Numerical Solution of Fredholm Integral Equations Using Hosoya Polynomial of Path Graphs

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1 America Joural of Numerical Aalysis, 7, Vol. 5, No., -5 Available olie at Sciece ad Educatio Publishig DOI:.69/aja-5-- Numerical Solutio of Fredholm Itegral Equatios Usig Hosoya Polyomial of Path Graphs H. S. Ramae, S.C. Shiralashetti, R. A. Mudewadi,*, R. B. Jummaaver Departmet of Mathematics, Karatak Uiversity, Dharwad, Idia P.A. College of Egieerig, Magalore, Idia *Correspodig author: Abstract he mai purpose of this paper is to develop the graph theoretic polyomial to solve umerical problems. We preset a ew method for the solutio of Fredholm itegral equatios usig Hosoya polyomials obtaied from oe of the stadard class of graphs called as path. Proposed algorithm epads the desired solutio i terms of a set of cotiuous polyomials over a closed iterval [,]. However, accuracy ad efficiecy are depedet o the size of the set of Hosoya polyomials ad compared with the eistig method. Keywords: Fredholm itegral equatios, Hosoya polyomial, path Cite his Article: H. S. Ramae, S.C. Shiralashetti, R. A. Mudewadi, ad R. B. Jummaaver, Numerical Solutio of Fredholm Itegral Equatios Usig Hosoya Polyomial of Path Graphs. America Joural of Numerical Aalysis, vol. 5, o. (7): 8-. doi:.69/aja Itroductio I graph theory, as i discrete mathematics i geeral, ot oly the eistece, but also the coutig of objects with some give properties, is of mai iterest. Each area itroduces its ow special terms for shared cocepts i discrete mathematics. he oly way to keep from reivetig the wheel from area to area is to kow the precise mathematical ideas behid the cocept beig applied by these various fields. Graph theory is rapidly movig ito the maistream of mathematics maily because of its applicatios i diverse fields which iclude biochemistry, electrical egieerig (Commuicatios etworks ad codig theory), computer sciece (algorithms ad computatios) ad operatios research (schedulig) [,]. Itegral equatios have motivated a large amout of research work i recet years. Itegral equatios fid its applicatios i various fields of mathematics, sciece ad techology has bee studied etesively both at the theoretical ad practical level. I particular, itegral equatios arise i fluid mechaics, biological models, solid state physics, kietics i chemistry etc. I most of the cases, it is difficult to solve them, especially aalytically []. Aalytical solutios of itegral equatios, however, either does ot eist or are difficult to fid. It is precisely due to this fact that several umerical methods have bee developed for fidig solutios of itegral equatios. Cosider the Fredholm itegral equatio: y ( ) = f( ) + k( t, ) yt ( ) dt, t, () where ff() ad the kerels kk(, tt) are assumed to be i L (R) o the iterval, tt. We assume that Eq.() has a uique solutio y to be determied. here are several umerical methods for approimatig the solutio of Fredholm itegral equatios are kow ad may differet basic fuctios have bee used. Such as, Lepik et al. [4] applied the Haar Wavelets. Malekejad et al. [5] applied a combiatio of Hybrid taylor ad block-pulse fuctios, Ratioalized Haar wavelet [6], Hermite Cubic splies [7]. Muthuvalu et al. [8] applied Half-sweep arithmetic mea method with composite trapezoidal scheme for the solutio of Fredholm itegral equatios. I this paper, we proposed a umerical method for the solutio of Fredholm itegral equatios usig Hosoya polyomial of paths as basis.. Properties of Hosoya Polyomial A graph G cosists of a fiite oempty set V of poits (vertices) together with a prescribed set X of m uordered pairs of distict poits of V. Each pair = ( uv, ) of poits i X is a edge of G. If the poits u ad v are joied by a edge, the we say that u ad v are adjacet poits. Let v, v,,v be the vertices of G. he path P is a graph with vertices v, v,, v, where v i is adjacet to v i +, i =,,,. he legth of a path is the umber of edges i it. A graph G is said to be coected if every pair of poits of G is joied by some path. he distace betwee the vertices v i ad v j i G is equal to the legth of the shortest path joiig them ad is deoted by du ( i, v j). For more details about the graph theory oe ca refer the book [9]. he Wieer ide W(G) of a coected graph G is defied as the sum of the distaces betwee all uordered pairs of vertices of G, that is, WG ( ) = du ( i, vj). < i j

