1792 1.09483758192 4854
ݕଶ ()ݐ 2.00000000000 0000 2.02000000000 0000 2.04000000000 0000 2.06000000000 0000 2.08000000000 0000 2.10000000000 0000
Approximate Solution
ݕଷ ()ݐ 0 0.02040677465 3090 0.04165508265 8333 0.06378896722 9099 0.08685536639 1426 0.11090429511 8327
ݕଵ ()ݐ 1.00000000000 0000 1.01979933998 6798 1.03919077383 5181 1.05816654523 0148 1.07671906411 6142 1.09484090973 2961
ݕଶ ()ݐ 2.00000000000 0000 2.02000000000 6678 2.04000000001 3377 2.06000000002 0100 2.08000000002 6844 2.10000000003 3611
ݕଷ ()ݐ 0 0.02040959349 4214 0.04166104123 1650 0.06379841591 7822 0.08686868750 2286 0.11092190550 0630
Table 04: Calculating error by comparing the results of exact and approximate solution for h=0.01 Node T 0.01 0.02 0.04 0.06 0.08 0.1
Error calculation, | | = | , − , |; = 1,2,3. |ଵ | |ଶ | |ଷ | 0 0 0 0.000000666626887 0.000000000006678 0.000002818841124 0.000001332987469 0.000000000013377 0.000005958573317 0.000001998815499 0.000000000020100 0.000009448688723 0.000002663844350 0.000000000026844 0.000013321110860 0.000003427808107 0.000000000033611 0.000017610282303
5. CONCLUSION In this paper, we applied an application of repeated Simpson’s quadrature rule for solving the system of linear Volterra integral equations of the second kind. According to the numerical results which obtaining from the illustrative example, we conclude that for sufficiently small h we get a good accuracy, since by reducing the step size length the least square error will be reduced. The same approach can be used to solve other problems like: - Nonlinear Volterra integral equations. - Linear Fredholm integral equations with Cauchy kernel, Abel kernel. - Volterra- Fredholm integral equations. REFERENCES 1. Mostafa Safavi, A New Technique for Numerical Solution of System of Volterra Integral Equations of the first kind via Quadrature Rule, ACMA, (2012). 338
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2. A.J. Jerri, Introduction to Integral Equations with Applications, second edition, John Wiley and Sons, (1999). 3. A.D. Polyanin, A.V. Manzhirov, Handbook of Integral Equations, CRC Press, (2008). 4. C.T.H. Baker, G.F. Miller, Treatment of Integral Equations by Numerical Methods, Academic Press Inc., London, (1982). 5. L. M. Delves, J. L. Mohammad, Computational methods for integral equations, Cambridge University Press, (1985). 6. R. P Kanwal, Linear integral equations theory and technique, Academic Press, New York and London, (1971). 7. R. Kress, Linear Integral Equations, Springer-Verlag, Berlin, Heidelbarg, 1989. 8. P.K Kyte, P. Puri, E. Computational methods for integral equations, Birkhauser, Boston, (2002). 9. M. Nadir, A. Rahmoune, P.P. Zabrejko, Modified Method for solving linear Volterra integral equations of the second kind using Simpson’s rule, Int. J. Math. Manus., 1(1), 141-146 (2007). 10. G. M. Philips, P. J. Taylor, Theory and applications of numerical analysis, Academic press, New York, (1973).
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