2 America Joural of Numerical Aalysis his ide was put forward by Harold Wieer [] i 947 for approimatio of the boilig poits of alkaes. he effect of approimatio was surprisigly good. From that poit forward, the Wieer ide has attracted the attetio of chemists. he Hosoya polyomial of a graph is a geeratig fuctio about distace distributig, itroduced by Hosoya [] i 988. For a coected graph G, the Hosoya polyomial deoted by HGλ (, ) is defied as HG (, λ) = dgk (, ) λ k () k where dgk (, ) is the umber of pairs of vertices of G that are at distace k ad λ is the parameter. he coectio betwee the Hosoya polyomial ad the Wieer ide is elemetary [,]: WG ( ) = H'( G,), where H'( G, λ) is the first derivative of HGλ (, ). Hosoya polyomial of tress [,4], composite graphs [5], bezeoid graphs [6,7], tori [8], zig-zag opeeded aotubes [9], armchair ope-eded aotubes [], zigzag polyheaotorus [], Fiboacci ad Lucas cubes [] are reported i the literature. he paths P, P ad P are depicted i Figure. Figure. Path graphs P, P ad P he Hosoya polyomial of a path P is: H( P, λ) = + ( ) λ+ ( ) λ + + [ ( )] λ + [ ( )] λ. I particular, H( P, λ) = H( P, λ) = λ+ H( P, λ) = λ + λ+.. Fuctio Approimatio A fuctio u() = f(), u'() = f '() is epaded as: f( ) = ch i ( Pi, ) = C HP( ), i= () where C ad HP ( ) are matrices give by: ad C = [ c, c,, c ], (4) H ( ) = [ H( P, ), H( P, ),, H( P, ),]. (5) P 4. Hosoya Polyomial Method (HPM) Cosider the Fredholm itegral equatio, y( ) = f ( ) + k(, t) y( t) dt,, t (6) o solve Eq. (6), the procedure is as follows: Step : We first approimate y() as trucated series defied i Eq. (). hat is, y ( ) = C H ( ) (7) where C ad HP ( ) are defied as i Eqs. (4) ad (5). Step : Substitutig Eq. (7) i Eq. (6), we get, C HP( ) f( ) k(, t)[ C HP( t)] dt P = +. (8) Step : Substitutig the collocatio poit i =,,..., i Eq. (8), we obtai, C HP( i) f( i) k( i, t)[ C HP( t)] dt. hat is, i.5 i =, = + (9) C ( HP( i) Z) = f, where Z = k( i,) t HP() t dt. Step 4: Now, we get the system of algebraic equatios with ukow coefficiets. C K = f, where K = ( HP( i) Z). Solvig the above system of equatios, we get the Hosoya coefficiets C ad the substitutig these coefficiets i Eq. (7), we get the required approimate solutio of Eq. (6). 5. Numerical Results I this sectio, we cosider the some illustrative eamples from the literature to demostrate the capability of the method ad error fuctio is preseted to verify the accuracy ad efficiecy of the umerical results: E = Errorfuctio Ma ye i ya i ye i ya i i= ( ) ( ( ) ) = ( ) = ( ). where, yy ee ad yy aa are the eact ad approimate solutio respectively. We cosidered the eamples give i [7]. Usig the preset techique, the umerical solutios with eact solutios preseted i able ad the maimum error aalysis compared with eistig method [7] give i able. Eample. Cosider Fredholm itegral equatios, y( ) si( π ) cos ( ) y( t) dt, = + () which has the eact solutio yy() = si(ππππ).

3 America Joural of Numerical Aalysis able. Compariso of HPM solutios with eact solutios, for = 8 Eample Eample Eample HPM Eact HPM Eact HPM Eact able. Maimum error aalysis of HPM with the eistig method [7] Method [7] Hosoya Polyomial Method (HPM) + Eample Eample Eample Eample Eample Eample 4.84e-.84e-.e-.55e-5.e e-6 8.8e-.8e-.79e- 4.e-.9e-.55e-5 6.9e-4.e-4.6e- 6.47e- 6.8e- 6.94e-4.e-4.e-4 4.8e e- 9.e-.44e- Solutio: First we substitute y ( ) = C HP ( ) i Eq. () we get C HP( ) = si( π ) + cos( )[ C HP( t)] dt. herefore for = C H( ) cos( ) H( t) dt + C H( ) cos( ) H( t) dt + C H( ) cos( ) H( t) dt = si( π ). Net, we substitute the Hosoya polyomials as Net, C cos( ) dt + C ( + ) cos( )( t + ) dt + C ( + + ) cos( ) ( t + t + ) dt = si( π ). 5cos( ) C[ cos( ) ] + C ( + ) cos( ) + C ( + + ) = si( π ). Substitutig the collocatio poits, we get the system of three equatios with three ukows as, 5cos( ) C[ cos( ) ] + C ( + ) cos( ) + C ( + + ) = si( π ), 5cos( ) C[ cos( ) ] + C ( + ) cos( ) + C ( + + ) = si( π ), 5cos( ) C[ cos( ) ] + C ( + ) cos( ) + C ( + + ) = si( π ). Solvig these systems we obtai the three ukow Hosoya coefficiets C = 6.495, C =.598, C =. Substitutig these coefficiets i the approimatio, [ ] [ ] [ ] y( ) = C H ( ) + C H ( ) + C H ( ), we get the approimate values y =.866, y =, y =.866. Maimum Error aalyzed for = is.55e-5 ad for = 4, 6 & 8 are show i the able. Eample. Cosider Fredholm itegral equatios, ( ) = ( π ) + ( + ) () y si t t y( t) dt, has the eact solutio yy(tt) = si(ππππ). Eample. Cosider Fredholm itegral equatios, ( ) = + + ( + ) () y t t y() t dt, which has the eact solutio yy() = +.

4 4 America Joural of Numerical Aalysis Numerical Solutio Hosaya Polyomial Method (HPM) Eact Figure. Numerical solutio of eact ad HPM Eample, for = 8 Numerical Solutio Hosaya Polyomial Method (HPM) Eact Figure. Numerical solutio of eact ad HPM Eample, for = 8 Numerical Solutio Hosaya Polyomial Method (HPM) Eact 6. Coclusio he Hosoya polyomial method is applied for the umerical solutio of Fredholm itegral equatios. he preset method reduces a itegral equatio ito a set of algebraic equatios. For istace i Eample, our results are higher accuracy with eact oes ad eistig method [7]. Subsequetly other eamples are also same i the ature. he umerical result shows that the accuracy improves with icreasig of, the order of a path P, for better accuracy. Error aalysis justifies the accuracy, efficiecy ad validity of the preset techique. Refereces [] L. Caccetta, K. Vijaya, Applicatios of graph theory, Ars. Combi., (987), -77. Figure 4. Numerical solutio of eact ad HPM Eample, for = 8 [] F. S. Roberts, Graph heory ad Its Applicatios to the Problems of Society, SIAM Publicatios, Philadelphia, 978. [] A.M. Wazwaz. A First Course i Itegral Equatios. WSPC. New Jersey [4] U. Lepik, E. amme, Applicatio of the Haar wavelets for solutio of liearitegral equatios, i: Dyamical Systems ad Applicatios, Atala. Proce. (4) [5] K. Malekejad, Y. Mahmoudi, Numerical solutio of liear Fredholm itegral equatio by usig hybrid aylor ad Block- Pulse fuctios, App.Math. Comp. 49 (4) [6] K. Malekejad, F. Mirzaee, Usig ratioalized Haar wavelet for solvigliear itegral equatios, App. Math. Comp. 6 (5) [7] K. Malekejad, M. Yousefi, Numerical solutio of the itegral equatio ofthe secod kid by usig wavelet bases of hermite cubic splies, App. Math.Comp. 8 (6) [8] M. S. Muthuvalu, J. Sulaima, Half-Sweep Arithmetic Mea method withcomposite trapezoidal scheme for solvig liear fredholm itegral equatios, App. Math. Comp. 7 () [9] F. Harary, Graph heory, Addiso Wesley, Readig, 968.

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